Exploring harvest regulations of the New Zealand paua abalone

Transcription

Exploring harvest regulations of
the New Zealand paua abalone
(Haliotis iris) via population
modelling.
Gayle Somerville
a thesis submitted for the degree of
Master of Science
at the University of Otago, Dunedin,
New Zealand.
28 February 2013
Abstract
Limiting the harvest of Haliotis species by the use of shell length restrictions has been a common practice for many years. However these length
restrictions are often based more on the need to protect a proportion of
the population, rather than an analysis of best harvest practice. Increased
emphasis on mātaitai reserves (closed to commercial fishers), and the use
of individual transferable quotas allows increased specificity of the harvest
through the use of localised regulations on both shell length and harvest
rates.
Information is needed on the best harvest system for specific Haliotis populations to aid in maximising returns, whilst maintaining healthy populations. One promising method of exploring the effects of changing the shell
harvest length regulations is via matrix population modelling. Matrix models can be used to explore the normal minimum shell length harvests, as
well as a variety of slot type (e.g. 100-127 mm) harvests. This method
of population modelling allows inclusion of the stock-recruit relationship
and population growth rates, whilst negating the need for knowledge of
population numbers.
Here I investigated a theoretical homogeneous midrange population based
on H. iris measured at Kaikoura in 1968-70 to find the shell length restrictions and harvest rates that provided the highest sustainable annual
harvest. Annual harvest was measured both in terms of numbers taken,
and biomass yield. H. iris shell lengths can vary from 79 mm to 163 mm in
different locations around New Zealand, and so a midrange population with
an average maximum length of 146.2 mm was used. I found that a large
slot type harvest system consistently maximised the numbers that could
ii
be sustainably harvested, however the total biomass yield was maximised
from a minimum shell length longer than the current 125 mm, reflecting a
trend towards the longer minimum shell lengths being introduced in many
commercial H. iris catchments.
Several other results I have included are also of interest, besides the shell
length recommendations. The preliminary calculation of a population growth
rate of 16% for one region, based on Ministry of Primary Industry H. iris
publications provides a starting point for the analysis of healthy Haliotis populations, however there were concerns about the accuracy of some
baseline data. Due to uncertainty about the population growth rate values ranging from 0% to 16% were examined. Although absolute values of
the elasticities and sensitivities were sensitive to changes in the population growth rates, their overall rankings remained the same. Although the
optimal harvest lengths changed markedly at different population growth
rates, the recommended proportion in the harvestable class remained constant. The suggested changes in harvest length had a mixed effect on harvester workload, with decreases in findability (proportion of adults in the
harvestable class) often tied to increases in bodyweight (more biomass per
animal harvested).
Distribution error in the matrix model was largely removed by the use of
a spline function in ’R’. This was verified by multiple integrations based
on equations containing exponentials and the construction of a set of consistently accurate matrices. These integration methods may also be useful
outside matrix modelling. Consistently using just three classes created a
manageable number of biologically relevant matrix elasticities which were
then separated from elasticity measures influenced by the matrix construction. And finally the new terms of ’promotion’ and ’relegation’ were introduced to describe movement between matrix classes.
Many wild abalone fisheries around the world have severely declined or
ceased, possibly due to poor management (Braje et al., 2009; Searcy-Bernal
et al., 2010; Plagányi et al., 2011), and research into specific shell harvest
length is sadly lacking. The main aim of this research was to identify possible modelling methods that could be used to refine the setting of harvest
length regulations. The importance of wild Haliotis populations economiiii
cally, recreationally, biologically and culturally, both in New Zealand and
internationally means further work in this area is important, and could lead
to increases in sustainable yield whilst maintaining, or even increasing the
long term stability of Haliotis populations.
iv
Acknowledgements
A number of people have contributed to this thesis in one way or another.
I would like to thank:
• My primary supervisor, Martin Krkosek, who has a keen interest in measuring and improving the sustainability of aquatic animals affected by marine harvesting. He has helped teach me the mathematical theory underpinning this model and encouraged me in exploring my interests.
• My second supervisor Chris Hepburn has taught me about the elusive
blackfoot paua H. iris, included me in a great group of marine researchers,
and increased my involvement with the local iwi. Thank you to both of
them for their enthusiasm, advice, and timely feedback.
• The lecturers at Otago University for their wholehearted teaching and encouragement in the fields of zoology, mathematics, statistics, and computer
languages. My heartfelt thanks to all of them.
• The programers who have helped develop and make ’R’ (R Development
Core Team, 2008) and ’LaTeX’ both easy to use (Wilkins, 1995) and freely
available.
v
Contents
1 Introduction
1.1 General overview . . . . . .
1.2 Biology . . . . . . . . . . . .
1.2.1 Taxonomy . . . . . .
1.2.2 Life cycle . . . . . .
1.2.3 Vital rates . . . . . .
1.2.4 Population Structure
1.3 Harvest . . . . . . . . . . .
1.3.1 History . . . . . . . .
1.3.2 The harvesters . . .
1.3.3 Markets . . . . . . .
1.4 Aims of this thesis . . . . .
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1
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2 The Linear Population Model
Introduction . . . . . . . . . . . .
Method . . . . . . . . . . . . . .
Matrix design . . . . . . . .
Population parameters . . .
Matrix analysis . . . . . . .
Results . . . . . . . . . . . . . . .
The matrix A . . . . . . . .
Discussion . . . . . . . . . . . . .
Summary . . . . . . . . . . . . .
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3 Recommendations to increase the minimum harvest length of Haliotis
iris are affected by population growth rate
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Density dependent population modelling . . . . . . . . . . . . . . . . .
Harvest . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Maximum sustainable harvest . . . . . . . . . . . . . . . . . . . . . . .
vi
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Robustness . . . . . . . . . . . . . . . .
Results . . . . . . . . . . . . . . . . . . . . . .
Density dependent population modelling
Harvest . . . . . . . . . . . . . . . . . .
Maximum sustainable harvest . . . . . .
Robustness . . . . . . . . . . . . . . . .
Discussion . . . . . . . . . . . . . . . . . . . .
Maximum sustainable harvest . . . . . .
Robustness . . . . . . . . . . . . . . . .
Summary . . . . . . . . . . . . . . . . . . . .
4 General Discussion, conclusions and
General discussion . . . . . . . . . . . .
Relevance of the matrix analysis . .
Maximising harvest systems . . . .
Different population growth rates .
Conclusions . . . . . . . . . . . . . . . .
Limitations . . . . . . . . . . . . . . . .
Recommendations (areas of future work)
References
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recommendations
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A Appendix: Harvest tables
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vii
List of Tables
2.1
2.2
2.3
2.4
Historical and current commercial harvests . . . . .
Parameters of a midrange H. iris population . . . .
Population demographics, observed verses simulated
Elasticity measures of the population parameters .
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3.1
Yield predictions, harvest lengths and population growth rates . . . . .
61
A.1
A.2
A.3
A.4
A.5
A.6
Fixed harvest parameters . . . . . . . . . . . . . . . . .
Variable harvest parameters . . . . . . . . . . . . . . .
Maximum sustainable yield calculations, PGR of 2.5%
Maximum sustainable yield calculations, PGR of 5% .
viii
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List of Figures
2.1
2.2
2.3
2.4
Yearly calendar for the modelled population . . . . . . . . . .
Life cycle diagram . . . . . . . . . . . . . . . . . . . . . . . . .
Fecundity of H. iris at Kaikoura 1967-1969 . . . . . . . . . . .
Different population growth rates affect the elasticity measures
3.1
3.2
3.3
3.4
Density constrained population projections . . . . .
Maximum sustainable yields at a population growth
Profiles of selected harvest systems . . . . . . . . .
Maximum sustainable yields at a population growth
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63
Chapter 1
Introduction
1.1
General overview
The Blackfoot pāua Haliotis iris Martyn 1784 are large edible molluscs in the family Haliotidae (Will et al., 2011). H. iris are endemic to New Zealand and are found throughout the
country, including Stewart Island and the Chatham Islands. The word pāua is from the New
Zealand Māori language, a people to whom they have been an important customary resource
for at least 800 years (Smith, 2011a,b). Currently, as well as significant cultural, recreational
and illegal harvests, this shallow water delicacy plays an important part in New Zealand’s
commercial fishery where it largely supplies an international market, in which they are more
commonly called abalone (Hooker and Creese, 1995; Cook and Gordon, 2010).
Large declines in many overseas harvests of wild Haliotidae species are widespread, and have
been linked to poor management (Braje et al., 2009; Searcy-Bernal et al., 2010; Plagányi
et al., 2011). This international decline is so severe that large wild commercial harvests are
now confined to Australia and New Zealand (Cook and Gordon, 2010). Because of a lack
of sustainability via self-seeding, many commercial harvesters have voluntarily raised the
minimum shell harvest length, based on allowing two years post-emergent prior to entering
the harvested class (Paua Industry Council Ltd, 2006; Pickering, 2012; Mayfield et al., 2012).
This chapter looks at both historical and present measurement and regulation of the blackfoot
pāua Haliotis iris harvest. The focus is on the biology, assessment, and importance of H. iris
in New Zealand.
1.2
1.2.1
Biology
Taxonomy
Haliotis species range in size, with average adult shell lengths between 20 and 200 mm. They
are generally found at depths from intertidal to 30 metres on reefs and rocky shores, and
inhabit tropical through to temperate zones (Lindberg, 1992; Degnan et al., 2006). There are
three New Zealand species: blackfoot pāua Haliotis iris; yellowfoot pāua Haliotis australis;
and whitefoot pāua Haliotis virginea (Ministry of Fisheries, 2011b).
1
The focus of this report is the most common New Zealand species, the blackfoot or ordinary
pāua, H. iris that are genetically distant from all abalone outside New Zealand (Coleman
and Vacquier, 2002; Degnan et al., 2006). At least four phylogenetic breaks exist, although
the overall amount of genetic variation is lower in H. iris than in other New Zealand coastal
marine invertebrates (Will et al., 2011), despite their ancient lineage (Degnan et al., 2006).
1.2.2
Life cycle
H. iris are slow growing, long-lived, broadcast spawning gastropods. The H. iris life cycle
can be divided into three stages: initially, they are pelagic larvae; then they are juvenile and
cryptic; and finally, an emergent stage when they are sexually mature and typically form
aggregates.
The minute larvae usually settle at around 7-9 days of age (Moss and Tong, 1992). Settlement
statistics from the field are unlikely to be accurate due to the small size and broad distribution
of larvae.
Juveniles move into cryptic habitats at around 5 mm in length (McShane, 1995; McShane and
Naylor, 1995a), a behaviour possibly due to increased predation (Shepherd and Turner, 1985;
Francis, 1996). They continue to live beneath rocks and boulders for 3-5 years (Schiel, 1992b),
until they reach the size of around 80 mm (McShane and Naylor, 1997). They are not sexually
mature, and would be prone to predation if exposed at this stage (Hepburn, pers. comm.
2012). Habitat selection during this period is important, and crowding and sedimentation
influence survival and growth (Schiel, 1992a; Phillips and Shima, 2006). Although more
sheltered sites have less wave disturbance, they are more prone to sedimentation (Aguirre and
McNaught, 2011) and expose the sedentary juveniles to a lower amount of water movement,
and thus less drifting food.
Adult H. iris have different shell shapes and maturity rates at different locations (Poore,
1972c), which may be linked to differing shell growth rates (Prince et al., 2008; Poore, 1973).
They move more when a lack of food is limiting growth (Poore, 1972a) and usually aggregate
in groups of 2-200+ individuals (Haist, 2010). These groups are sometimes referred to as
patches, and in a New Zealand-wide survey 10% were found to be solitary (Haist, 2010).
Movement of adult H. iris does occur (McShane and Naylor, 1995d), and aggregations have
been described as ‘unstable in time and space’ (Naylor and McShane, 2001). Higher levels
of congregation are linked to spawning (McShane, 1992) in some Haliotis species and have
been observed in H. iris (Hepburn, pers. comm. 2012). Occasional differences from a 1:1
female: male ratio in H. iris have been found, both positive (Wilson and Schiel, 1995) and
negative (Hooker and Creese, 1995). However difficulty in assigning gender after spawning
(Gnanalingam, 2012) may be influencing these apparent differences.
1.2.3
Vital rates
Fecundity
Both the frequency and volume of spawnings from H. iris are variable; adults are diocious
broadcast spawners, and the spawning triggers are not fully understood (Hooker and Creese,
1995; McShane, 1995). The influence of group dynamics, including the use of broadcast
2
chemical triggers is probable, as correlation in spawning between grouped animals implies
some causality (Kabir, 2001). Hooker and Creese (1995) found three spawnings in one year,
and none the next year, with only one leading to local recruitment. Reproduction is site
specific, and varies from year to year (Poore, 1973).
Fertility increases with length, weight, and amount of shell fouling, and age may be the
critical factor in sexual maturity, rather than shell length (Prince et al., 2008). Egg size does
vary between individuals in several species of Haliotis and is specific to different individuals
and their diet. Size-related competition for preferred sites (Officer et al., 2001) in H. iris
may result in larger females obtaining better food, and producing larger eggs (Graham et al.,
2006; Huchette et al., 2004; Aguirre and McNaught, 2012). Larger eggs are advantageous
at lower sperm densities (Huchette et al., 2004), and as females spawn after males in many
invertebrates (Levitan, 2005) they possibly control fertilisation density. The possibility exists
for the positive maternal effects of larger females found in many finned fish species (Hsich and
Yamauchi, 2010) to also occur in Haliotis, although difficulties in identifying and observing
natural spawning times will make diagnosis difficult.
Fertilisation success may be correlated to cluster size at low densities in Haliotis (Shepherd
and Partington, 1995). In the short term, both aggregation and density are reduced by fishing,
as fishers target the larger patches (McShane and Naylor, 1996). This raises the possibility
that the reduction in recruitment caused by harvest may be greater than a linear reduction
due to the reduction in cluster size, this is known as the Allee effect, and is discussed further
below. However Officer et al. (2001) found that patches of Haliotis returned to pre-harvest
aggregation levels within 10 weeks. Also the cluster size decreases that occur with decreases
in density could occur only at very low densities due to compensatory clustering that may
occur at intermediate densities (Lundquist and Botsford, 2011).
An ideal density to maximise fertilisation success exists (Shepherd and Partington, 1995),
with higher aggregation at lower densities increasing the compensatory density effect in sea
urchins (Lundquist and Botsford, 2011). However aggregation rates may be different for
small adults and larger adults; small clean shell H. iris have been observed forming separate
aggregations in shaded spots (Hepburn, pers. comm. 2012) and smaller Haliotis are much
less likely to join adult aggregations (Shepherd, 1986). This may amplify any reproductive
differences due to maternal effects, although this effect decreases at lower densities (Lundquist
and Botsford, 2011)
The Allee effect describes a reduction in fitness with decreasing density greater than that
expected from a linear relationship. At very low densities populations of Haliotis in the
field have failed to produce any recruits (Neuman et al., 2010). This is known as the strong
Allee effect (Shepherd and Partington, 1995) and occurs due to the broadcast nature of their
spawning. Gametes are released at indeterminate intervals and aggregations are important
in fertilisation success. Critical levels, below which recruitment is zero were measured at 0.34
adults per m2 in H. cracherodii during the collapse of the Californian abalone fishery (Neuman
et al., 2010). The more subtle weak Allee effect causing reductions in fecundity when H. iris
density is marginally lowered may be undetectable (Lundquist and Botsford, 2004), but is of
concern to H. iris population modellers (Breen et al., 2003; Kahui and Alexander, 2008).
3
Recruitment
Typically, recruitment levels of free-living larvae of benthic marine invertebrates are very low
(Pechenik, 1999), and H. iris is no exception (McShane and Naylor, 1996). Indications exist
that recruitment of H. iris is mainly philopatric (McCowan, 2012), with the spread of pelagic
larvae influenced by headlands and bays (Stephens et al., 2006). A 20-year study of Haliotis in
Australia found only a few years had good levels of recruitment (Prince and Shepherd, 1992)
A snapshot study of H. iris recruits at seven sites found three with consistent recruitment
(McShane et al., 1994), but only one of those sites had sufficient recruits to maintain the
local population. Recruitment rates are often best estimated from calculating backwards
from population changes in adult H. iris numbers (Shepherd, 1990).
Shell Growth
Juvenile growth rates can vary depending on several factors that include; season, locations,
substrates and density (Sainsbury, 1982a; Poore, 1972c; Kawamura et al., 1998; Heath and
Moss, 2009). The use of a reverse logistic equation was found to be superior to both the
von Bertalanffy (Day and Taylor, 1981) and the Gompertz growth equations when modelling
abalone growth from a young age (Helidoniotis et al., 2011).
Adult growth varies in response to several factors, including topography, location, age, and
food availability (McShane and Naylor, 1995d; Fu and McKenzie, 2010a,b). McShane (1995)
found faster growth on headlands as opposed to bays; there tends to be less H. iris on
the headland, where they are bigger (Sainsbury, 1982a). Adult groups with average shell
lengths as small as 78 mm and up to 163 mm have been reported, with larger sizes and
faster shell growth associated with lower water temperatures (Naylor et al., 2006). H. iris
are opportunistic feeders, growing better when on headlands and high energy reef systems,
which provide the water movement, aeration and food levels needed for H. iris to maintain
a faster shell growth rate. This means that different groups within the same population will
have different shell growth rates (McShane et al., 1994). However losses of H. iris from these
more water swept areas are higher (Naylor et al., 2006). Although McShane and Naylor
(1995d) found movement of adult H. iris does occur, they could find no preference for sites
with better growth rates, although movement into areas of lowered density has been recorded
(Officer et al., 2001). Density did not effect adult growth rate in farmed H. iris until very high
levels of density were reached, beyond what would be found currently in any wild populations
of H. iris (Wassnig et al., 2009). Growth slows as H. iris age, and later shell growth may
be slowed by earlier rapid increments (Sainsbury, 1982b), however little work has been done
on this in the field, as H. iris are difficult to track over any length of time. This difficulty
in tracking H. iris was measured by Naylor (2006) who found an average recovery rate of
tagged H. iris of 12.4% (s.d. = 9.3) after 12 months.
A review of the different methods of measuring Haliotis growth was conducted by Day and
Fleming (1992). Counting rings of shell discolouration to calculate age, although useful in
some Haliotis species (Prince et al., 1988), has been found to be inaccurate as an age indicator
for H. iris (Schiel and Breen, 1991), although this area of assessment is being revisited (Paua
Industry Council Ltd, 2010). Many populations show natural movement (McShane and
Naylor, 1995d) and changing environments that would also affect size-frequency estimations.
The preferred method of measuring growth in H. iris is tag recapture, however it is labour
intensive (Burch et al., 2010). Due to the large variation in shell growth rates (Breen et al.,
4
2003; McShane, 1995) and the difficulty of determining the age in sampled H. iris, they are
classified by size (Breen et al., 2003). Thus the use of age terms is seldom encountered in
work relating to H. iris, and length in mm, measured along the base of the shell as per harvest
regulations (Ministry of Fisheries, 2011b) is the standard measure of growth. Increases in
both the breadth: length and height: length ratios have been recorded in older H. iris (Poore,
1972c), which could affect both the length: age and length: fecundity relationships.
Adult Haliotis growth is usually modelled using the von Bertalanffy or Gompertz equations
(Day and Fleming, 1992; Troynikov and Gorfine, 1998; Helidoniotis et al., 2011), although
problems exist when using tag-recapture data to formulate these average growth curves (Haddon, 2001). Several alternative body growth models were considered by Haddon (2001).
Mortality
Mortality rates of post-settlement Haliotis larvae are high, but difficult to quantify. Rates
vary depending on many factors including: substrates; location (wave effects); conspecific
adults; sedimentation; depth; and predation (McShane and Naylor, 1995c; Nash et al., 1995;
Naylor and McShane, 2001). Naylor and McShane (2001) found that adult H. iris actively
graze (or bulldoze) settled larvae in the field and caused a decrease in numbers to below half
that on ungrazed concrete blocks. Other studies have found a positive relationship between
adult and juvenile numbers in the field (McShane et al., 1994), which is possibly due to
increased reproduction at higher population densities (Shepherd and Brown, 1993).
Adult mortality rates vary among different Haliotis species, with H. iris having comparatively
high survival rates (Rosetto et al., 2012). Their main predator in some locations is the starfish
Astrostole scabra (Kyle, 2012). Deaths in returned sub-legal Haliotis can occur as they usually
bleed to death if cut (Rogers-Bennett and Leaf, 2006), and have limited self righting skills
(Ahmed et al., 2005). However deaths from cuts and handling may be less in H. iris than
in other Haliotis species (Gerring et al., 2003). Mortality in the early stages of life has a big
influence on H. iris numbers, however adult losses due to natural mortality are insignificant
when compared to the rates of harvest and poaching, and in well controlled marine reserves
the greatest loss of H. iris may be from storms (McShane and Naylor, 1997). Although
disease can be a problem in farmed H. iris, this has not happened in wild populations, and
may have been due to higher water temperatures (20o C) in the growing tanks (Diggles et al.,
2002).
Shepherd and Breen (1992) list several different methods of estimating adult Haliotis mortality, and give a calculated rate of less than 0.1%. A later study covering nine months
found variance in annual mortality rates for H. iris of between 0.02 and 0.08% (McShane
and Naylor, 1997). Edwards and Plagányi (2008) used interview data to calculate levels of
Haliotis poaching in South Africa. Unfortunately information about levels of recreational
and customary catch and poaching in New Zealand are inaccurate (Ministry of Fisheries,
2011c), however in both countries annual and spatial variability in poaching is large.
1.2.4
Population Structure
The definition of a population of H. iris is equivocal, as the recruitment of the minute pelagic
larvae is difficult to study (McShane, 1995), and movement of the mostly sedentary adults
5
does occur (Poore, 1972b). Current thinking is each bay or reef may be a single population
(Breen et al., 2003), although the New Zealand-wide magnitude of genetic differentiation in
H. iris is lower than that identified in other coastal marine invertebrates (Will et al., 2011).
This lack of genetic variation along the coast implies larger populations; with the genetic
impacts of movement of H. iris by pre-European Māori undetermined. Their shell shape
increases drag compared to other molluscs (Tissot, 1992), and this suggests the possibility
of local differences in shell shape (Saunders et al., 2008) having a hydrodynamic effect on
regional adult dispersal rates during storms.
1.3
Harvest
1.3.1
History
H. iris has been important to the people of New Zealand since the arrival of the tāngata
whenua (Māori, or people of the land) over 800 years ago (Smith, 2011a). Due to a shortage
of huntable land mammals and extinction of the large flightless moa, kiamoana (seafood),
and particularly the large H. iris were a staple food of coastal Māori (Smith, 2011a). H. iris
played a significant role in manaakitanga ki ngā manuhiri (hosting of visitors), and besides
being an important component of traditional everyday diet (Smith, 2011a), they were also
dried and traded with inland tribes and used as a source of decorative shell (Gibson, P. on
behalf of Ngāti Konohi, 2006). This further increased both their management needs, and
taonga (treasured value) to the community.
The monitoring, care and harvest of H. iris has long been a primary concern, and each
iwi (tribe) and sometimes hapū (named sub tribe) had specific tikanga (rules, rituals and
protocols) for the care of seafood that were set by kaumātua (senior people in the kin group)
(Booth and Cox, 2003). There is evidence in old middens of large variation in the harvests
of H. iris taken since at least 1400 CE, with harvests of larger and greater numbers possibly
linked to increases in shell length during the little ice age (Wilson et al., 1979), especially
in the warmer North Island (Smith, 2011a; Anderson, 1981). There is also evidence of an
increase in periodic harvesting over time, particularly in the more populated north (Smith,
2011b). In more recent records, Gibson, P. on behalf of Ngāti Konohi (2006) prepared a
report on traditions important to the Ngāti Konohi people. He found that interviewees were
all taught to leave some seafood for the future, a popular comment was of having been
directed to leave the biggest/greatest spawners. By refraining from harvesting the H. iris
after they reach a certain size these larger more fecundant animals (Poore, 1973; Ministry
of Fisheries, 2011c) were traditionally left to freely reproduce, utilising a system known as a
slot type harvest.
The arrival of Europeans had little initial impact on H. iris numbers, as for many years
Pakeha (non Māori) did not see them as the edible luxury they are considered to be today
(Johnson, 2004), and H. iris were easily gathered from the rocks and used for bait (Johnson,
2004). Initial commercial harvests of H. iris from 1944 were purely for their saleable iridescent
shell (Schiel, 1992b).
The discovery of a satisfactory bleaching method for H. iris meat in 1968 lead to the establishment of a canning and exporting factory in 1969 (France, 1982); and coupled with a
lifting on the ban of exporting frozen meat, led to a large increase in exports (Cunningham,
6
1982), and the beginning of commercial H. iris diving (Brown, 1982). This caused large
increases in prices and harvest, with the Wellington region quickly depleted, due to the good
H. iris stocks available close to a large population centre (Moore, 1982). New regulations
were brought in in 1972, and the export quota scheme was introduced in 1973 as a major
attempt to restrict harvesting for the international market.
Individual transferable quotas (ITQ) allocated to specific areas were in place throughout
New Zealand by 1986 with a total allowable catch (TAC) set for each area, and commercial
fishers began feeling an ownership of the resource. In the 1990’s, quota owners formed an
association and became concerned with quota limits, policing, and reseeding (Johnson, 2004).
The value of exports continued to increase and poaching became more of a problem (Ministry
of Fisheries, 2011c). In 2010 exports of H. iris were worth $55 million dollars, the commercial
catch having remained relatively stable, both in tonnage and value, since 2003 (Statistics New
Zealand, 2010).
1.3.2
The harvesters
Cultural harvest
The Fisheries Act 1996 contains several tools designed to support the rights guaranteed
to tāngata whenua under the Treaty of Waitangi. Tangata Kaitiaki and Tangata Tiaki
(guardians) represent the interests of local iwi and hapū groups and can issue permits under
Regulation 27a for the customary harvest of seafood, including H. iris. This is to enable the
collection of food to feed whānau (family) or manuhiri (guests), especially for events such as
tangi (funerals), hui (gatherings) and blessings, that are important to the cultural heritage of
tāngata whenua (Ministry of Fisheries, 2011a,d). Kaikiaka also have a role in kaitiakitanga
(guardianship, protection) and can set up a rāhui to temporally restrict or ban local harvests.
Three types of special management areas can be established within traditional fishing grounds,
firstly mātaitai reserves (closed to commercial fishers), aimed at promoting customary management practices and food gathering. Within these reserves guardians can bring in changes
to the rules by setting new bylaws. These can be in relation to closures, number and size restrictions affecting both customary and recreational fishers. Secondly taiāpure (local fisheries
areas), which are under the management of a local iwi or hapū who see the area as customarily significant. Regulations in taiāpure are more difficult to change, requiring a Ministerial
signature. All fishing (including commercial fishing) can continue in a taiāpure (Ministry of
Fisheries, 2009). Finally closures and method restrictions may also be applied temporarily
(Ministry of Fisheries, 2009).
Recreational harvest
Several regulations concerning recreational harvest currently exist. They allow the harvest
of up to 10 H. iris per person, with a minimum shell length (MHL) of 125 mm (measured
over the greatest length of the shell), except in parts of Taranaki where MHL= 85 mm.
Accumulation limits are also applied, where the maximum amount of H. iris that one person
can have in their possession at any one time is limited (Ministry of Fisheries, 2008a). There are
also restrictions on the use of underwater breathing apparatus (UBA) (this does not include
7
snorkels), so that no person may take H. iris using UBA, nor be in possession of H. iris while
in possession of UBA (Ministry of Fisheries, 2009). Finally areas can be temporally closed
due to overfishing (Ministry of Fisheries, 2011c).
Poaching
In New Zealand poachers can be divided into two groups, those who break the regulations
on size, number and/or UBA, to obtain H. iris mainly for personal use (Ministry of Primary
Industries, 2012c); and professional and semi-professional black market harvesters who sell
illegally harvested H. iris (Ministry of Primary Industries, 2012d). Prices realised for poached
pāua are around $20-$30 per kg for meat, and $8 per kg for the shells (Beaumont, 2008; Fox,
2011). The markets for these sales include workplaces, clubs, hotels and restaurants (Fox,
2011), as well as overseas. People arrested have included gang members, business people
and restaurant owners (Beaumont, 2008). Sales of black market Haliotis have been linked to
organised crime and drug trafficking in South Africa, (Kiley, 2007), where black market sales
may be twice the legal commercial catch (Lopata et al., 2002). Similar findings also exist for
Australia where black market sales are linked to illegal drugs, outlaw motorcycle gangs and
organised crime figures (Tailby and Gant, 2002) and made up around one fifth of commercial
sales in 2002 (Haas, 2009). In 2002 New Zealand’s legal harvest, live weight was 1153 tonnes,
and illegal production was estimated at 400 tonnes (Haas, 2009). The importation of illegally
harvested Haliotis into China is accepted as a market force strong enough to lower prices
(JLJ group, 2010), and although regulations do exist to curtail international trade in illegally
harvested Haliotis, they are ineffective (Plagányi et al., 2011). Tests using DNA and other
biochemical and molecular techniques have been used to identify specific Haliotis species and
aid in prosecutions (Lopata et al., 2002; Tropea, 2006).
Commercial Harvest
Commercial harvesting of H. iris is controlled by quotas, which are restricted to specific areas
and can be traded, with minimum and maximum limits on the amount of quota held. Harvest
is not restricted by season, but tends to be concentrated in the summer after the season opens
on 1st October (Ministry of Fisheries, 2011c). Commercial harvesters are also banned from
using underwater breathing apparatus (Ministry of Fisheries, 2009), which limits the harvest
of deeper beds (up to 20 m in places (Schiel, 1992b)), which can then act as a breeding
reserve. The commercial fishery is divided into 10 zones. Because H. iris grow faster and
larger in the colder southern and eastern zones (Naylor et al., 2006), the majority of the catch
is from the sea around Stewart Island and the Chatham Islands; the South Island and the
lower part of the North Island. (Ministry of Fisheries, 2011c). This has been linked to lower
water temperature (Naylor et al., 2006) and may be helped by the cool, fresher, relatively
nutrient-rich Sub-Antarctic waters, which come as far north as the Chatham Rise (Delizo
et al., 2007).
Within each quota management area there is a management area council that includes representatives from many interested groups. There are also regional representative groups under
the national umbrella of the Pāua Regional Council with a majority mandate from fishing
and non-fishing quota owners, ACE (annual catch entitlement) holders, permit holders, processors and exporters to “protect and grow the property rights of (local) quota owners, and
8
to preserve and expand access to (local) areas” (The New Zealand Seafood Industry Council,
2011; Paua Industry Council Ltd, 2013). They are described by Gary Cameron, Executive
officer of PāuaMAC4 as “stakeholder group(s) that can responsibly take collective action for
the long term benefit of the resource” (Bartram, 2010). Fine scale management is becoming
more popular, with voluntary localised different minimum shell lengths which increase gradually from 125 mm to 135 mm as you move south through the southern catchments (Ministry of
Fisheries, 2011c; Paua Industry Council Ltd, 2013). This is done in an effort to allow H. iris
two years as adults before entering the harvestable class (Paua Industry Council Ltd, 2006;
Pickering, 2012). “...divers are employing turtle loggers to record divers’ movements in order
to better understand the spatial nature of the fishery... and voluntarily take part in shell
sampling and shell tagging programmes to further help manage the resource. Bi-annual diver
input on the state of the fishery is now an integral part of management decisions giving divers
a key role in ensuring the sustainability of this unique resource” (Bartram, 2010). The use
of catch samples and (turtle) data loggers is continuing to grow, and is actively encouraged
in the annual operating plan produced by the regional representative groups (Paua Industry
Council Ltd, 2010, 2012). In several parts of New Zealand commercial harvesting takes are
well below maximum quota levels due to decisions made by these commercial stakeholder
organisations (Ministry of Fisheries, 2011c; Paua Industry Council Ltd, 2012), and a levy is
collected to aid in research and development (The New Zealand Seafood Industry Council,
2011). ACE agreements can have a requirement of successful completion of training standards relating to harvesting and handling H. iris, sustainable fisheries, and seafood work.
Although there does not appear to be any registered training organisations currently offering
these standards (New Zealand Qualifications Authority, 2011, 2013a), however high completions in 2006 − 2007 (New Zealand Qualifications Authority, 2013b) may have satisfied the
market. The use of fishery-dependent and fishery-independent data is used in quota setting
both in New Zealand and overseas (Ministry of Fisheries, 2011c; Chick and Mayfield, 2012;
Woodham, 2009).
1.3.3
Markets
Both the meat and the shell are sold locally and overseas. A large amount of the meat is
processed for export in canned form (Ngāi Tahu Seafood, 2011) and sold in the Singapore and
Hong Kong markets, with sales concentrated on Chinese New Year celebrations (Aotearoa
Fisheries Limited, 2011). Total overseas sales in 2009 were 776 tonnes for NZ$51.1 million,
averaging $65.72/ kg. Locally, small quantities of commercially harvested H. iris can be
bought in several forms live (in shell) animals ($100/kg), or chilled ($170/kg), or processed
($80/kg) (Solander Gourmet Seafood, 2011; Abalone Divers of New Zealand, 2011).
Larger companies are being established (Ngāi Tahu Seafood, 2011; Aotearoa Fisheries Limited, 2011), and links with Chinese markets are becoming easier to pursue, as over two thirds
of the world’s Haliotis harvest is consumed in China (JLJ group, 2010). H. iris shell is
also available in many on-line sites, both local, and overseas. Dried Haliotis is 10% of the
total market, and the highest demand internationally is for small amounts of high-end dried
Haliotis meat from Japan, worth $6000/kg, however the black colour on H. iris makes them
unsuitable for this market (National Research Institute of Fisheries, 2011). The growing
Chinese production of farmed Haliotis is aimed at the lower end of the market, where most
H. iris is sold (JLJ group, 2010). In China increases in locally produced supply are outstripping the growth in demand (JLJ group, 2010), however current prices for H. iris remain
9
stable (Statistics New Zealand, 2010).
1.4
Aims of this thesis
The main thrust of this thesis is to explore the possibility that changes in the permissible shell
lengths of harvested Haliotis could be used to increase the maximum sustainable yield. Currently shell length recommendations are simply a broad stroke regulation, designed to protect
individuals until they have a chance to breed. However I believe that recent developments
now allow us to move beyond that broad stroke approach to set more favourable regulations.
These recent developments include better knowledge of population parameters, increased
ownership of resources by fishers, better monitering of both harvest levels and Haliotis populations, and increases in computing and analytical skills. The gains to be made by a more
delicate approach to shell length harvest regulations include increases in the sustainable yield
from large commercially harvested wild populations, and the possibility of higher harvest
rates and better population recoveries in smaller populations. As well as these advantages, I
also aimed to generate an increased awareness of the importance of maintaining highly fecund
individuals in a population, and explore effects on both commercial and recreational fishers
of any suggested change in the regulations.
This first chapter outlined much of the current knowledge on H. iris. Background material,
sometimes in relation to conjoiner species, was included where necessary to make up for a
lack of material and give more breadth to the study. My aim here was to both identify
strengths in current knowledge, and outline any areas likely to benefit from further research.
The following chapters of my thesis aimed to build on this knowledge using an appropriate
population model in combination with some new modelling techniques.
Due to both time and length constraints uncertainty in relation to population growth rate
was the only stochastic factor investigated, however this did include trialling a variety of
values for the highly variable parameters of egg settlement (numbers of newly settled juveniles per egg) and juvenile survival. The robustness of the calculations was also thoroughly
explored. Further research including variability in the growth and fecundity functions, the
adult mortality rate, as well as poaching levels in and behavioural differences between small
and large adult Haliotis would need to be incorporated into the model before a specific shell
length recommendation could be generated for a practical situation.
Chapter two, the Linear Population Model
The matrix model developed in this study was used to analyse a theoretical homogeneous
H. iris population, primarily based on H. iris analysed at Kaikoura by Poore (1972a,b,c,
1973). A matrix analysis was conducted, including an investigation into how the sensitivity
and elasticity are influenced by changes in the matrix divisions. This investigation concluded
that large adult survival had the highest elasticity and was most important to the population
growth rate. The maturation rate of juveniles into adults was the parameter with the highest
sensitivity, related to adaptability to environmental change. An exploration of inaccuracies
in the analysis, and the robustness of the model was also undertaken. The use of a spline
function in ’R’ was found to be a simple and efficient method of integrating equations with
10
exponents, supported by consistencies between the eigenanalyses of different matrices based
on the same population.
Chapter three, the Harvest regulations
In Chapter three the model developed in Chapter two was refined to calculate the maximum
sustainable yields possible under different harvest length scenarios, at different population
growth rates. The first task undertaken was to increase the realism of the model by adding
a density effect to the population projection matrix. The insertion of harvest terms into the
matrices lead to the calculation that maximum number could be sustainably harvested with
a slot type system, however biomass yield was maximised with minimum harvest lengths
longer than the currently employed 125 mm. Population growth rate was found to have a
large effect on the recommended harvest lengths, however proportion of the adult population
in the harvest class remained consistent. The recommendations were to harvest from the
smallest 47-49% of adults for a slot harvest or from the largest 63-66% for a minimum length
harvest system.
Chapter four, the Conclusions
Chapter 4 contains the final synthesis that was conducted to identify any overarching conclusions, and relate this study to the future management of H. iris harvests. Limitations of
the analysis and some possible areas of future research are also identified.
11
Chapter 2
The Linear Population Model
Introduction
An improved understanding of the population dynamics of abalone species should allow better management of exploited Haliotis populations. However concern exists that the life
history traits of many commercially fished Haliotis species are not fully understood (Ministry of Fisheries, 2011c). Life history traits such as fertility and shell growth influence a
species ability to persist in a specific environment, recover from environmental changes, and
withstand increased mortality either as a result of fishing, or in response to environmental
changes (Guisan and Thuiller, 2005). Haliotis have complex life-cycles, including benthic and
pelagic stages with different life history traits, which makes understanding specific population dynamics very difficult. Increased knowledge of the life history traits and the selection of
appropriate population models could aid in predicting the dynamics of Haliotis populations
both currently and in response to any environmental or management changes.
Effective assessment of abalone populations is globally important, due to both the value of the
abalone harvest and the decline in many populations (Edwards and Plagányi, 2008; Haddon
et al., 2008). Globally legal wild fishery landings of abalone have decreased 55% between
the 1970s and 2008 (Cook and Gordon, 2010). In New Zealand the initial high H. iris
harvest quotas of the 1980s were steadily decreased up until 2006 (Table 2.1). Since then
the voluntary commercial quota shelving (an annually reviewed percentage drop in allowable
catch) has continued under the control of the largely self-governing Paua Industry Council
(Childs, 2012; Paua Industry Council Ltd, 2012).
Internationally most of the currently harvested wild abalone populations are assessed for
quota management by examining localised changes in animal densities, locations and shell
length (Gorfine et al., 2001; State of California. Dept of Fish and Game., 2010; Mayfield
et al., 2011). Several reasons exist for favouring this use of direct measures over modelling
as the main criteria for setting harvest levels. Firstly, Haliotis reef-based populations can be
considered to exist as separate entities with local spatial and temporal differences in size and
growth rates, which can be difficult to model (Jiao et al., 2010; Chen et al., 2003; Mayfield
et al., 2011; Naylor et al., 2006). Secondly, increased pressure from governments to define and
meet population reference points (Gorfine et al., 2001; Bunn et al., 2007) means demographic
assessments are a regular occurrence, and finally, large levels of unassessed poaching, as occurs
12
in many abalone fisheries (Cook and Gordon, 2010), can undermine a model’s reliability
(Ministry of Fisheries, 2011c). However there is room to alter the regulations covering the
spatial and temporal location of the harvest, as well as the characteristics of the permitted
catch, whilst staying within the set harvest levels. The use of structured population models
can provide useful predictions of the possible impacts on both the harvest yield and the
population of altering these regulations.
Alternative assessment methods for H. iris continue to be investigated. Gathering accurate
population estimates is undertaken with both fishery dependent (Fu and McKenzie, 2010b;
Paua Industry Council Ltd, 2013) and fishery independent researchers (Naylor, 2003). Analysis of these data to create population predictions has been completed using both Bayesian
analysis (Breen et al., 2003; Ministry of Fisheries, 2011c) and bioeconomic modelling (Kahui
and Alexander, 2008). However the effects of changing harvest length on recruitment levels
is not included in these studies, and uncertainty exists in the predicted outcomes.
The aims of fishery assessment have changed over the years, with an early emphasis on
maximising yield from an apparently inexhaustible resource, to more cautious approaches
that incorporate environmental, economic and social needs (Breen, 1992; Quinn and Collie,
2005). However, accurate information concerning population dynamics remain an important
component of abalone assessment. Population matrices, based on the original Leslie matrix
(Caswell et al., 1997), are a common method of structured population modelling (Haddon,
2001) that allow inclusion of different vital rates for different sectors of the population. The
alternative stage-based Lefkovitch matrix models are suitable for organisms where the age of
individual animals is unknown (Caswell, 2001). Matrices are useful when analysing management and conservation of abalone populations (Rogers-Bennett and Leaf, 2006; Button and
Rogers-Bennett, 2011), as they are easy to construct and simple to simulate (Caswell, 2001).
The population model chosen for this study is a stage-based birth pulse linear Lefkovitch
model, a useful model that does not need information about the size of either a harvested
or unharvested population (Ramakrishnan and Santosh, 1998; Caswell, 2001). A stage-based
model is necessary as the H. iris are unageable (Punt et al., 2013), and although birth pulse
models are easier to construct and solve (White, 1998), adjustments may be needed due to
the difficulty of quantifying the reproduction of H. iris.
Matrix design
The first step in matrix formation is dividing the life-cycle into stages, with the number
of stages equalling the number of columns and rows in the matrix. In matrix modelling
movement through the matrix happens at a specific point in the year, called the census
date, that needs to be chosen. Next, population parameters governing annual likelihood of
movement through the matrix (relating to survival, growth and fecundity) are calculated for
each stage and inserted into the matrix. Finally analysis of the completed matrix can then be
used to tell you both more about the population, and its theoretical response to alternative
simulations.
The division of abalone populations into discrete stages based on size is necessary for the
unageable H. iris populations, and although assignment of the divisions is somewhat arbitrary, selection of size classes for sexual maturity and different fecundity levels are needed
in the model (Caswell, 2001). Larger divisions can be chosen to fit measured data (RogersBennett and Leaf, 2006) or smaller divisions (down to 2 mm (Kahui and Alexander, 2008))
13
.
Table 2.1: Statistics from the main Haliotis iris catchment zones in
the South and Stewart Islands of New Zealand. The catchment zones
PAU5A (Fiordland), PAU5B (Stewart Island) and PAU5D (Southland/Otago) were created from PAU5 in 1996. PAU5A (Fiordland)
was split into North and South zones in 2010 for the purposes of
stock assessment. The g measurements are annual shell growth rate
at 75 mm and 120 mm; m95 gives the length when 95% are sexually
mature. TACC is total allowable commercial catch.
Catchment
zone
PAU3 (Canterbury,
specifically
Kaikoura)
PAU5A
North Fiordland
PAU5A
South Fiordland
PAU5B
(Stewart
Island)
PAU5D
(Southland /
Otago)
PAU7 ( Marlborough)
Commercial
voluntary
MHL changes
and year of
implementation
125 mm 1988
Years of
significant
TACC
changes
MOF
calculated
exploitation
rates
The predicted
changes in
H. iris
spawning
biomass
Length
parameters
Source
of the
data
Increased
1995
unknown
g75 = 22.1 mm
g120 = 8.1 mm
m95 = 100 mm
a,b
127 mm
2007,
125-132 mm
2010
Decreased in
2006, 10,000
kg moved to
PAU5A
South in
2012
Decreased in
2006
U2005 = 0.45
U2010 = 0.31
probably
stable, 1992
until at least
2007
decrease 7% by
2012
g75 = 25.2 mm
g120 = 6.9 mm
m95 = 109 mm
a,f,g
as above
a,f,g
Decreases in
1999, 2000,
and 2002
U2003 = 0.14
increase 3%
first year, 14%
by 2012
2003, should
increase,
balance tipped
at U2003 = 0.16
2012+ 53%
chance of
reductions
unknown
m95 = 92 mm
c
g75 = 26.1 mm
g120 = 6.9 mm
m95 = 133 mm
g75 = 19.6 mm
g120 = 8.2 mm
m95 = 93 mm
g75 = 15.4 mm
g120 = 5.7 mm
m95 = 102 mm
a,d,e,f,g
130 mm
2007,
132 mm 2010
135 mm 2010
125-130 mm
2010, 125 &
132 mm 2012
125-130 mm
2010
Decreases in
2003
Decreased in
2001, and
2002.
U2005 = 0.45
U2010 = 0.22
U1998 = 0.24,
U2001 = 0.17,
U2007 = 0.09
U2001 = 0.57,
U2007 = 0.79h
U2008 = 0.37,
U2010 = 0.25j
a
2008+
predicted to
increase
Ministry of Fisheries (2011c), b calculated from Poore (1972c) via integration of equation 2.8.
Kahui and Alexander (2008),d Breen et al. (2000), e Breen et al. (2003), f Paua Industry Council Ltd
(2010), g Paua Industry Council Ltd (2012).
h
A predicted increase due to falling stock, was made prior to the TACC changes.
j
Predicted decrease due to increasing stock
c
14
a,e,f,g
a
may be used, with calculated rates of fecundity, growth and mortality. Three distinct life
stages are exhibited by all species of abalone; larval, immature juvenile and adult. Scope
exists to include more than these three stages in the model, however due to difficulties in
counting the minute larval stage they are generally combined into the first of the juvenile
stages when the matrix model is assembled. A division at the age or length when abalone
change from cryptic juveniles into emergent adults is sensible, as survival and reproductive
rates change at this time, and a further division at harvestable length allows a change in
mortality to reflect harvest rates. Any further divisions are at the discretion of the modeller,
and numbers as diverse as 4, 5, 7 (years), 8, or even 50 stages have been used in an abalone
matrix population model (Bardos et al., 2006; Rogers-Bennett and Leaf, 2006; Kahui and
Alexander, 2008; Button and Rogers-Bennett, 2011).
When choosing the number of divisions to include in a matrix model minimising error should
be an important consideration. When using data sampled in the field increasing the number of
classes means it is more likely individuals are classified incorrectly, causing a loss of sensitivity
due to sampling error (Caswell, 2001). In an age-based matrix model both skewness and
kurtosis in the data sets are worse in smaller classes (Boucher, 1997). However decreasing
the number of classes will increase the distribution error as all individuals within a class are
considered to have indistinguishable characteristics. Therefore all individuals within a class
will be treated as if they are the same, although they are not. This increase in class width
could be a problem when modelling H. iris populations, particularly in calculating average
fecundity of the larger classes (Caswell, 2001). This means that modellers of Haliotis have
used up to 50 classes when using formula generated data, when sampling error is removed
(Breen et al., 2003; Kahui and Alexander, 2008). Picard et al. (2010) investigated total error
(sampling error plus distribution error) in Dicorynia guianensis (a tropical South American
tree living 15-25 years, (Matbase, 2012)) and found very little change in total error if between
2 to 10 classes were used. The influence of the number of classes on the matrix analysis is
another factor to be considered (Carslake et al., 2009).
Parameterisation
After designing an appropriate matrix the next task is to calculate the population parameters
that will make up the matrix. The parametrisation of a matrix population model involves
firstly the gathering of data, and secondly the calculation of reproduction, growth and mortality rates for the different size classes. Gathering data to use in stock assessment has been
undertaken by both fishers and scientists. The advantages of fisheries gathered (commercial)
data is that it is cheaper (as the fishers are already in the water) and targets locally harvested
areas, however fishers aims may be different to researchers. The advantage of scientifically
gathered (research) data is that data gatherers often have extensive training, and the sampling can be targeted to specific needs. An integrated approach for Haliotis assessment is
becoming more common as any annual changes in the total allowable catch (TAC) are based
on both scientific counts and local fisher reporting (Gorfine et al., 2001; State of California.
Dept of Fish and Game., 2010; Mayfield et al., 2008, 2011; Bartram, 2010).
Deficiencies in data quality can have a large effect on yield estimations (Chen et al., 2003),
and care is needed (Ministry of Fisheries, 2011c). Improvements in computing have led
to the increased use of tools designed to cater for variability in the data (McShane, 1995;
Zhang et al., 2009). Improvements are often suggested (Hillary, 2011), with no single best
method being recognised. Weighting of different measurements included in estimating the
15
same parameter in Haliotis species is possible (Breen, 1992; McDonald et al., 2001). That
being said, a model by Breen et al. (2003) incorporated two measurements of length frequency
as well as tag-recapture information in formulating shell growth rates, was found to have a
loss of sensitivity, probably due to over parametrisation, so care is needed in this area.
Unfished H. iris populations in their natural state are presumed to have a positive population
growth rate in order for them to both recover from fluctuations in abundance, and colonise
new areas. Thus, in order for populations to be restrained from continuous expansion some
density dependent constraints are logical, and could affect harvest levels. In contrast depleted populations can suffer from lowered population growth rates (due to distance between
spawners) in a process called the Allee effect (Lundquist and Botsford, 2011). Non-linear
matrix models (which allow for changes in fertility, mortality or shell growth rates, depending on population density) were found to be necessary in a hypothetical study of Australian
abalone by Bardos et al. (2006), although McShane and Naylor (1995b) found shell growth of
H. iris to be independent of density. As Bardos et al. (2006) used a population growth rate
of 98% per annum (by my calculation), those influences may be minimal in this unharvested
matrix analysis. If the population size is at equilibrium, or freely increasing, and affected
by density, matrix analysis results can be very similar to those from a density independent
model (Caswell and Takada, 2004).
Matrix analysis
The next task was matrix analysis, which is undertaken after inserting the calculated population parameters into the selected population matrix. Matrix use allows analysis of the possible
effects on the population of a change in any one of the population parameters. These examinations are completed by making minute changes in the matrix elements and parameters.
This type of perturbation analysis on the matrix generates sensitivity and elasticity figures
that give insights into both the importance of accuracy in estimating the parameters, and
can (with care) indicate the relative importance of each of these parameters to the population
growth rate (Caswell, 2001; Caswell and Takada, 2004; Caswell et al., 2004).
Sensitivity is a measure of the sensitivity of lambda (the population growth rate) to changes in
the matrix elements (Caswell, 2001). It aims to quantify how much an increase in (for example
fecundity) will increase the population growth rate. Trade-offs often occur in marine species,
for example between fecundity and growth (Tsikliras et al., 2007), however this interaction
is not considered in matrix sensitivity analysis. Sensitivity is still a useful tool, particularly
in comparing the similar matrix elements in different sectors of the population.
Elasticity is a measure of proportional sensitivity (Caswell, 2001). This enables comparisons
of different parameters on the same (0-1) scale. Elasticities of both individual life history
parameters and of the different size classes within the population can be calculated (van
Tienderen, 2000). The life history parameter with the greatest elasticity will theoretically
have the most influence on future population numbers. This can aid in planning fishery limits
and assisting commercial farming. However practical considerations sometimes mean greater
financial gains may be possible via alternative strategies (Kahui and Alexander, 2008).
Elasticity analysis of different size classes has been used to help choose the best management
and conservation strategies for the endangered white abalone H. sorenseni in California by
identifying the vital rates with the most influence on population growth (Rogers-Bennett
and Leaf, 2006). However Young and Harcourt (1997) suggested that population models
16
were more useful for sensitivity analysis, rather than predicting outcomes, and Benton and
Grant (1999) caution against implementing changes without experimental validation.
The effects of changes in the matrix construction on matrix analysis are also important.
What happens if the division between the classes is shifted? Any results from the matrix
analysis that change when the class divisions are shifted are not a measure of the population,
but instead measure some component of the matrix construction. This is because shifting
the class division simply analyses the same population with a different matrix.
Checks and balances
Finally the robustness of the model was explored. If there is uncertainty about any parameter,
then the analysis would become much less valuable if, at some future time, the parameter
estimations were superseded. It is possible to reduce this possibility, by varying the population
parameters with the highest degree of uncertainty. These include juvenile survival, fecundity,
and the population growth rate.
Method
In the following section the mathematical analysis was completed using the ‘R’ program (R
Development Core Team, 2008). Bold face type is used to denote vectors and matrices.
Matrix design
Census date
The construction of a linear matrix population model first requires the determination of an
annual time of census (Figure 2.1). I chose the census time to be shortly after the young
H. iris larvae settle, and at the end of the harvest season, in April of each year. Note
that adult survival rates will later influence class fecundity levels, as some adults do not
survive through the year until reproduction. Populations are counted annually just after
settlement, and the adult H. iris must survive from then through to the next spawning in
order to successfully reproduce. Thus the adult numbers are affected by annual mortality
prior to both spawning and harvest, and juveniles at their first census (conducted just after
settlement) are unaffected by the annual juvenile mortality rate.
Matrix components
This model is of a closed population of H. iris within a spatially uniform environment. Only
the female animals are included in the matrix; a common practice in sexually reproducing
animals with identical male and female vital rates as it limits the complexity of the model
(Williams et al., 2002). The selection of an annual time of census (Figure 2.1) and the division
of the H. iris population into three classes: juvenile (J), small adults (Y), and large adults
17
Figure 2.1: Yearly calendar including the life-cycle and current management of H. iris in New Zealand, with the chosen census date
shown.
(M) , allows the entire population to be written as a population vector Nt containing the
number of individuals in each size class at census time t
(Jt , Yt , Mt ) = Nt
(2.1)
The juveniles are in a separate class because they have different growth and mortality rates to
the adult H. iris, and are non-reproductive. The cryptic juveniles are also spatially separated
from the adults. The division of the adult H. iris into small adults and large adults classes
enabled me to later apply separate harvest rates to either the small or large adult classes.
The same three classes were also counted one year later, when time was t + 1:
(Jt+1 , Yt+1 , Mt+1 ) = Nt+1
(2.2)
The vital rates effect on how the numbers in these three classes change from one year to the
next can be represented diagrammatically (Figure 2.2).
Due to some ambiguity surrounding the terms ‘parameters’, ‘vital rates’ and ‘elements’ used
in matrix analysis (van Tienderen, 2000) my first task was to define the way I have used these
terms. The population parameters, or vital rates calculated in section 2 are hereafter called
parameters. The elements within the matrix were named for the event they simulated within
the population; promotion, stasis or fecundity. The probability of surviving and moving into
the next class in one year was called promotion, so there was promotion from the juvenile
class into the small adult class the next year, and also promotion from small adults into
the large adult class. The probability of surviving and remaining within the same class for
another year was called stasis, which occurred in each class, juvenile stasis, small adult stasis,
and large adult stasis. Continuing with these designations, I would use the term relegation
for negative or backwards growth, although it was not included here, despite its discovery
in shell length in Haliotis rufescens (Rogers-Bennett and Leaf, 2006), where 6-12% of the
largest adults lost around one or two millimetres. As only three classes are used here, any
18
Figure 2.2: Life cycle diagram for stage-structured H. iris population. J: juvenile; Y: small adults; M: large adults; SJ , SY , SM : annual
survival probability of each class; FY , FM : annual fecundity values
for the two adult classes, where F = E × SE representing the average
number of eggs per adult female (EY or EM ) multiplied by the annual
probability that an egg will result in a settled female larva SE . GJ :
probability of growing from a juvenile into a small adult in one year
and GY : probability of growing from a small adult into a large adult
in one year.
decrease in shell length, were it to occur, would be unlikely to cause much movement into a
shorter class.
The symbols included below are fully explained in the caption to Figure 2.2. Briefly the rates
are explained using S for survival, F for fecundity and G for growth into the next class.
The number of juveniles next year (Jt+1 ) was made up of juvenile stasis (Jt SJ (1 − GJ )), or
the proportion of last year’s juveniles that survived and did not grow into the small adult
class this year, plus newly settled larvae from last year’s small adult (Yt SY FY ) and large
adult (Mt SM FM ) spawnings.
Jt+1 = Jt SJ (1 − GJ ) + Yt SY FY + Mt SM FM
(2.3)
The number of small adults next year (Yt+1 ) was composed of juvenile promotion (Jt SJ GJ ),
or any juveniles from last year that survived and grew large enough to move into the small
adult class this year; plus small adult stasis (Yt SY (1 − GY )), last year’s small adults that
survived and did not grow into the large adult class this year.
Yt+1 = Jt SJ GJ + Yt SY (1 − GY )
(2.4)
The number of large adults next year (Mt+1 ) was made up of small adult promotion (Yt SY GY ),
or any small adults from last year that survived and grew large enough to move into the large
adult class, plus large adult stasis (Mt SM ), the proportion of last year’s large adults that
survived, and so remained in the large adult class for another year.
Mt+1 = Yt SY GY + Mt SM
19
(2.5)
Equations 2.3, 2.4 and 2.5 can then be arranged into a matrix (Caswell, 2001):





J
SJ (1 − GJ )
SY FY
SM FM
J
 Y 
 Y 
SJ GJ
SY (1 − GY )
0
=
M t
0
SY GY
SM
M t+1
(2.6)
With A as the name for this population projection matrix containing the parameters that
relate the population in vector N in year t to the population at time t + 1, one year later.
Nt+1 = ANt
(2.7)
In order to parametrise the matrix A for H. iris population analysis there are several population parameters (used to calculate the elements in A) that need to be determined.
Population parameters
My aim was to use relevant information to assemble as solid an image as is possible of
a single population of H. iris to parameterise the matrix. The collection of much of the
available information on the biology of the endemic blackfoot paua H. iris was assembled in
Chapter 1, however due to the large variability that exists between different populations this
thesis is primarily based on H. iris analysed at Kaikoura over two years by Poore (1972a,b,c,
1973). The large amount of information he researched from a small geographical area within
a reasonably tight time frame made his study very useful, together with the information in
Chapter 1 in portraying a theoretical homogenous midrange population. Poore’s studies are
still a major source of information about H. iris at Kaikoura in PAU3 (Canterbury) used
by the Ministry of Primary Industries in quota management (Ministry of Fisheries, 2011c;
Ministry of Primary Industries, 2013a), and I was grateful to be able to use his information
in my thesis.
Survival
Survival rates of H. iris of any age at Kaikoura were not calculated by Poore (1972c), and in
long term studies of H. iris survival rates were difficult to calculate (McShane and Naylor,
1997). I calculated a survival rate for the juvenile H. iris in this study (SJ ) by combining rates
from McShane and Naylor (1995c); Roberts et al. (2007) and Sainsbury (1982a). McShane
and Naylor (1995c) calculated cumulative mortality for H. iris several times as they passed
between the ages of two weeks and four months. I used a general linear model (log scale)
to extrapolate their data out to six months of age. This calculation gave an estimated
survival from two weeks post settlement through to six months of 0.049. Roberts et al. (2007)
calculated an average survival from 6-24 months of 0.14, and Sainsbury (1982a) estimated
survival after 24 months at 0.9 per annum. Combining these three figures (whilst accounting
for the length of each trial) gives an average annual juvenile survival from two weeks of age
until joining the adult population at 100 mm of SJ = 0.29. Although Schiel (1993) found
transplanted juvenile survival rates as high as 0.72 per annum, his study of Chatham Islands
H. iris involved extensive searches for good juvenile habitat to improve seeding success,
whereas here I am looking for data from average sites that are capable of supporting juvenile
H. iris, the aim being to model an average data set, rather than to maximise reproduction.
20
Here I use an age-independent (Shepherd and Breen, 1992) natural adult survival rate of
0.94 per annum for both the small adult (SY ) and large adult (SM ) classes. This was based
on the Sainsbury (1982a) figure quoted above of estimated annual adult survival of greater
than 0.9, as well as a later study covering nine months, which found a range in annual adult
survival rates from 0.92 to 0.98 (McShane and Naylor, 1997).
Shell growth
Average shell growth rates were calculated by Poore (1972c) from the H. iris measurements
he took at Kaikoura over two years (1967-69). Poore found that these H. iris emerged from
their cryptic habitat with a shell length of approximately 100 mm, and that shell growth
(above 50 mm) followed the von Bertalanffy growth equation:
Lt = 146.2(1 − e−0.3104(t−0.636) )
(2.8)
Which implies the average maximum shell length of 146.2 mm. The inverse of this equation
is
t = −3.222 ln(−0.00561(−146.2 + L))
(2.9)
The need to determine shell growth rates of juvenile H. iris below 100 mm was avoided here
by combining all juveniles into a single class, whereby average age at 100 mm (TJ ) was the
sole measurement required. The average age of 4.3 years when the H. iris shell length reached
100 mm (equation 2.9) makes TJ = 4.3. The annual rate at which juveniles became small
adults (GJ ) was taken to be the inverse of this:
GJ
= 1/TJ
= 1/4.3
= 0.23
(2.10)
The proportion of small adults that moved into the large adult class annually (GY ) was
calculated assuming average shell growth followed the same equations. I also assumed that
any unevenness in shell length that was caused by the birth pulse was dissipated before the
Haliotis iris reached adulthood, and that the shell growth rate was independent of gender
(Button and Rogers-Bennett, 2011). This equation 2.9 gave an average age of 4.3 years to
reach adulthood at 100 mm and 6.8 years to reach 125 mm, which is the current minimum
harvest length. A shell length of 125 mm was used as the division between small adults and
large adults, so the average time spent in the small adult class (TY ) was 6.8 − 4.3 = 2.5 years.
The parameter defining the rate that small adults were promoted into the large adult class
(GY ) was dependent on the length of time in a class, the survival, and the population growth
rate. It followed an equation from Caswell (2001):
Gi =
Si
λ
Ti
Si
λ
Ti −1
−
Ti
Si
−1
λ
(2.11)
Placing the information that each individual spent an average of 2.5 years in the small adult
class (TY = 2.5), with a population growth rate of 5% (giving lambda (λ) = 1.05) and an
21
average annual adult survival of 0.94 (SY = 0.94) into equation 2.11, with i = Y gives:
0.94 2.5
0.94 2.5−1
−
1.05
1.05
=
2.5
0.94
−1
1.05
= 0.37
GY
(2.12)
A similar calculation to the one completed for GY (using equation 2.11) was not conducted
for GJ because juvenile survival (SJ ) varied over the time in class (Kawamura et al., 1998;
Rodriguez et al., 1993). Moreover as TY changes under different harvest systems, the forthcoming comparative analysis in Chapter 3 will benefit from GY having a higher degree of
accuracy, compared to the constant GJ .
This minimum shell harvest length of 125 mm for H. iris is currently in use throughout New
Zealand by recreational fishers, however it is less than voluntary minimum harvest sizes used
by commercial fishers in several management zones with faster growing H. iris populations
(Fu et al., 2010). Poore’s suggestion that a L∞ greater than 170 mm may be more realistic
for this Kaikoura population was possibly based on faster summer growth (Sainsbury, 1982a).
Egg numbers
Links between egg numbers and adult age, weight and length are equivocal, with site-specific
assessments needed for the best H. iris egg number calculations (Nash, 1992; McAvaney
et al., 2004; Naylor et al., 2006; Ministry of Fisheries, 2011c). Accordingly I used Poore’s
(1973) measurements (shown in Figure 2.3, from Fig. 6, p79 in Poore (1973)) of egg numbers
versus shell length at Kaikoura in 1967-69, to calculate an average egg numbers (E) power
function:
E = aLb
(2.13)
relating egg numbers to shell length, based on the same Kaikoura H. iris as the shell length
growth curve in equation 2.8. This means that the considerable variations in egg numbers
that exist between different H. iris populations (Hooker and Creese, 1995; McShane, 1995)
will be negated by retaining site-specific estimates. A natural log conversion gave:
ln(E) = ln(a) + b ∗ ln(L)
which was then solved for the egg numbers data (Poore, 1973) using linear regression analysis.
This yielded the equations
ln(a) = −29.4 ± 7.5
(2.14)
b = 6.3 ± 1.5
(2.15)
I assumed average data for this population, and inserted average values from 2.14 and 2.15
into equation 2.13, which gave:
E = 1.7x10−7 L6.3
(2.16)
These egg numbers values could then be used to calculate average egg numbers of each length
class using integration of equation 2.16. However I was concerned that integration does not
consider the unequal distribution of H. iris within each class. As H. iris grow, their shell
22
Figure 2.3: Linear regression of data from Poore (1973), used to calculate an average egg numbers equation for the H. iris at Kaikoura.
The dots shown are an approximation of the data gathered by Poore
(1973)
23
growth rate slows, resulting in more individual H. iris at the upper end of each length class,
and a corresponding underestimation of the fecundity of that class with integration. This
effect was more pronounced, the larger the class became. The first step in overcoming this
difficulty was converting the equation with egg numbers proportional to length (equation 2.16)
to one with egg numbers proportional to age, and by inserting equation 2.8 into equation 2.16
I obtained:
E = 1.66x10−7 (146.2(1 − e−0.3104(t−0.636) ))6.284
(2.17)
which simplifies to:
E = 6.7x106 (1 − e−0.31(t−0.64) )6.3
(2.18)
An alternative method to find average fecundity, avoiding the complexities of integrating
exponential equation 2.18, was implemented. First, I generated a large data set containing
1000 sequential H. iris ages, which I called t. The first 999 ages used were evenly spread
between maturity (at 4.34 years) and when numbers reach low levels due to mortality (taken
to be after 30 years). Due to the loss of any birth pulse effect on shell length measurements
above 100 mm (Poore, 1973), I assumed age was evenly distributed throughout the year,
thereby allowing this vector to be evenly distributed between the ages of 4.34 and 30 years.
Finally, for the last (1000th ) entry in t the very old age of 100,000 years was used, which makes
sure all the older H. iris were included in the calculations. One thousand corresponding egg
numbers figures were generated for these 1000 ages in t using equation 2.18 thereby creating
a data set of fecundity figures F(t) . These two sets of numbers (t and F(t) ) were interpolated
using a spline function (completed using the splinefun command in the ‘stats’ package in
‘R’). The equation obtained via this interpolation was then integrated to produce average
egg numbers per unit of age E(F(t) , t).
R t2
E(F(t) , t)(t1 →t2 ) =
t1
F(t) .dt
(2.19)
t2 − t1
However distribution of adult H. iris within the small and large adult classes will also be
influenced by the numbers of adults that die as the population ages. So the final step I
initiated was designed to weight egg numbers based on relative survival. One thousand
survival values were generated for the ages in t using the annual mortality rate of 0.06:
st = 0.94t
(2.20)
with survival of those very old H. iris in the 1000th age bracket equal to zero; sT [1000] = 0.
The results were then summed for all ages that corresponded to an average length between
100 mm and 125 mm for small adults, combining equations 2.19 and 2.20.
L=125
Xmm EY =
E(F(t) , t)(t1 →t2 ) st1
L=100 mm
L=125
Xmm
(2.21)
st
L=100 mm
I used a similar equation to generate EM from L = 125 mm to L = 146.2 mm for the large
adults. The egg number parameters are expressed as an annual figure for females, and I
assumed a 1:1 spawning ratio in males and females (Hooker and Creese, 1995; Litaay and
De Silva, 2003).
24
Egg settlement
Egg settlement (SE ) was defined here as the probability that an egg will be spawned and
successfully fertilised, forming a female larva that settles, then survives two weeks. Two
weeks was chosen as the egg settlement end date, as this is the age from which H. iris
juvenile survival (SJ ) was calculated. Unfortunately survival rates per egg in the field are
unknown, but may be very variable (McShane, 1995), both temporarily (Hooker and Creese,
1995) and spatially (McShane and Naylor, 1995d). However with egg and larval survival
as the only unknown within this matrix parametrisation, calculation of an egg settlement
rate was possible if all other population parameters are known, including the population
growth rate. Because spawning was included as part of the calculation this also accounted
for irregularities in H. iris breeding cycles.
If the population growth rate of an unharvested H. iris population is known, then this can
be used to estimate egg settlement. Several early papers estimated low natural recruitment
levels (Sainsbury, 1982a; Tong et al., 1987). McShane et al. (1994) surveyed eight sites and
recorded only one that was mathematically found to be self-sustaining, with a cryptic: adult
ratio of 1:4. With 1/3 of these cryptic H. iris large enough to become adults within 12
months (an annual ratio of 1:12 or 7.7%), they would firstly replace the 6% lost to natural
mortality, and the remainder gives a net population growth rate of only 1.7% per annum.
Another option to calculate the population growth rate was to consider the wealth of information that is available about H. iris commercial harvest areas (Ministry of Fisheries, 2011c).
If a wild paua population harvest is sustainable, then the tabulated exploitation rate will
not exceed the calculable maximum sustainable harvest rate. This harvest rate (HM ), can
then be used to calculate lambda from a matrix similar to A, leading to a population growth
rate, and thus an egg settlement rate. I tabled the Ministry of Fisheries estimations of the
sustainability of exploitation levels (U ) in Table 2.1, for several South and Stewart Island
H. iris commercial catchment areas, along with descriptions of other population parameters
(Table 2.1). In examining these records I assumed that the quoted exploitation levels equated
to total catch weight (commercial landings + recreational, cultural and illegal take) as a percentage of legal weight (all H. iris above the specified shell length). If I assume all H. iris
harvested are a representative sample of the legal harvest class then the weight percentages
can also be used as number percentages without change. These Ministry of Fisheries (2011c)
figures are based on long running standardised catch per unit effort (CPUE) population studies. The constant and increasing CPUE levels over the last 5-6 years (Ministry of Fisheries,
2011c) point to the sustainability of many of these harvest levels.
Exploitation levels are not ideal, but are seen here as a possible measure of current H. iris
population dynamics. The ‘length parameters’ (Table 2.1) of the Kaikoura H. iris align them
most closely with H. iris in catchment area PAU5A (Fiordland). Unfortunately the Canterbury zone PAU3, including Kaikoura, has insufficient biomass data to create a realistic
model (Ministry of Fisheries, 2011c). Naylor et al. (2006) postulated that differences between
H. iris populations from different parts of New Zealand were minimised if the populations
were classified based on their shell growth rates. In PAU5A an exploitation level between 0.22
(MHL=130 mm) and 0.31 (MHL=127 mm) appears to give maximum sustainable yield (Ministry of Fisheries, 2011c). These two sets of figures were incorporated with a matrix similar
to A, but parametrised to reflect the fished H. iris population (see Chapter 3) in Fiordland.
Iteration using this data concludes population growth rates of 15% (MHL=130 mm) and 17%
(MHL=127 mm) when the populations had reached equilibrium. However adult Haliotis are
25
difficult to find, and population counts may be underestimated (Hesp et al., 2008) (Hepburn,
pers. comm. 2013). If the population is larger than assumed, then the MOF calculated exploitation rates need to be lowered, which will lower the estimated population growth rates.
For example, if the population is twice as large as is estimated in (Ministry of Fisheries,
2011c), then the population growth rates required to support the current sustainable harvest
in Fiordland would drop to 10.5% (MHL=130 mm) and 11.5% (MHL=127 mm).
This wide disparity in calculable population growth rates, with 1.7% from McShane et al.
(1994) and up to 17% for Ministry of Fisheries (2011c) for similar H. iris populations, was difficult to resolve. Consulting published peer reviewed literature containing matrix population
models of Haliotis species yielded several papers that used a mathematical reference value
of λ = 1 (implying a population growth rate of zero) (Chen and Liao, 2004; Rogers-Bennett
and Leaf, 2006); I also found an undiscussed value of λ = 1.95 (my calculation) in Bardos
et al. (2006). Clearly there is still much uncertainty in this area of Haliotis assessment.
One alternative to this difficulty in identifying the correct population growth rate was to
examine results from a range of population growth rates. To this end I calculated the egg
settlement rates necessary to support annual population growth rates of 2.5, 5, 10, and 15%.
These results were applied to the original matrix A, and iteration yielded the corresponding
egg settlement rates of 4.49 × 10−7 , 6.25 × 10−7 , 1.04 × 10−6 and 1.55 × 10−6 (with the matrix
divisions where shell lengths are 100 mm and 125 mm).
Fecundity
Fecundity was expressed as the number of two weeks post-settlement female individuals produced by each adult female annually, counted at census time. Fecundity (F ) combined average
egg numbers (E) and estimated egg settlement (SE ) into a single parameter, proportional to
shell length.
F = E × SE
(2.22)
The mathematical modelling of fecundity was based on several assumptions. Firstly, a 1:1
gender ratio in the resulting H. iris offspring, and a similar number of males and females
maintained in each class of the population. The second assumption stated that there was an
equal likelihood of spawning, fertilisation and settlement of offspring for all adult females,
with numbers of settled offspring strictly proportional to total egg numbers. F was used
to weight the relative contributions of small adults FY and large adults FM to numbers of
female juveniles the next year, Jt+1 .
Matrix analysis
Many of these calculations were based on work by Caswell (2001). After I calculated the
equations and ‘R’ codes necessary to complete these calculations, many were also calculated
using the ‘popbio’ package (Stubben and Milligan, 2007) in ‘R’, to check their values.
26
Eigen analysis
The population growth rate can be calculated from the matrix A, where the eigenvalues (λ)
of matrix A are solutions to:
det|A − λI| = 0
(2.23)
where I is the 3 × 3 identity matrix, and lambda is not equal to zero. The population growth
rate is equal to the largest eigenvalue λ1 .
The eigenvector ν that solves:
Aν = λ1 ν.
(2.24)
corresponding to this dominant eigenvalue is called the (primary) right eigenvector and represents the relative numbers of individual H. iris in each population class. This equation 2.24
is true for many values of the right eigenvector, since it was only the relative numbers that
were relevant. For this reason ν was scaled so that the sum of the elements of ν equaled one,
and in this form it is known as the asymptotic stage structure, or stable stage distribution,
and renamed w. The stable stage distribution showed the proportion of the population that
would eventually be in each class (Caswell, 2001). Rearranging equation 2.24:
ν 0 A = ν 0 λ1 .
(2.25)
This equation yielded the (primary) left eigenvector (ν 0 ), where the prime indicates the
transpose vector. The vector was scaled so that ν 0 [1] = 1, and in this form it is known as
the reproductive value vector and renamed v. The elements of v represented the relative
value of each of the length classes to population growth, once the stable stage distribution
was reached.
Sensitivity
The sensitivity (s) of lambda (λ1 , hereafter simply called λ) to changes in each of the elements
in A (aij ) can be expressed as change in λ with respect to changes in aij :
sij =
v i wj
δλ
=
δaij
hw, vi
(2.26)
where aij is the ith element of the jth row of matrix A, and hw, vi is the scaler product of w multiplied by v (Caswell, 2001). A small change in each element, and in λ was
affected by subtracting a small ratio from each. The three ratios examined for δaij were
0.0001aij , 0.00001aij and 0.000001aij (with the same ratio used for δλ):
The division between the small and large adult H. iris classes was at the shell length
of 125 mm, which is the currently used minimum harvest length. However as this is an
unharvested population this division is somewhat arbitrary, and resulted in very few adult
H. iris (around 15%) in the small adult class with the remaining 85% in the large adult class
of an unharvested population. Altering the division between small and large adults allowed
a more even consideration of how the two adult classes influenced the sensitivities. This
alteration was achieved by shifting the division between small and large adults to 140 mm
so that around 50% were in each adult class. I then recalculated the average adult growth,
fecundity, and sensitivity of each class, and slightly altered the egg settlement rate SE (from
6.248 × 10−7 to 6.348 × 10− 7 so that the population growth rate remains stable). This lead
27
to the formulation of a new matrix, still containing the same population of H. iris. As this
new population matrix was of the same unfished population, any change in the sensitivities is
a reflection of the model, rather than the population itself, which was unchanged by shifting
the class division from 125 mm to 140 mm.
Elasticity
Matrix elasticities
Elasticity analysis is a study of the ratio of the proportional change in λ to the proportional
changes in each of the elements of matrix A (Caswell, 2001):
a δλ logδλ
ij
=
(2.27)
eij =
λ
δaij
logδaij
Elasticity (e) was a useful tool because the parameters in A were measured with different
scales, for example growth had a scale from zero to one, but fertility took values higher than
one, which made the sensitivity values of different elements difficult to compare. Elasticity
measured the relative influence of each element aij (and parameter ap ) on λ through equations 2.27 (and 2.31), and thus allowed a direct comparison of the influence of the different
elements (and parameters) of A on λ.
The total elasticities were calculated for juveniles (e(J)), small adults (e(Y )) and large adults
(e(M )) by summing relevant figures (van Tienderen, 2000) from equations 2.27:
e(SJ (1 − GJ )) + e(SJ GJ ) = e(J)
(2.28)
e(SY FY ) + e(SY (1 − GY )) + e(SY GY ) = e(Y )
(2.29)
e(SM FM ) + e(SM ) = e(M )
(2.30)
As changes in shell harvest length were examined later in this thesis, the impact on the elasticities of these different harvest regulations was explored by once again compiling matrices
with different points of division between the small and large adults.
Parameter elasticities
Changes in the elasticity of one element in a matrix are often tied to changes in elasticities
of the other matrix elements, and many matrix parameters appeared more than once in the
matrix. Therefore it was desirable to consider changes in the individual matrix parameters.
The elasticity of the individual parameters of A (ap ) could only be measured by direct
calculation, as this was not a part of the the ‘popbio’ package (Stubben and Milligan, 2007).
a δλ p
ep =
(2.31)
λ
δap
The results from equations 2.31 were summed, giving elasticity totals for e(F ), e(G) and e(S)
(Caswell, 2001):
e(FY ) + e(FM ) = e(F )
(2.32)
e(GJ ) + e(GY ) = e(G)
(2.33)
e(SJ ) + e(SY ) + e(SM ) = e(S)
(2.34)
28
Robustness
Note that the parameter λ that was minutely varied in the process of calculating the sensitivities and elasticities of the matrix A (containing GY ) was also in the equation 2.11 used
to calculate GY . This fact was ignored by treating the λ in equation 2.11 as a constant when
calculating both the sensitivities and the elasticities.
Robustness looked at how elasticities changed due to changes in the parameters, reflecting
uncertainty in their estimation. This gave an indication of whether any conclusions drawn
from this analysis were still viable if the parameters were different. The parameters examined
were firstly juvenile survival, which is very variable and difficult to quantify (McShane et al.,
1994). Fecundity, specifically SE , a measure of the number of eggs released and surviving
through to two weeks post settlement (as female larvae), was also uncertain. Finally a range
of different populating growth rates from 1% to 16% were also explored.
I firstly examined the changes in the elasticities as I changed the survival of the juvenile H. iris
(SJ ). Two egg settlement rates were also chosen, one used the value calculated in section 2.2.2
(SE ), and the second egg settlement rate used was fives times that value (5 × SE ). A rate five
times the original SE value was chosen because this enabled me to explore annual juvenile
survival rates (SJ ) ranging from 0.033 to 0.60 whilst maintaining the desired population
growth rates. The two values that I changed (SE and SJ ) are difficult to calculate, in part
due to the cryptic habitat of these tiny H. iris. There are also large amounts of uncertainty
surrounding survival per egg, including the timing and volume of spawnings. Manipulating
FY and FM (via changes in egg settlement SE ) and SJ while holding the other parameters
constant has the added bonus of allowing examination of the changes in elasticities of the
elements of A at different population growth rates. This was useful due to the large amount
of uncertainty surrounding the population growth rate.
Results
The parameters calculated using the methods outlined in section 2 are listed in Table 2.2,
with the division between Y class and M class set at 125 mm and a population growth rate
of 5% per annum.
The matrix A
The parameters from Table 2.2 are inserted into matrix A as outlined in equation 2.6.


0.22 0.44 1.64
A =  0.07 0.60 0.00 
(2.35)
0.00 0.34 0.94
29
.
Table 2.2: Parameters I calculated to compile the stage structured
population matrix A with the shell length division between small
and large adults at 125 mm and λ = 1.05
Symbol
SJ
Description
Survival of juveniles
SY
Survival of small adults
SM
Survival of large adults
GJ
Growth of juveniles into small
adults
GY
Growth of small adults into
large adults
EY
Egg production by the small
adults
Egg production by the large
adults
egg settlement
EM
SE
a
b
c
Explanation
Probability of juveniles
surviving for one year
Probability of the small adults
Probability of the large adults
Probability that a juvenile will
become a small adult in any one
year
Probability that a small adult
will become a large adult in any
one year
Number of eggs per female
small adult
Number of eggs per female large
adult
Probability of a two week post
settlement female larvae
resulting from an egg
Values
0.29a
0.94a
0.94a
0.23a
0.37b
740000c
2800000c
6.2 × 10−7c
See section 2.2.2 Population parameters.
From equation 2.12.
From equation 2.21.
.
Table 2.3: Haliotis iris counts from Kaikoura 1967-1968 (Poore,
1972c), compared with figures from the matrix stable stage distribution calculated previously
source
Low water November 1967a
Sub-tidal November 1967a
Low water May 1968a
Sub-tidal May 1968a
Low water total
Sub-tidal total
Matrix, stable stage distribution (w)b
a
Poore (1972c)
b
From equation 2.38
small adults:large adult counts
34 : 8
39 : 35
27 : 3
11 : 10
61 : 11
50 : 45
9 : 29
30
small adults:large adult ratios
4:1
10 : 9
10 : 1
11 : 10
6:1
10 : 9
3 : 10
Matrix analysis
Eigen analysis
Calculating the eigenvalues first involved finding the determinant |A − λI|, whereby values
from equation 2.35 were inserted into the standard three dimensional matrix determinant
equation:
det|A − λI| = (SM − λ) ((SJ − SJ GJ − λ) (SY − SY GY − λ))
+ (SM − λ)(SM FM SJ GJ SY GY − SY FY SJ GJ )
(2.36)
which is solved algebraically:
det|A − λI| = −λ3 + 1.69λ2 − 0.79λ + 0.11 = 0
(2.37)
to give three eigenvalues; λ1 = 1.05; λ2 = 0.36 + 0.059i, and λ3 = 0.36 − 0.059i. The
population growth rate for this matrix is thus 5%, equating to λ1 = 1.05. This implies the
population can persist and grow, and so will support a fishing harvest. When the primary
eigenvalue λ1 was inserted into equations 2.24 it yielded the stable stage distribution
w = (0.62, 0.09, 0.29)
(2.38)
and using equation 2.25 gave the reproductive value vector of
v = (1.0, 12.3, 14.9)
(2.39)
In comparing these calculated values to measurements from the field, juveniles are not considered as they were uncountable due to their cryptic habitat and very small sizes. Poore
(1972c) recorded two surveys of H. iris at Kaikoura in 1967 and 1968. I halved his count in
the 120 mm to 130 mm class at 125 mm. His results are included in Table 2.3, along with
my calculated values in w from the matrix analysis.
Comparing the matrix generated stable stage distribution w to the counts from Kaikoura
shown in Table 2.3 we see that there were comparatively less large H. iris found at Kaikoura
than would be expected from an unrestrained unharvested population. The low numbers of
large adults was pronounced in the low water, and more pronunced still in the low water (but
not in the deeper sub-tidal zone) after the summer.
Sensitivity
Using a multiplicative factor of of 0.0001 in equation 2.26
sij =
δλ
λ − λ × 0.0001
=
δaij
aij − aij × 0.0001
(2.40)
through the computer program ‘R’ returned sensitivity figures close to those generated by the
‘popbio’ package (Stubben and Milligan, 2007), with better results when the lower multiple of
0.00001 was used. However a multiplicative factor of 0.000001 returned values of δλ = 0, and
was thus below the sensitivity requirements of ‘R’ . This limitation means the multiplicative
factor of 0.00001 was used for this analysis.
31
In the sensitivity analysis, when the division between small and large adults was at 125 mm
(the current minimum shell length)


0.10 0.02 0.05
s(A125 ) =  1.26 0.19 0.00 
(2.41)
0.00 0.23 0.71
it appeared that changes in population numbers were twice as sensitive to changes in juvenile
promotion as they were to changes in adult stasis. That is, s(SJ GJ ) = 1.26, was nearly twice
as important to population change as s(SM ) = 0.71.
However, with the stable stage distribution for 125 mm division being (0.62, 0.092, 0.288) the
higher number of H. iris in the large adult class probably influenced the sensitivities. The
effects of changing the population model, so that the stable stage distribution has more equal
numbers of H. iris in both the adult classes (0.62, 0.19, 0.19) , when the shell length division
between small and large adults is 140 mm, were a reproductive value vector of (1, 12.3, 17.0),
and:


0.10 0.03 0.03
s(A140 ) =  1.23 0.39 0.00 
(2.42)
0.00 0.54 0.51
This change meant that H. iris had to get larger before they left the small adult class, so this
class became bigger, and contained around half the adults. The sensitivity of promotion of
juveniles dropped slightly, all the sensitivity measure of small adults have more than doubled
in size, with a smaller but noticeable drop in the large adult sensitivities. This means that
the small and large adult sensitivities are now similar in several areas, although juvenile
promotion retains the highest sensitivity.
Elasticity
Matrix elasticities
The elasticity analysis painted a somewhat different picture (compared to the sensitivities)
regarding the most important elements of the matrix. With the division between small and
large adults at 125 mm, changes in the stasis of adults in the large adult class had an elasticity
of around 0.64, which was 6.5 times greater than the elastic of any other element of A.


0.02 0.01 0.07
e(A125 ) =  0.08 0.11 0.00 
(2.43)
0.00 0.07 0.64
Next I examined the effects of changing the population model, this time to reflect a suggested
division between small and large adults of 135 mm. The stable stage distribution became
(0.62, 0.16, 0.22) and the reproductive values were (1.0, 12.3, 17.4). The elasticities of this
matrix were:


0.02 0.02 0.06
e(A135 ) =  0.08 0.23 0.00 
(2.44)
0.00 0.06 0.53
Another change to the population model, this time to reflect a suggested slot harvest, with
the shell length division at 143 mm yielded the stable stage distribution of (0.62, 0.23, 0.15)
32
and the reproductive values of (1.00, 12.3, 16.5).

0.02

0.08
e(A143 ) =
0.00
The elasticities of this matrix were:

0.04 0.04
0.39 0.00 
(2.45)
0.04 0.38
I also examined the elasticities with the class division at 132 mm as this was the point where
GJ is close to equalling GY , and so is where the relative time in class J and class Y is equal.
This allowed better analysis of the elasticities of promotion from each class.


0.02 0.01 0.07
e(A132 ) =  0.08 0.18 0.00 
(2.46)
0.00 0.07 0.57
These elasticities have some factors influenced by the makeup of the matrix (as it is a three
by three‘progression’ matrix (Carslake et al., 2009)):
e(SJ GJ ) = e(SY GY ) + e(SY FY )
(2.47)
e(SY GY ) = e(SM FM )
(2.48)
The proportional elasticity displayed in these three matrices suggested that the lower two of
the three matrix elements on the diagonal, reflecting adult stasis, had the largest elasticity.
The effects of changing the shell harvest length were primarily felt through the resultant
change in the proportion of adult H. iris in the small adult and large adult classes, and to
a lesser extent through the decreasing importance of adult promotion (SY GY ), as H. iris
spend an increasing proportion of their life in the small adult class. As the division shifted,
elasticity of fecundity and stasis in small adults increased at a rate equal to the decrease in
fecundity and stasis of the large adults, meaning that total adult fecundity and stasis stayed
constant as shell harvest length changed. The elasticity of the elements relating to juvenile
H. iris (in the left column of the matrices), remained relatively unchanged by the changes in
division between small and large adults. The promotion of juveniles had a higher elasticity
than their stasis, with both juvenile stasis and small adult fecundity having the lowest levels
of elasticity.
The elasticity values in the matrices 2.43 , 2.44, 2.45, and 2.46 all added to one:
3
X
eij = 1
(2.49)
i,j=1
However these calculations of the importance of an increase in the adult promotion (that
is, H. iris moving faster into the longer class) do not include any consideration that, as
this promotional rate increases, the number of H. iris remaining behind will decrease. That
is, as SY GY increases SY (1 − GY ) must decrease. A similar but opposite consideration
applies to stasis. The calculation of the elasticity of the parameter GY (and similarly GJ )
using equation 2.31 was a better way to get reliable estimations of the overall importance of
growth, as this considered the impacts of both increasing the rate of promotion out of the
smaller class, and decreasing stasis within the class.
33
.
Table 2.4: Elasticities of the parameters of A where the shell lengths considered
correspond to those used in equations 2.43, 2.44 and 2.45. The harvest systems
suggested in Chapter 3 are shaded.
Shell length
(mm)
125+
135+
100-143
e(SJ )
e(GJ )
e(FY )
e(SY )
e(GY )
e(FM )
e(SM )
0.10
0.10
0.10
0.07
0.07
0.07
0.01
0.02
0.04
0.19
0.31
0.48
0.01
0.02
0.01
0.07
0.06
0.04
0.71
0.59
0.42
Shell length
(mm)
125+
135+
100-143
e(T otalS)
e(T otalG)
e(T otalF )
1.0
1.0
1.0
0.09
0.09
0.09
0.08
0.08
0.08
Where the parameter of A (fully explained in Table 2.2) are:
SJ = Survival of juveniles
GJ = Growth of juveniles into small adults
FY = Fertility of small adults
SY = Survival of small adults
GY = Growth of small adults into large adults
FY = Fertility of large adults
SM = Survival of large adults
T otalS = SJ + SY + SM
T otalG = GJ + GY
T otalF = FY + FM
34
Table 2.4 looks at the elasticity of the individual parameters ap of A using the same class
divisions as are in matrices 2.43, 2.44 and 2.45.
On examination of the elasticities of the parameters generated using equation 2.31, and
displayed in Table 2.4 I found that these two parameters GY and GJ were the only two
parameters that generated new information. This is because I found experimentally that
three of the elasticities calculated using equation 2.31 for this analysis (listed in table 2.4)
could be obtained from summing the elasticities listed in the matrices, 2.43 - 2.45 via equations
2.28 - 2.30:
e(SJ GJ ) + e(SJ (1 − GJ )) = e(J) = e(SJ )
(2.50)
e(SY GY ) + e(SY (1 − GY )) + e(SY FY ) = e(Y ) = e(SY )
(2.51)
e(SM ) + e(SM FM ) = e(M ) = e(SM )
(2.52)
And the two fecundity elasticities in table 2.4 were equal to the fecundity elasticities in the
matrices 2.43 - 2.45:
e(SY FY ) = e(FY )
(2.53)
e(SM FM ) = e(FM )
(2.54)
A mathematical proof for these was written by van Tienderen (2000). Note that in equation 2.52 the e(SM ) on the left relates to the element in the lower right corner of the matrix A
(stasis of the large adult class), whereas the e(SM ) on the right was the total elasticity for
survival of the large adults, and these two elasticities are not equal. One final result from my
calculations was:
e(SJ ) + e(SY ) + e(SM ) = 1
(2.55)
Equation 2.55 is the sum of equations 2.50, 2.51 and 2.52, which follows logically from equation 2.49.
The highlighted (or grey) entries in Table 2.4 are of the class of H. iris that is harvested
under different scenarios. The small adults in 100-143 mm have the lowest elasticity readings
of the three harvest options.
Robustness
Here I examined the impacts of changing juvenile survival (SJ ) and the corresponding population growth rate, on the elasticity analysis. As SJ increased so did the population growth
rate, albeit at a slower rate than the increase in SJ . The effects on the elasticities were
examined only in so far as they related to the matrix parameters of survival, fecundity, and
growth. The elasticities of the matrix elements were not specifically calculated here, as they
are sensitive to the placement of the class division.
At a population growth rate (PGR) of 5% (shown by the vertical lines in Figure 2.4) the most
important elasticity was e(SM ) (graph 3 in Figure 2.4, which can also be seen as the value
of 0.71 in column eight in Table 2.4), followed by e(SY ). These two population parameters
(in fact the same parameter, as average adult survival is set at 0.94 per annum for both SY
and SM ) account for 90% of the elasticity of survival.
I included a five-fold increase in the egg settlement parameter (moving from the dashed line
to the dotted line in the double line graphs in Figure 2.4), thereby exploring average annual
35
.
Figure 2.4: The elasticities of the parameters, and how they are affected by
varying both the population growth rate (x axis), and the egg settlement
(two options chosen). The vertical line shows the presumed population
growth rate of 5% per annum. The graph in the upper right corner is not
of an elasticity measure, but instead shows how changing juvenile survival
has been used to increase the population growth rate as egg settlement is
held constant. The dashes use the calculated egg settlement parameter,
while the dotted lines in these double line graphs were generated with an
egg settlement parameter five times larger, to allow the inclusion of lower
juvenile survival figures. The headings on graphs 1-10 are explained in
Table 2.4
36
juvenile survival across the range from 0.03 to 0.60. The left hand side of the dashed line and
the right hand side of the dotted line both have the same juvenile survival value, and any
differences between these two points thus reflected the response to changing egg settlement.
Following on from this, such a change in egg settlement had a nearly identical effect on
the elasticities as changing the juvenile survival. The gap between the two lines is often
undetectable at population growth rates less than 5%.
Across the entire range of population growth rates (PGR), from 1% to 16% per annum, it
appeared that although the elasticity of large adult survival decreased with increasing PGR,
it remained the most important parameter in maintaining the current population growth rate.
Also, as PGR increased, juvenile survival and growth become more important in maintaining
PGR, so there was a decrease in the spread of survival elasticities at higher PGR. The total
elasticities for growth and fecundity behaved similarly at both values of SE , and showed
a gradual steady increase with the increase in SJ . This meant that at higher PGR the
importance of all the other parameters to PGR increased, relative to large adult survival,
which decreased over the same range.
Discussion
This eigen analysis of a population of the endemic New Zealand blackfoot pāua Haliotis
iris found that measures of the sensitivity and elasticity of the static population parameters
were insensitive to the placement of the matrix division between the small and large adult
classes. They were also largely unchanged by alterations in the egg settlement verses juvenile
survival rates at a constant population growth rate. This is in contrast to many of the
matrix elements which varied when the matrix configuration changed. Accordingly these
dynamic measures such as the reproductive values were assessed with equal numbers in the
small and large adult classes. When this is done the reproductive values of (1,12.3,17) show
only minor comparative gains afforded to population growth by the large (versus the small)
adult class, with both much more important than the potential reproduction of the juveniles.
The calculated population growth rate was also largely uninfluenced (lambda was accurate
to 2 decimal places) by shifting the division between the small and large adults. Across a
wide range of population growth rates I found that survival had higher elasticity than either
fecundity or shell growth, and that large adult survival was the population parameter with
the highest elasticity.
Matrix parameterisation for any species with so many unknown life history characteristics
was always going to be challenging. However the severely declining numbers, and slow (or
even non-existent) recovery of many overfished Haliotis species (Braje et al., 2009; Prince and
Delproo, 1993; Plagányi and Butterworth, 2010) makes any attempt to analyse the genera
worthwhile, assuming the aforementioned limitations are diligently evaluated.
The use by early researchers of a von Bertalanffy growth equation to model the shell growth
of abalone (Poore, 1972c) was improved to better incorporate the growth of juvenile abalone
by the use of a Gompertz model (Troynikov and Gorfine, 1998) and again later, using inverse
Bertalanffy-logistic models (Haddon et al., 2008; Helidoniotis et al., 2011). However as access
37
to Poore’s original data was not sought, and because the growth curve of juveniles below
100 mm was not needed, I considered the original von Bertalanffy equation formulated by
Poore (1972c) to be the best choice for this study.
Whilst the parameters of adult survival, shell growth, and egg numbers were comparatively
easy to ascertain, large amounts of uncertainty surrounded the measurements of both egg
settlement and juvenile survival. The mathematics I designed to account for the irregularities
commonly found in H. iris breeding cycles enabled me to formulate a workable matrix,
however the emphasis was then shifted to a focus on uncertainty in the population growth
rate. From a wide literature search my estimations of a feasible population growth rate ranged
from less than 2% to as high as 17% per annum. The estimation of a population growth rate
for any benthic animal is difficult due to the difficulty of accessing undersea environments.
The cryptic nature of the juvenile Haliotis species increases assessment difficulties, as does
the plasticity of their growth and maturity rates (Marsden and Williams, 1996; Heath and
Moss, 2009). Problems with assessment are further compounded by spatial patchiness, and
the unageable shell growth of this particular Haliotis species (Punt et al., 2013). However
as H. iris have been harvested by Māori for many hundreds of years (Smith, 2011a) the
population must have been able to withstand some level of fishing pressure. The general
consensus seems to be that population growth rates of H. iris are very low (Sainsbury, 1982a;
Tong et al., 1987; McShane et al., 1994), however current apparently sustainable fishery
exploitation levels (U ) above 0.25 per annum (Table 2.1) are in conflict with this assumption.
The general assumption of low population growth rates may have been influenced by Poore
(1972c) whose early underestimation of a life expectancy of H. iris of ‘beyond ten years’
overestimated the number of recruits required for population maintenance, and Sainsbury
(1982b) who failed to find sufficient recruits for population replacement. However Schiel
(1993) identified the difficulty in finding cryptic H. iris, which could cause an underestimation
of juvenile numbers, and McShane (1992) postulated subpopulations that lacked juvenile
H. iris may be stocked by movement of smaller adults from sheltered ‘nursery’ areas. Low
estimations of population growth rates still exist, and are influenced by the very low recovery
rates for decimated populations (Tegner, 1993), however continued poaching (Hobday et al.,
2001; Huchette and Clavier, 2004; Raemaekers and Britz, 2009), or the weak Allee effect
(Lundquist and Botsford, 2011; Mendez et al., 2011) may be more likely causes of slow
population recovery (Button and Rogers-Bennett, 2011; Plagányi and Butterworth, 2010).
In this H. iris population analysis size classes were used (as opposed to age), and included
only females. This is common in matrix population modelling (Caswell and Takada, 2004)
and was based here on the assumption that the sexes are identical in growth, maturity, and
reproduction (Button and Rogers-Bennett, 2011). A recent study found no differences in
spawning aggregations between abalone genders (Seamone, 2011), however unequal male:
female ratios were found in one H. iris study (Wilson and Schiel, 1995), but this result may
have been affected by the difficulty in visually assigning gender to adult H. iris outside their
fertile period (Gnanalingam, 2012).
Accuracy of the parametrisation
This matrix parametrisation uses a theoretical homogeneous midrange population based on
H. iris measured by Poore (1972c, 1973). My assumption that they were representative of an
entire population may be erroneous. Large numbers of H. iris were removed from Kaikoura in
1946-47; significant harvesting for shell was carried out in the 1950s, and pāua was harvested
38
under licence around Kaikoura from the 1960s (Johnson, 2004). Poore (1972c) acknowledged
a possible lack of large H. iris, and postulated it may be due to mortality, human exploitation,
or possibly that larger animals moved to deeper water outside the sample site. Alternatively
if population growth or reproduction is restrained in some way then a density effect may be
changing the population ratios at Kaikoura.
Several instances have been reported of slower growth and lower maximum shell length in
more sheltered bays, with faster growing and larger H. iris being found on nearby wave swept
headlands (McShane and Naylor, 1996; Naylor et al., 2006; Donovan and Taylor, 2008) where
they have migrated in search of better access to food. This implies the population analysed
here may be just part of a larger population, where satellite groups of larger H. iris are
maintained by emigration. The relative level at which these satellite populations contributed
to the juvenile numbers would be influenced by distance from the sheltered area, as well
as water movement and egg settlement rates. Reported recruitment levels can vary widely,
and if settlement rates are higher in more sheltered bays, the possibility exists that the
smaller H. iris adults more prevalent in the bays have a larger than expected fecundity levels
(where FY = EY SEY ), due to a larger relative egg settlement rate, making SEY 6= SEM .
Although information on mortality was unfortunately missing from that study, this may be
unimportant to the results. This is because several H. iris studies have found similar adult
mortality rates (Sainsbury, 1982a; McShane and Naylor, 1997) and because survival is so
large it is relatively unaffected by changes in mortality (Nilsen et al., 2009) (for example a
50% increase in mortality (from 6% to 9%) only lowers survival by 3% (from 94% to 91%).
Adult mortality levels may differ in different locations (for example smothering by sediment
in sheltered areas (McShane and Naylor, 1995c; Sainsbury, 1982b), or alternatively, being
swept away in storms from more exposed positions (McShane and Naylor, 1997)), and thus
SY and SM may also differ within a population.
If these measurements by Poore (1972c, 1973) are not estimated from an entire population,
then the shell growth (G) and egg numbers (E) parameters I used are more uncertain, and
will limit the reliability of my analysis. Error in the calculated fecundity curve shown in
Equations 2.14 and 2.15 was large, and error in estimating the K and L∞ parameters of the
von Bertalanffy growth equation are likely (Haddon, 2001; Nollens et al., 2003). However
Kahui and Alexander (2008) found mortality and recruitment were more important than the
K and L∞ parameters in H. iris modelling, so any effects of the error identified by Haddon
may be minimal in my analysis.
Adult H. iris should behave in a way that maximises their reproductive potential, however if
this behaviour evolved to fit selection pressure where mortality was very low (prior to human
arrival) or periodic (under Māori harvests) then reproductive success may be constrained
by adaptive behaviour inappropriate to the more regular and widespread current harvest
systems. This could result in an unknown and depressive effect on the parameters of a
population being harvested.
Matrix analysis
The assumption of a large unrestrained population allowed me to ignore both density dependence (including the Allee effect) and local demographic stochasticity (Caswell, 2001).
39
Eigen analysis
Primary eigen analysis enables calculation of the stable stage distribution and the reproductive values. The population growth rate is also normally calculated via matrix analysis, but
rather than being a point of interest produced in this analysis, it was instead a component
used to finalise the parametrisation of the matrix (see section 2).
The stable stage distributions calculated from the population matrix A are very different
to the ratios of adult H. iris gathered at Kaikoura by Poore (1972c), on which this matrix
parametrisation was based. That Poore’s low (shallow) water colony counts contained proportionally less large adults in May (post summer) compared to November (pre summer) is
consistent with either summer human exploitation, or movement of larger adults to deeper
water. This casts doubt on the assumption that Poore’s study was representative of a complete unfished population, and allowed the assumption that the matrix generated stable stage
distribution is a viable portrayal of an unrestrained unfished population of H. iris at Kaikoura, were one to exist. An alternative explanation for the differences between calculated and
measured stage distributions is episodic recruitment, which could cause short term changes
in population ratios. Episodic recruitment is common in many Haliotis species, including
H. iris (McShane, 1995). Any decrease in the numbers of larger adults may increase the
inaccuracy in the growth and fecundity calculations (particularly in relation to the scarce
larger adults) used in this matrix parametrisation.
The reproductive value vector of (1, 12.3, 17), gives a calculated ratio for the small to large
adults of around 1 : 1.38. This appears at first glance to be too level for an animal with such
a large difference between small and large adult fecundity, as shown in Figure 2.3. These
reproductive values are defined as the probable contribution of an individual in that class, via
reproduction, to population growth from their current age onward, with higher value given
to the more immediate prospect of progeny (Fisher, 1930; Caswell and Takada, 2004). This
definition explains the slightly higher reproductive value of the large adult class, even when
numbers in the two adult classes were equal. The reproductive values of the larger adult
class are greater, as, although they probably have less reproductive life left, their short term
egg numbers rates were higher, compared to the generally younger, less fecund small adults.
Compared to both adult classes the juveniles have a very low value, meaning a low future
value to reproduction. This would be influenced by their lower survival rates.
Sensitivity
The sensitivity analysis gave some insight into evolutionary pressures, assuming a large unrestrained population. Caswell (2001) claimed that making predictions of evolutionary pressures based on sensitivity analysis of stage-based (as opposed to age-based) population models
was more difficult because the organisms within a class are a range of ages, and so they will
have different vital rates, and a history of different selection pressures. This effect is mitigated
by the consideration of intraclass growth and mortality I used when calculating average class
fecundities, and by a recent analysis by Barfield et al. (2011), who showed that evolutionary
predictions can be made successfully from stage-based models.
Equation 2.42, where the two adult classes contain equal numbers of adults, shows that the
sensitivity of fecundity declines slightly with age, whilst the sensitivity of stasis increases with
age. Life characteristics whose sensitivities decreases with age have an equal evolutionary
40
selection pressure for larger increases in the later stage and smaller increases at the earlier
stage (Caswell, 2012), implying evolutionary pressure for higher fecundity in the larger adults.
The increase in the sensitivity of stasis with age is tied to a phenomenon labelled negative
senescence, and identified in some in molluscs by Vaupel et al. (2004), who describe it as
a phenomenon whereby fertility and fitness increase with age, as mortality decreases. The
later term ’nonsenescent’ meaning constant or declining mortality coupled with constant or
increasing fertility, as used by Braudisch and Vaupel (2013), seems better suited to Haliotis.
Although the sensitivity of fertility of the small and large adults did decline with age, the
results for the two adult classes were quite similar, along with similarities in their reproductive
values (found in the reproductive value vector). This may be unusual considering the large
increases in fecundity that occur with length, however reproductive values also consider future
value to the population. Kawecki and Stearns (1993) suggest equality of young reproductive
value spatially if genetic material is shared, I am suggesting that it may also occur across
age groups to some extent in broadcast spawners. From an evolutionary viewpoint equal
genetic gains from small and large adult reproduction (implied by their equal sensitivies)
means selection pressure is applied equally to both age groups.
Elasticity
Matrix elasticities
Transition matrix elasticity analysis was used to investigate the relative role of the annual
rates of movement through the matrix. They included: survival and growth between classes
(promotion); survival without reaching the next class (stasis); and surviving and reproducing
(fecundity) in determining the estimated rate of population increase (Caswell and Takada,
2004). The number and position of divisions between classes is important in elasticity analysis
when comparing different models (Enright et al., 1995). The equalities in equations 2.47 and
2.48 are a feature of three by three ‘progression’ matrices, but do not necessarily occur if the
number of stages changes (Carslake et al., 2009). I question, does this have a biological basis,
or is it an artefact of the population projection model chosen? That is, biologically does
changing SY GY (proportion of small adults promoted) have the same impact as changing
SM FM (relative large adult fecundity) on population numbers? Assuming no environmental
effects, as I have in this study, biological reasoning seems logical. This is an example of the
loop elasticities of matrices, where the elasticity of matrix elements increasing a population
component is equal to the elasticities of the matrix elements influenced by that component
(Caswell, 2001). If together, the parameters that increase a component of the population
increase (or decrease) at the same rate, then the population produced by that component
will collectively increase (or decrease) in a similar way. This means that changes in the current
population growth rate will be consistent across identical changes in the matrix parameters,
so giving them equal elasticity.
The different matrix models (2.43 through 2.46), created by changing the class sizes, showed
that the time spent in each adult class and the composition of adults within the class had a
large effect on their relative elemental elasticities. The use of elemental elasticities to draw
conclusions about the importance of different sectors of a population to the population growth
rate has been found to be problematic if the number of classes is changed (Enright et al.,
1995; Carslake et al., 2009). In this thesis I went further, and found similar problems from
simply altering the position of one of the class divisions, whilst maintaining the same number
41
of classes. Unlike Picard et al. (2010), where recruitment was constant, the changes that
occur in relative egg numbers as the class divisions shift can be seen, leading to changes in
the elasticities of the matrix elements (Table A.1).
Matrix 2.46 was designed so that the probability of growth is equal, so that roughly GJ = GY ,
and is therefore a matrix where the relative time in class J and class Y is equal. This indicated
the importance of relative growth into the next class (Enright et al., 1995). A look at the
elasticities of promotion from each class shows that e(SJ GJ ) ≥ e(SY GY ), and with not a
lot of difference between the two elasticity figures it would be prudent to say promotion out
of the juvenile class is around as important to maintaining the population growth rate as is
promotion from the small into the large adult class. This is possibly because TJ and TY (time
in each class) were calculated from the same equation 2.8. In this same matrix the elasticity
of stasis increases steadily with age, probably influenced by improvements in survival between
the juvenile and small adult classes, and by increased time in the class between the small and
large adult classes.
With a consistent population growth rate of 5%, the stasis of the two adult classes (e(SY (1 −
GY )) and e(SM )) consistently has the highest elasticities, while the elasticities measured in
relation to fecundity (e(SY FY ) and e(SM FM )) remained low across the different matrix combinations. A review by Benton and Grant (1999) found support for this, with the magnitude
of the difference between stasis and fecundity greater in species with longer generation times.
I also found that as the proportion of adult H. iris within a class increased, the elasticities of
that class increased. To be successful a species such as H. iris that has intermittent reproductive episodes and slow growth needs to survive for a long time. Similar life strategies are
often observed, as this is a typical effective strategy of many long-lived species (Campbell,
2006).
Concerning promotion between classes, e(SY GY ) is always less than or equal to e(SJ GJ ).
Biologically this is because of the declining numbers in each year group as the H. iris age.
The growth of older H. iris into the next class becomes less important to the population
as they are a lower proportion of the population, and there are proportionally less of them
moving into the longer class each year.
In matrix models 2.43 to 2.46 and in Table 2.4 the elasticities relating to the juvenile class
are consistent across the four different length classification models, suggesting this model is
robust to the change in class size. Because juvenile numbers, growth and survival rates were
unaffected by these different matrix formations, I was not expecting the juvenile elasticities
to change between the different matrix models.
In Table 2.4 the elasticity of adult growth into the next class e(GY ) is unaffected by changing
the matrix, as are the totalled elasticities of adult survival e(T otalS) and adult fecundity
e(T otalF ). This implies these elasticities are measured independently of the adult class
division selected, and thus reflect more closely the H. iris population being measured, rather
than the matrix being used. By extending this assumption to all the population parameters
in Table 2.4 I re-examined the results discussed above: The elasticities relating to survival
are the most important (as were the elasticities of stasis discussed previously), and indicate
the importance of surviving to the current population growth rate of this H. iris theoretical
population. In the juvenile class elasticity of stasis, e(SJ (1 − GJ )) is around one fifth the size
42
of the elasticity of survival e(SJ ), whereas in the adult classes elasticity of stasis and survival
are much closer in size. Juveniles exhibiting stasis have the highest likelihood of mortality in
the following year (70%), whereas around 23% of juveniles that survive are promoted, and
so have much lower mortality the next year (6%). This accounts for the lower elasticity of
juvenile stasis, compared to juvenile survival, because stasis includes not growing, as well as
survival. The two lowest parameter elasticities were growth and fecundity of the small adults,
(particularly fecundity in the first matrix, when the small adult group contained only adult
H. iris below 125 mm in length). These compare well with similar elasticities measured for
other marine invertebrates, and are consistent with a long lifespan (Linares et al., 2007). The
small size of these two elasticities implies these two matrix parameters were having little effect
on population growth in an unrestrained, unharvested H. iris population. Aside from the
small numbers in e(A125 ) small adult class, another possible reason for this is given by Pfister
(1998, p.213), who found that “variable life history stages tend to contribute relatively little
to population growth rates” and postulated this may be due to “natural selection altering life
histories to minimise stages with both high sensitivity and high variation.” H. iris populations
have been recorded with variable adult growth (Naylor et al., 2006) and fecundity, especially
during adolescence (when they are small adults) (McShane and Naylor, 1995c). Natural
selection against high variation presents one possible explanation for their low elasticity and
relative unimportance to changes in the current population growth rate (Pfister, 1998) when
they are in the small adult class.
Juvenile H. iris spend over four years in a cryptic habitat, unable to reproduce and limited
for space. On becoming adults and leaving the cryptic habitat survival is the most important
parameter, and in any single year spawnings are often missed altogether (Hooker and Creese,
1995; Poore, 1973). These life history characteristics are reflected in the elasticity readings,
with large adult survival the most important factor in maintaining the current population
growth rate, followed by small adult survival.
Robustness
Calculating the sensitivities and elasticities of the matrix A was problematic because λ was
both used to calculate GY ( in equation 2.11) and was also part of the sensitivity and elasticity
calculations. This was overcome by treating the λ in equation 2.11 as a constant when
calculating the sensitivities and elasticities of the matrix. This is justified as the elasticity
of GY is only one per cent of the total elasticity, which means that changing λ in 2.11 will
have a very small effect on λ in the Matrix A, and therefore a very small effect on both the
sensitivities and the elasticities.
This section examines how robust the model is to changes in parameter measurements with
high levels of uncertainty in their estimation. A five fold increase in egg settlement was
coupled with a similar decrease in juvenile survival to maintain the same population growth
rates for both trials. The choice of changing egg settlement or juvenile survival had little
impact on the elasticities, with changes in the population growth rate more important than
which parameter was changed. As the change in egg settlement caused a nearly identical
response in the elasticities as changing the juvenile survival, the method of changing the
populating growth rate appears to be unimportant. To raise the population growth rate
I increased both the egg settlement and juvenile survival, with increases in egg settlement
having a similar effect on elasticities to increases in juvenile survival, so that there was very
little distance between the two lines in Figure 2.4. This suggests the model is robust to
43
changes in these parameters over the range examined, assuming that the population growth
rate is known.
With a wide range of population growth rates to explore, variation in the elasticities of the
parameters was inevitable, due to the changes in the population proportions. This can be
explained by the changing dynamics of a rapidly increasing population. As faster growing
populations have a lower average age, this increase in the relative size of the juvenile cohort
probably accounts for much of the increase in their elasticity. However over the entire range
of population growth rates explored, from 1% to 16%, the relative ranking of the different
elasticities did not change, and survival of large adult H. iris consistently had the greatest
elasticity, and is predicted to have the greatest impact on the current population growth rate
(Caswell and Takada, 2004) when the large adult class shell length begins at 125 mm. At
higher population growth rates adult growth and fecundity became more important, relative
to survival.
For all the matrices the population growth rate was largely unaltered by the placement of
the division between the small and large adults. Picard et al. (2010) found with constant
mortality and recruitment that changes in class width did not effect the population growth
rate. Therefore my calculation of a consistent population growth rate (with constant mortality) implied total recruitment (SY FY + SM FM ) was consistent across the different matrix
models. Also both the sensitivity and elasticity measures relating to the juvenile class were
unaffected by changing the division between the small and large adult classes, as were the
elasticities of the population parameters. These two facts support an assumption that the
model is robust.
Summary
Population growth rate was more important to determining elasticity than either settlement
rate or juvenile survival. I calculated a population growth rate of 16% for one region, based
on Ministry of Primary Industry H. iris data.
Distribution error was largely removed from the matrix model despite using a length dependent fecundity power function. This was completed via computer analysis, producing an
accurate matrix with only three classes.
Biologically relevant matrix elasticities were separated from elasticity measures responsive
to matrix construction. When the same population was analysed with a different matrix,
dynamic elasticities were isolated as not true reflections of the population. By this process
of elimination I isolated the most relevant results, and commented briefly on how they could
be used to learn more about the population.
The new terms of ’promotion’ and ’relegation’ were introduced to describe movement through
a matrix. This leaves the terms ’positive growth’ and ’negative growth’ to be used specifically
as population parameters.
44
Chapter 3
Recommendations to increase the
minimum harvest length of Haliotis
iris are affected by population
growth rate
Introduction
Protecting the sustainable harvest of living marine resources by limiting fisher access to breeding stock is vital to their effective and sustainable management (Pulvenis de Sèligny-Maurel
et al., 2010). Almost all harvests are limited in some way, with restrictions aiming to limit
the time, location, volume and/or section of the stock harvested (Pulvenis de Sèligny-Maurel
et al., 2010). The aim of these restrictions is to protect the long-term viability of the harvest,
without unduly restricting the fishers. However these limitations are often based on incomplete population analysis, and so may fail to meet the dual targets of adequately protecting
the species whilst maximising the yield. An understanding of the dynamics of any marine
species is vital when aiming to optimise a sustainable harvest, but many fished species have
not been adequately investigated (Cook and Gordon, 2010). This lack of understanding undermines the reliability of many long term harvest plans. As a result of this the development
of methods of more accurately analysing the effects of alternative regulations is an important
undertaking in fisheries management.
Benthic marine invertebrate fisheries are often constrained by protecting a section of the
pre-breeding and breeding stock from harvest via body length restrictions. Body length
restrictions are used because of the difficulty of ageing harvestable stock. A further factor
supporting the use of length restrictions is that for many species landing size can be accurately
implemented by fishers (Punt et al., 2013) and undersize animals can be either successfully
avoided in the harvest, or returned to their environment with little harm. Unfortunately for
the population modeller this protection of smaller animals, as well their often cryptic and
patchy distribution, limits their monitoring and assessment (Culver et al., 2010; Dichmont
and Brown, 2010), thereby making population modelling more difficult.
Abalone are in the family Haliotidae, which contains only one extant genus Haliotis. Abalone
45
are found throughout most of the world, mostly in the shallow sub-tidal zone (Geiger, 1998).
The largest populations were historically in the colder climates of New Zealand, southern
Australia and South Africa in the south, and the west coast of North America and Japan
in the north (Lindberg, 1992; Geiger, 1998; Degnan et al., 2006). However large wild commercial harvests are now limited to New Zealand and Australia. Several different methods
of restricting the harvest have been used with abalone stocks, usually in combination with
length restrictions. These include limits on number, biomass, time of harvest, and the implementation of closed or restricted areas (State of California. Dept of Fish and Game., 2010;
Chick and Mayfield, 2012; Hesp et al., 2008; Ministry of Fisheries, 2011c; Woodham, 2009).
The specific applications of several of these has been questioned (ARMP, 2005; Prince et al.,
2008; Froese et al., 2008; Edgar and Barrett, 1999).
Questions on whether the current length restrictions applied to harvested Haliotis are ideal
to maximise yield are seldom quantified, although length restrictions have been regularly
changed (Chick et al., 2012; Ministry of Fisheries, 2011c; Hesp et al., 2008). Changes in
minimum shell length were probably introduced in an effort to prevent small scale overexploitation which could deplete breeding numbers, rather than as a tool to maximise sustainable yield (Mayfield et al., 2011; Ministry of Fisheries, 2011c). Larger Haliotis were found
further from boat ramps in Tasmania, implying fishers targeted easier access sites. However
smaller legal Haliotis had apparently moved to even out the distribution, possibly mitigating
any impacts on fertilisation success (Stuart-Smith et al., 2008). Several models used in the
past have focussed on shell length, however the relationship between shell length and reproductive output has seldom been included by modellers (Breen et al., 2003; Mayfield, 2010;
Ministry of Fisheries, 2011c). Leaving the larger more fecundatant adults unharvested in a
slot type harvest system shows promise, as traditional Māori harvesters have been known to
target the smaller adult Haliotis, and leave larger adults unharvested (Gibson, P. on behalf
of Ngāti Konohi, 2006).
The most common New Zealand abalone, called blackfoot pāua or Haliotis iris, are fished
throughout New Zealand, with the majority of the catch from around Stewart Island and
the Chatham Islands, the South Island and the lower part of the North Island (Ministry of
Fisheries, 2011c). Harvesters are classified as cultural, recreational, illegal and commercial
(Ministry of Fisheries, 2011c), with the majority of the harvest removed and exported by
commercial harvesters. Annual export earnings of around $50 million dollars have remained
stable, both in tonnage and value, since 2003 (Statistics New Zealand, 2010).
Stock assessment of H. iris in several quota management areas is completed using relative
abundance estimates in a similar way to many overseas fisheries, by checking for population
changes (Gorfine et al., 2001; State of California. Dept of Fish and Game., 2010; Mayfield
et al., 2011). In New Zealand measurements are primarily based on standardised CPUE data
(catch per unit effort) (Ministry of Fisheries, 2011c). CPUE is based on the premise that
if a fisher can catch more in a shorter amount of time then there is more stock in the area.
Research by Fu and McKenzie (2010a) found CPUE to have a variable relationship to density,
and to be only useful in seeing trends over time. Breen et al. (2003) described CPUE as
‘possibly misleading’. H. iris stocks in the larger New Zealand fisheries are quantitatively
assessed using a Bayesian length-based model, and include CPUE, as well as other stock
measures (Ministry of Fisheries, 2011c). Limitations of this Bayesian stock analysis include;
46
dependence on an uncertain estimated catch history; the lack of a stock-recruitment relationship; and local variations in population parameters, although the use of smaller assessment
areas, and extensive use of sensitivity trials and probability estimations in the model outputs
are going some way to improving accuracy (Ministry of Fisheries, 2011c). However none of
these assessments specifically quantify the effect of changes in shell length restrictions on
reproductive output.
In addition to changing harvestable shell lengths I explored what would happen if it were
impossible to maintain ideal control over both the volume and shell length of harvested
H. iris; if one or other of these restrictions (on size and number taken) was unenforceable.
Firstly, if unrestrained harvesting of a class where to occur, does a class exist that makes
this sustainable? In close proximity to large human populations, even when people limit
their harvest to 10 per person, an over-harvest may occur. Indeed it is common to find areas
were not a single H. iris, of harvestable length can be found (Hepburn, pers. comm. 2013).
Secondly, what happens with no control over the minimum or maximum shell harvest length?
In this case adults of any length are taken, but there is control over volume. This may apply
if a high level of poaching existed, but was limited to specific areas, or times of the year, if
H. iris could be successfully protected within reserves, or by the use of an open season.
The elasticity measures I calculated in Chapter 2 implied that a slot system may maximise
the sustainable yield. Accordingly my aim here is to investigate both slot and minimum
harvest length regulations, in an attempt to quantify the best harvest length for maximising
the sustainable yield from a population of H. iris. I also wanted to examine the magnitude of
the effect on fisher effort of changing the harvest length, and so have quantitatively included
some measurements relating to poaching and the provision of reserves.
Method
The following section contains an outline of the mathematical framework used in this analysis.
The mathematical analysis was once again completed using the ‘R’ program (R Development
Core Team, 2008).
Bold face type is used to denote vectors and matrices. Many of the symbols used in this
chapter were explained in Chapter 2, however a brief review is provided here: Jt was the
number of juveniles at time t, Y was small adults and M was large adults. The population
parameters within the matrix are annual rates: S for survival, G for growth to the next class,
and F for fecundity.
The first task undertaken in this Chapter was to choose and implement a density dependent
effect on this same population matrix developed in Chapter 2, which constrained the population growth to stop when it reached the carrying capacity. For a population at carrying
capacity the equilibrium population ratios could then be calculated.
47
Formatting the density dependent matrix Aφ
The density dependence required to constrain population growth could theoretically affect
any of the population vital rates included in the matrix. I chose to apply a density dependent
effect solely to juvenile mortality, for the following reasons. Firstly, high levels of juvenile
mortality are linked to juvenile density in many hard shelled benthic marine invertebrates
(Hunt and Scheibling, 1997), including H. iris (McShane, 1995). Secondly, fished Haliotis
populations have been shown to increase in number when cryptic habitats are increased
in situ (Davis 1995), leading to the assumption that juvenile density is a critical factor in
population growth and that density dependent mortality affects the youngest cohort (Connell,
1985). Finally juvenile H. iris inhabit a spatially separate environment to the adults, so the
effects of density in the juvenile class are assumed to be solely due to the numbers of H. iris
within that juvenile class. Hunt and Scheibling (1997) found no evidence of a relationship
between juvenile and adult mortality in benthic marine invertebrates. Accordingly, density
effects were applied solely to survival of the juvenile class.
The relationship between density dependent mortality (φt ) and juvenile abalone numbers
operates mathematically in a similar way to a stock-recruitment relationship and different
forms of the same equations can be used to model either relationship. The two main mathematical forms of these relationships widely modelled in the fisheries industry are Ricker and
Beverton-Holt (Kimura, 1988). A Beverton-Holt type equation:
φt = 1/(1 + αJt )
(3.1)
was found to be better suited to juvenile fish mortality (Shephard and Cushing, 1980). Therefore I incorporated this density dependent equation into the matrix A, and created a new
density constrained matrix Aφ .
Nt+1 = Aφ Nt


SJ φt (1 − GJ )
SY FY
SM FM
J
J
 Y 
 Y 
SJ φt GJ
SY (1 − GY )
0
=
M t+1
0
SY GY
SM
M t



(3.2)
For the purpose of this study the density dependent term alpha (α) in equation 3.1 was set at
one. This is a method of non-dimensionally scaling the density dependent mortality, without
a loss of generality (Krkosek, pers. comm. 2013). Density effects on juvenile shell growth are
assumed to be negligible in this model.
Calculating equilibria
With a density constraint in the matrix, the modelled population would reach some maximum
carrying capacity. I assumed a spatially and temporally uniform environment and an unfished
population, so at carrying capacity the population would remain constant and:
Nt = Nt+1
48
(3.3)
This equality also applies to each class of H. iris. Thus, average class size was constant,
year to year, as I assumed all other factors were constant, and so by algebraic analysis of
equation 3.2, I obtained three equilibria stage distributions:
Jt = Jt+1 = J ?
Yt = Yt+1 = Y
(3.4)
?
?
Mt = Mt+1 = M .
(3.5)
(3.6)
The vector N? was created to combine these three class size estimations:
N? = (J ? Y ? M ? )
(3.7)
Harvest
Two methods of calculating the maximum sustainable harvest rate were considered. Firstly,
by exploring the harvest of the smaller adults, that is harvesting those with shells longer
than the emergent length of 100 mm, but below a longer length, for example harvesting some
of the small adult H. iris, with shell lengths from 100-125 mm. Secondly, I calculated the
maximum sustainable harvest if a minimum harvest length was enforced; an example of this
would be the current system of harvesting the larger adult H. iris, with shells longer than
125 mm. Note that in this report the term harvest refers to the proportion of the harvestable
stock caught annually.
In deciding where to apply the harvest terms in the matrix, I considered that the commercial
H. iris harvest in New Zealand begins when the new season opens on the first of October
and runs until the quota is filled (Ministry of Fisheries, 2011c). Therefore H. iris harvest
would affect both annual survival and fecundity rates, as the majority of H. iris are caught
before spawning occurs (taken to be in late summer or autumn (Poore 1973, Figure 2.1)).
By introducing the term HY for the harvest of small adults, the parameter (1 − HY ) (the
rate that small adults survive after harvest) could be incorporated into the matrix where it
modified all the small adult numbers, and so the matrix Aφ then contained (1 − HY ) in three
positions and became Ahy :




J
SJ Φt (1 − GJ ) (1 − HY )SY FY
SM FM
J
 Y 
 Y 
S J φt G J
(1 − HY )SY (1 − GY )
0
=
M t
M t+1
0
(1 − HY )SY GY
SM

(3.8)
The possibility of harvesting the large adults (for example those above 125 mm, as is currently the regulation), whilst leaving the small adults unharvested was also investigated. By
introducing the term HM for the harvest of large adults, the matrix Aφ could alternatively
include (1 − HM ) and thus became:
49




J
SJ Φt (1 − GJ ) SY FY
(1 − HM )SM FM
J
 Y 
 Y 
SJ φt GJ
SY (1 − GY )
0
=
M t
0
SY GY
(1 − HM )SM
M t+1

(3.9)
Then, by introducing a generic term for the density construed matrix (Aφ ) that also included
the harvest of either small or large adults I created a new matrix Ah , that could satisfy
following matrix multiplication:
Nh ,t+1 = Ah Nh ,t
(3.10)
which created a new equilibrium population N?h . The unfished population N? would be fished
down to this new level, where it is assumed to be able to consistently maintain itself under
some fishing pressure, and stabilise. A drop in the numbers is necessary because of the very
low population growth rate close to the carrying capacity, which is shown in figure 3.1. The
three components of this population when it reached equilibrium were once again calculated
for Ah , so that in each instance:
Jt = Jt+1 = Jh?
Yt = Yt+1 =
Mt = Mt+1 =
Yh?
Mh?
(3.11)
(3.12)
(3.13)
These class estimations Jh? ,Yh? , andMh? were also arranged into a vector:
N?h = (Jh? Yh? Mh? )
(3.14)
which described the equilibrium state of the population under harvest. In all instances the
stated harvest length was shell length at census time. For this model to function I assumed a
precise selectivity between shell length classes during harvest, and that harvesting occurred
immediately after the census date. This meant to be harvestable in this model the H. iris
needed to be more than one year post emergent.
Maximum sustainable harvest
Maximum sustainable yield was calculated for the harvest of H. iris from the small adult (Y)
class or the large adult (M) class. The accuracy of these calculations depended on both the
accuracy of the parameters estimated in Table 2.2, and the appropriateness of the selected
stage-based (here length-based) matrix model for the data analysis.
Maximising sustainable harvest: numbers
If it is possible for a population to reach a positive equilibrium whilst being harvested, then
the highest sustainable harvest (MSY) would be
Highest sustainable harvest (MSY) = (Aφ N?h − N?h )
50
(3.15)
with N?h showing the population classes at equilibrium under this harvest method. Therefore
to find the best harvest rate for H. iris measuring 100-125 mm (a slot harvest) from this
theoretical population I firstly created a vector (HY ) containing 10,000 different harvest
rates. I used computer simulation and trialed each of these harvest rates (HY ) in the matrix
in equation 3.8. For each value in (HY ) I then calculated the solutions to equations 3.11,
3.12, and 3.13, and used these to compile a new vector N?h via equation 3.14, that showed the
equilibrium population that would result under that harvest system. This gave me 10,000
different N?h vectors. Then finally, using equation 3.15, I recorded the highest sustainable
harvest (MSY) for each of these different N?h vectors. The harvest rate (or HY value) that
produced the largest MSY was identified as the maximum sustainable harvest rate for this
model.
The next task was to conduct a similar analysis by creating a vector of HM values to trial in
equation 3.9. I again used equations 3.11 through to 3.15 to find both the largest sustainable
harvest, and the corresponding recommended maximum sustainable harvest rate for the large
adults. Thus I identified the ideal rate for harvesting H. iris that have shell lengths greater
than 125 mm from this theoretical population.
The largest sustainable harvest was calculated in the same way for harvests involving different
shell length regulations, for example harvesting H. iris with shell lengths above 128 mm rather
than those above 125 mm. Harvested animals could still be taken from either class Y or class
M, with the shell length that divided class Y and M altered to reflect differences in these
harvest regulations. A total of 38 different length classes were created for both Y and M
classes, giving a total of 76 different harvest systems. These alterations effectively resulted in
a new matrix each time the harvest length was changed. These new matrices represented an
analysis of the same population, but it was analysed in a different way as there were individual
H. iris that were now in different adult classes. Therefore the growth and fecundity rates were
recalculated for each of the different matrices using equations 2.12 and 2.21, and a new egg
settlement rate was used to give λ = 1.0500 for each unharvested matrix. The new matrices
were harvested, once again using equations 3.9 through to 3.15, and lead to the calculation
of maximum sustainable yield and the harvest level at which it occurred, for each of the new
Y and M classes.
Maximising sustainable harvest: biomass
The calculation of an estimated biomass or meat weight harvested is an important consideration, particularly in the commercial and export trade where H. iris data is usually calculated
as a meat weight (Ministry of Fisheries, 2011c; Statistics New Zealand, 2010). Poore (1972
Table 8 p 554) calculated an allometric relationship between individual animal weight (Wt)
and shell length (L) for measuring H. iris at Kaikoura, with a value of 3.22 for the coefficient.
I assumed the relationship was of the form W t = aLb + k (Schiel and Breen, 1991), and the
relationships between the total weight of a H. iris and their body (excluding shell) weight
was linear (Poore, 1972c). As I was using the same population to parametrise my matrix,
I used this coefficient to derive a comparative body weight equation, based on the average
shell lengths of H. iris measured at Kaikoura, within each size class:
Body weight ∝ L3.2
= kL3.2
51
(3.16)
I calculated a comparative average weight for individual H. iris in each class using integration
under the curve produced from equation 3.16, with a similar consideration of the unequal
distribution of H. iris within each class as is included under the heading ‘egg numbers’ in
section 2. For the small adult class of lengths 100-125 mm the equation I used was:
L=125
Xmm W tY =
Wt(L(t) , t)(t1 →t2 ) st1
L=100 mm
L=125
Xmm
(3.17)
st
L=100 mm
where W tY is the average body weight of an individual H. iris in class Y, Wt(L(t) , t)(t1 →t2 )
is a vector of average H. iris body weights at each unit of age and the vector st contains
the average survival values for H. iris at each unit of age. W tM was calculated in the same
way, in this case using lengths between 125 mm and the average maximum shell length of
146.2 mm. This was then repeated for the other 76 harvest classes, in each case giving the
average weight of an individual H. iris in that class.
The average body weight for individual H. iris in each class was also calculated from the
average egg numbers for each H. iris in the class, using the inverse of the egg numbers - shell
length equation (2.16):
L = 12 × E 0.16
(3.18)
body weight = kE 0.51
(3.19)
combined with equation 3.16 to give:
This extra method of computation of average bodyweight gave an indication of the accuracy
of the model, via a comparison of the most diverse values for average body weights originating
from the two equations 3.17 and 3.19.
A total biomass harvested for both the small adult and large adults classes was then calculated
from total number harvested from equation 3.15 multiplied by average body weight from
equation 3.17. Equation 3.17 was justified over equation 3.19 using Occam’s razor.
biomass = (Aφ N?h − N?h )W tY
(3.20)
These biomass calculations assume that any H. iris within the class being harvested are
equally likely to be harvested irrespective of their shell length. This means that the average
body weight used for the harvested cohort taken from the class will be equivalent to the
average body weight of all individual H. iris within the class.
Less controlled exploitation
I then completed an investigation of the outcomes that are sustainable if less regulated harvesting occurred. Previously the analysed harvest systems assumed ideal control over both
52
the volume and shell length of harvested H. iris, whereas here I looked at some of the possible
outcomes if one or other of these restrictions were unenforceable.
Firstly, I wanted to see which were the largest sustainable harvest rates of H. iris, and what
were the shell measurements of H. iris within that class. This was achieved by locating the
highest sustainable harvest rates, which are in columns two and three of Tables A.3 - A.6. If
the harvest rate is written as a 1.000 in these Tables, then 100% harvest from that class is
achievable for this theoretical population. Next I looked at a narrow slot, harvesting from the
smallest adults. Although it would not allow 100% harvest, any harvesting from the smallest
adults would allow a higher harvest rate due to the smaller number in the slot.
Secondly, I conducted an examination of what happens with no control over the minimum or
maximum shell harvest length. I wanted to determine what level of harvesting was sustainable
with no length restrictions. These results were generated when I took 146.2 mm as the largest
slot size. This is because this harvest class contains all the adults.
Changes in population size
Changes in population size are often expressed as changes in the number of individuals,
however due to the relationships between biomass and both fertility and commercial return
in H. iris studies, the option was taken to calculate the theoretical change in population
size (from unharvested to harvested populations) as a change in biomass. This is also a
measurement often used in New Zealand’s Ministry of Primary Industries calculations, where
it is sometimes called the fishdown level (Ministry of Fisheries, 2008b). This equation looked
at the impacts of harvesting from the small adult class in a slot type system. A reminder that
both the Yh? and Mh? values in equation 3.21 are obtained from a calculation of equilibrium
beginning with numbers generated from equation 3.8.
FishdownY,W t =
(Yh? × W tY + Mh? × W tM )
Y ? × W tY + M ? × W tM
(3.21)
Changes in the total population biomass due to a minimum length harvest system were also
investigated using an equation for FishdownM,W t , which was calculated in a similar way to
equation 3.21 (using values generated by equation 3.9).
Catchability and workload
In this section I examined the impacts of alternative harvest systems on the work of the
harvesters. First, the ‘catchability’ of the harvested cohort was the amount of the total adult
population that was in the harvested class, and gave an indication of the probability that if
a harvester found a H. iris, was it of the correct length to be harvested?
CatchabilityY =
Yh?
Yh? + Mh?
(3.22)
Secondly, the workload analysis compared the body weight of H. iris to be harvested under
the different harvest systems. A slot type harvest system means more animals must be taken
for the same biomass yield, thereby increasing the workload of the harvesters if they want
53
the same meat weight. For example, if the maximum sustainable yield were from harvesting
the small adults below 135 mm:
workloadY =
W tM (125)
W tY (135)
(3.23)
This indicated the extra amount of work required for the same biomass of meat harvested,
when compared to the harvest of H. iris above 125 mm. The inverse of equation 3.23 is the
average body weight gain (or loss) when comparing an average animal from the suggested
harvest system with one taken under the current (greater than 125 mm shell length) harvest.
change in average animal body weightY =
1
workloadY
(3.24)
Robustness
The final part of this chapter details my attempts to ensure, as much as possible, that the
results are immune from error due to uncertainty in the calculations, thereby making the
results more robust. The first method I used was to ensure that the population growth rate
remained unchanged as I reconfigured the matrix each time I applied a different harvest
system. Secondly I considered a range of alternative population growth rates, as I was
unable to ascertain in Chapter 2 which population growth rate (from 1% to 16%) was most
biologically reasonable.
Refining the model
I examined the parameters to identify any way to make the comparative analysis as fair
as possible across the different shell harvest lengths. That is, as I changed the division
between small and large adults, (with the aim of identifying the harvest system that maximised sustainable productivity), did all the static parameters remain static? To this end,
a comparison of the population growth rates (PGR) calculated from the unharvested population matrices was examined. Unfortunately changes in the matrix formation caused small
unwanted changes in the unharvested PGR, and here I was trying to establish as much as
possible a level playing field, where all alternative systems started from the same unharvested
population.
As a way to maintain a consistent PGR across the range of matrix formations I decided to
make small changes in one of my calculations. There were several alternatives, including
the option of a gradual change in one or more of the matrix parameters. Any such change
was artificial, and had no biological basis, so I was wary of changing any parameter which
would change the balance between the small and large adult classes, and possibly impact
preferentially on the comparative yield analysis. Fecundity was seen as the parameter having
least influence on numbers in the small and large adult classes, which meant a change in
EY , EM , and/or egg settlement rate, SE was considered. On the basis of Occam’s razor SE
was chosen as the parameter to alter, with a view to levelling the PGR across the different
harvest systems. By using this recalculation SE became SEi with i ∈ {1, 40}, and gave the
new equations:
54
FY i = EY i × SEi
(3.25)
FM i = EM i × SEi
(3.26)
These recalculated FY i and FM i values were then used in equations 3.2 - 3.20 in place of FY
and FM .
Population growth rates (PGR)
Due to the large number of different harvest classes included in the analysis, I decided to
examine only four representative population growth rates: Two extreme values of 2.5% and
15% were added to the value of 5% already used. Then an intermediary value of 10% was also
included, which arose due to uncertainty surrounding the true number of H. iris in analysed
populations (Ministry of Fisheries, 2011c). As there was also a large amount of uncertainty
surrounding both the egg settlement rate and the population growth rate, (and as the matrix
elasticites were largely unaffected by a change in the relativity of egg settlement and juvenile
survival in Chapter 2), I varied population growth rate by changing the egg settlement rate.
The full range of these new SE values is shown in the appendices, in columns three, five,
seven and nine of Table A.2. The values of GY will also change as the population growth rate
changes, as the population growth rate is included in the calculation of GY via equation 2.11.
The new GY parameters are included in columns two, four, six and eight of Table A.2. The
effects of the four population growth rates on the above computations (described in sections
3 through to 3) are included in the results.
Results
The density dependent matrix Aφ
The matrix Aφ was used to generate population predictions for 100 years assuming no fishing, and density dependent mortality in the juvenile (J) class. The shell length division
between the small adult class (Y) and the large adult class (M) was at 125 mm, with a
population growth rate of 5%. The three length classes (J, Y and M) converged to the same
stable equilibrium population known as N? . This is described as an ergodic matrix model
(Cohen, 1979), as the same end demography was reached independently of initial conditions
(Figure 3.1a and b).
Calculating equilibria
The matrix multiplication shown in equation 3.2 was expanded to give three equations:
Jt+1 = Jt SJ Φt (1 − GJ ) + Yt SY FY + Mt SM FM
(3.27)
Yt+1 = Jt SJ Φt GJ + Yt SY (1 − GY )
(3.28)
Mt+1 = Yt SY GY + Mt SM
55
(3.29)
b.
2.5
2.5
a.
2.0
1.5
0.0
0.5
1.0
1.5
1.0
0.0
0.5
Ratio in each length class
2.0
Juveniles
Small adults
Large adults
Population total
0
20
40
60
80
100
0
Year
20
40
60
80
100
Year
Figure 3.1: Trajectory of the H. iris theoretical population, predicted
for 100 years with no fishing and density dependent mortality in the
juvenile class. There was convergence to an equilibrium population,
reguardless of whether the initial population distribution starts low
(as in Figure a) or high (as in Figure b). In both a and b the number
of H. iris in each length class is shown, along with the population
total of all three classes.
56
Solving the three equations 3.27, 3.28 and 3.29 simultaneously (assuming the conditions
outlined in equations 3.4 through 3.6 inclusive), resulted in the equations:
1
J? =
(k2 − 1)
(3.30)
α
(k2 − 1) SJ GJ
(3.31)
Y? =
αk1 k2
(k2 − 1) SJ GJ SY GY
M? =
(3.32)
αk1 k2 (1 − SM )
where k1 and k2 are defined as:
k1 = (1 − SY + SY GY )
SJ GJ SY GY SM FM
SJ GJ SY FY
+
k 2 = S J − S J GJ +
k1
k1 (1 − SM )
These class size estimations J ? , Y ? , and M ? produced from equations 3.30, 3.31 and 3.32
were arranged into a vector N? (equation 3.14). With the parameters from Table 2.2 inserted
into 3.30, 3.31 and 3.32 the long term stable H. iris population ratio (equation 3.7) which
fits this population model was developed, and used in the harvest analysis.
Harvest
If adult H. iris are harvested from this population, then the fecundity, growth and survival of
the population will be affected. Here I assumed that if the small adults were harvested then
HY has a positive value and HM = 0, and if the large adults were harvested then HY = 0 and
HM has a positive value. The equations 3.30, 3.31 and 3.32 were then rewritten to include
both harvest terms (this time assuming the conditions outlined in equations 3.11 through
3.13 inclusive):
1
Jh? =
(k2h − 1)
(3.33)
α
(k2h − 1) SJ GJ
Yh? =
(3.34)
αk1h k2h
(k2h − 1) SJ GJ SY GY (1 − HY )
Mh? =
(3.35)
αk1h k2h (1 − SM (1 − HM ))
where k1h and k2h were defined as:
k1h = 1 − (SY (1 − GY )(1 − HY ))
SJ GJ SY FY (1 − HY ) SJ GJ SY GY (1 − HY )SM FM (1 − HM )
+
k2h = SJ − SJ GJ +
k1h
k1h (1 − SM (1 − HM ))
and this was then used via iteration to calculate both the maximum sustainable yield and
the harvest level at which this occurred. I did this for each of the 76 trialed harvest systems,
at each of the four population growth rates.
I calculated the maximum sustainable harvest for this population, in terms of both maximising the number harvested, and also maximising the biomass yield.
57
Using the parameters in Table 2.2 (based on a population growth rate of 5%), with a stage
classified population model lead to the implication that yields from the analysed population
could be increased with a change from current harvest regulations. The results of this analysis
to identify which of the examined shell harvest length regulations gave the greatest sustainable
yield (in numbers of H. iris) are shown in detail in the appendices in Table A.4, and visually in
Figure 3.2a. Note that these recommendations are limited to a theoretical population based
on the H. iris examined at Kaikoura by Poore (1972a,b,c, 1973), with an average maximum
shell length of 146.2 mm.
The current regulations allowing only H. iris above a shell length of 125 mm to be harvested
limited the sustainable yield (in number of H. iris) to 0.93 of the maximum achievable,
according to my model. This point is shown where the lines crossed near the peak of the
dotted line in Figure 3.2a. This 125 mm limit is thus nearly at the maximum achievable
when the larger sizes were harvested. However the maximum number of H. iris harvestable
was achieved if H. iris below a shell length of 143 mm were harvested (that is a slot harvest,
between 100 mm and 143 mm), and the largest H. iris (from 143 mm to the average maximum
length of 146.2 mm) were left to freely reproduce. This point is shown in Figure 3.2a where
the dashed line reached a maximum at the average age of 13 years, which was equivalent to
a length of 143 mm. The lighter columns in Figure 3.3 show a small selection of the more
interesting number results for all four population growth rates, with the full results in Tables
A.3 - A.6.
When maximum sustainable yield was considered in terms of harvested biomass, a different
shell harvest length was recommended compared to those harvest lengths that maximised
the number harvested. Results for the population growth rate of 5% are shown visually in
Figure 3.2b with the full analysis in Table A.4. Once again these results are specific to a
theoretical population with a maximum length of 146.2 mm, based on the H. iris measured
at Kaikoura by Poore (1972a,b,c, 1973). The maximum biomass yield was achieved if H. iris
above 135 mm were harvested, and the current practice of harvesting H. iris above 125 mm
yielded 0.97 of that maximum sustainable biomass. Once again, Figure 3.3 contains a small
selection of the more relevant results, with comparative weights in the darker columns. Full
results from all the population growth rates are again included in Tables A.3 - A.6.
It was possible to examine the predictions in two less regulated situations. Firstly, if there
was little or no control over the volume taken, but there is control over the lengths. The
annual harvest of all H. iris above a certain length was sustainable. The shortest minimum
shell harvest length regulation that allowed 100% harvest if the population growth rate is
5% is 146.05 mm, a measurement that is only 0.15 mm away from the theoretical maximum
shell length of 146.2 mm, and consisted of the longest 3.0% of the population being removed
annually, limiting yield to only 70% of the theoretical maximum number, or 81% of the
maximum biomass yield. This result can be seen in the dotted line in Figure 3.2 as the
58
Figure 3.2: Corresponding to a consistent population growth rate of 5%
for the H. iris measured at Kaikoura in 1967-1969. A slot harvest from
100-143 mm (the peak of the the dashed line) in Figure 3.2a maximises the
number harvested, while in Figure 3.2b harvesting H. iris above 135 mm
(the peak of the dotted line) maximises the biomass yield. Both graphs
have the same x axis, estimated age. The intersection points of the straight
lines show yield of both the current (lower left) and recommended (upper
right) harvest systems in each graph. Each graph shows yield under the
current harvest system (above 125 mm) as the lower left point of intersection with the dotted line, and the gains that were made with a new harvest
system as the upper right point where the lines intersect.
59
point of inflection at the age of 22.8 years (equivalent to a harvest length above 146.05 mm).
At a population growth rate of 2.5% none of the investigated harvest systems generated
a sustainable 100% harvest, the best was 21% harvest rate from the largest 1.2% of the
population. For the other population growth rates sustainable 100% harvest rates are shown
as ‘ones’ in column three in Tables A.5 and A.6.
With a narrow slot of 100-121 mm harvest rates varied from 10.5% to 36.5% as the population
growth rate varied from 2.5% to 15%. A narrow slot, targeting small emergent adults of 100121 mm allows a higher harvest rate, but limits yield to around half the number sustainably
achievable under the much larger 100-143 mm slot system. Comparative yields from these
narrow slot harvests are shown in Figure 3.3, and are listed in Tables A.3 - A.6.
Secondly, if no shell length restrictions were applied and all adults (100 mm to 146.2 mm)
were sustainably harvested. With a population growth rate of 5% the sustainable harvest
rate was 2.3% and the yield in this situation was 0.85 of the maximum achievable number, or
0.81 of the maximum achievable biomass. This point of less regulated harvest is visible on the
both graphs in Figure 3.2 as the first point on the dotted line (harvesting all H. iris above
the minimum length of 100 mm) and also as the last point on the dashed line (harvesting all
H. iris below the maximum length of 146.2 mm). These measurements are also visible for
all the examined population growth rates as the last numbers in columns two, four and six
in Tables A.3 - A.6.
The effect of sustainably fishing this H. iris population on the numbers of H. iris remaining in
the population was also examined (the fishdown level). The changes in biomass I calculated
for the recommended harvest regulations (compared to a similar unharvested population) are
shown in columns 4, 11 and 15 in Table 3.1. Both the current and recommended harvest
regulations keep the fishdown level above 0.340 in the harvestable class, with higher total
population biomass needing to be maintained at the lower PGR. At the same population
growth rates the slot type systems and the minimum length systems recommend very similar
fishdown levels.
Under the current system of harvesting H. iris above 125 mm the catchability (probability
of any single adult H. iris being in the harvestable class) decreased as the population growth
rate increased. These results are in column three of Table 3.1. However the catchability
remained relatively constant when exploring systems that maximise yield, both in terms of a
maximising numbers harvest system (column 10), and the harvest regulation that maximised
biomass harvest (in column 14). I also compared the numbers and biomass that could be
sustainably harvested above 125 mm at varying population growth rates, compared these to
the numbers harvested when collected under a maximising biomass model, these results are
in columns five and six of Table 3.1, also show in the heights of the bar graphs in Figure 3.3.
Column seven shows the change in average harvested animal weight, and gives an indication
of the extra work involved; from harvesting H. iris from this population above 125 mm
rather than using the biomass optimisation harvest system outlined in columns 12 though
15.
60
.
Table 3.1: Shell harvest lengths that gave maximum sustainable yield
(MSY) at different population growth rates (PGR) for the H. iris at
Kaikoura. The upper section (columns two through seven) details
the current system (minimum shell harvest length of 125 mm), in the
lower left (columns eight through eleven) are maximum shell harvest
lengths (slot type system) that maximised sustainable yield (numbers) and finally on the lower right (columns 12 through 15) is the
harvest length regulation that maximised sustainable yield (biomass).
Catchability shows how much of the total adult population was in the
harvested class.
Pop.
growth
rate
1.
The current regulation, harvesting H. iris with a shell
length greater than 125 mm
(125+)
2.
3.
4.
sustainable catchability population
harvest
drop due
rate
to fishing
(biomass)
2.5%
5%
10%
15%
1.4%
2.7%
5.1%
7.4%
0.83
0.80
0.75
0.71
0.45
0.42
0.37
0.34
Disadvantages due to harvesting H. iris 125+, as opposed to the harvest systems
suggested below
5.
6.
7.
Decrease
Decrease
Drop
in yield in yield in indi(number) (biomass) vidual
bodyweight
0.92
0.96
0.95
0.93
0.97
0.96
0.94
0.98
0.97
0.95
0.99
0.98
Pop.
growth
rate
1.
Alternative regulations designed to
maximise numbers harvested
Alternative regulations designed to
maximise biomass yield
8.
slot shell
length
(mm)
9.
10.
sustainable catchability
harvest
rate
12.
shell
length
(mm)
13.
14.
sustainable catchability
harvest
rate
2.5%
5%
10%
15%
100 − 144
100 − 143
100 − 142
100 − 141
2.8%
5.5%
9.6%
17.2%
137+
135+
132+
129+
1.7%
3.2%
5.8%
8.0%
0.49
0.47
0.47
0.47
11.
population
drop due
to fishing
(biomass)
0.45
0.42
0.38
0.35
61
0.66
0.65
0.63
0.63
15.
population
drop due
to fishing
(biomass)
0.45
0.42
0.38
0.34
Figure 3.3: Of the 76 harvest systems explored six were graphed for each
population growth rate, with a label stating the section of the population
harvested under that system, and the maximum sustainable harvest from
that section. From left to right the six pairs of columns are: Firstly, a small
slot system, harvesting adults one to two years after emergence. Next, the
best two slot harvest systems, the first pair maximises the number that can
be harvested, and the second pair maximises weight. The fourth set illistrates the current system of harvesting above 125 mm. The last two pairs
are the best recommendations generated if harvesting above a minimum
length, the first pair maximise number, and second pair maximises weight.
All 76 harvest systems are shown visually for the population growth rates
of 5% and 15% in Figures 3.2 and 3.4 respectively.
62
Figure 3.4: Corresponding to a consistent population growth rate of 15%
for a homogeneous theoretical H. iris population, with a maximum shell
length of 146.2 mm. A slot harvest between 100-141 mm (the peak of the
the dashed line) in Figure 3.4a maximises the number harvested, while in
Figure 3.4b harvesting H. iris above 129 mm (the peak of the dotted line)
maximises the biomass yield. Both graphs have the same x axis, estimated
age. The intersection points of the straight lines show yield of both the
current (lower left) and recommended (upper right) harvest systems in
both graphs. Each graph shows yield under the current harvest system
(above 125 mm) as the lower left point of intersection with the dotted line,
and the gains that were made with a new harvest system as the upper
right point where the lines intersect. Figure 3.4a shows that the maximum
suitable yield in numbers of H. iris was obtained by a slot harvest of the
smaller adults from this theoretical Kaikoura population, from 100-141 mm
in shell length. Figure 3.4b includes relative maximum sustainable yield
(biomass), which was achieved with harvesting H. iris with shells longer
than 129 mm.
63
Robustness
Refining the model
The task I undertook here was to make small changes in the egg settlement (SE ) values to
ensure that no matter how I divided the unharvested population (where I put the division
between small and large adults) the population growth rate remained the same. This was
achieved by raising the SE associated with lower lambda values, and lowering the SE generating higher lambda values. The changes in SE required to level the population growth rate
were less than 5%. These results are in columns three, five, seven and nine in Table A.2.
Population growth rates (PGR)
Some of the more interesting results for all four population growth rates are shown in Figure 3.3, with the full results in Tables A.3 - A.6. As expected, increasing the population
growth rate increased the recommended harvest rates. As the population growth rate increased the theoretical recommended shell harvest lengths to maximise sustainable yield
consistently decreased, although the percentage in the harvestable class remained constant.
To visually illustrate the differences in the results that occur as a result of changing the
population growth rate, I have graphed results for PGR= 15% in Figure 3.4, which I compared
with Figure 3.2 (where PGR= 5%). Studying Figure 3.4a, the numeric gains made from
changing the harvest system from the current minimum shell length of 125 mm to the advised
slot (100-141 mm) were comparatively less at the higher population growth rate, compared
to the gain at a population growth rate of 5%. Studying Figure 3.4b, the gains in biomass
harvested which were made from increasing harvest length from 125 mm to 129 mm are very
small. This shows that the recommended weight maximisation harvest length gets closer to
the current system (harvest above 125 mm) at higher population growth rates. The point of
inflection in the dotted line is less pronounced in Figure 3.4 than in the previous Figure 3.2,
and is at a younger age of 14.1 years. This inflection point represents the shell length of
144 mm, and 100% harvest above this length yields 82% of the maximum sustainable weight.
Although any gains to be made from changing the harvest length at a higher population
growth rate are less (as the horizontal lines are closer together), the steepness of both the
dotted and the dashed lines in the graphs is greater than in Figure 3.4.
Discussion
There was a noticeable impact on many of the results from examining different population
growth rates. Accordingly the effects of these different population growth rates have been
included where relevant throughout the discussion, rather than under a separate heading, as
was done in the Method and Results sections.
64
Choosing the correct equation to model the density dependence, and selecting the appropriate
class/es of the population to restrain with a density effect can have a significant impact on a
matrix population model and alter the stable stage distribution (Kimura, 1988; Bardos et al.,
2006). Bardos et al. (2006) found that the sensitivity of population stability to changes in
harvest rates was extremely responsive to the choice of density dependent equation. However
as he used a population growth rate λ = 1.98 (by my calculation), I am unsure whether this
large λ simply inflated the effects of the different density models, making the results easier
to see in his graphs, or whether the results were compounded, making them less useful.
Common methods of qualifying the density dependence in abalone populations include firstly
the Cushing equation, which models a population extremely resilient to fishing; secondly
the Ricker equation, whereby the population crashes at even moderate fishing levels; and
thirdly the Beverton Holt equation, the effect of which is intermediate between the two
(Kimura, 1988). H. iris fishing practices in New Zealand, both past and present, imply that a
moderate level of fishing is sustainable (Gibson, P. on behalf of Ngāti Konohi, 2006; Ministry
of Fisheries, 2011c). Therefore the Beverton Holt equation to model density dependent
juvenile mortality was chosen as the best option of the three to use in this analysis.
This ergodic matrix model can be specifically classified as strongly ergodic, as the age-specific
vital rates are kept constant over time (Cohen, 1979). When a population projection matrix
exhibits ergodic population growth, that population growth rate will equal lambda (λ) of the
matrix A (Caswell, 2012). I applied this theorem throughout my thesis.
This investigation of a theoretical population of H. iris suggested that shell length played an
important part in selecting harvest regulations to maximise sustainable yield. A major factor
found to influence these shell length recommendations was the population growth rate.
The sustainable harvest rates (shown in Table 3.1) increased at higher population growth rates
(PGR), with the increases in harvest rate being roughly proportional to the increase in PGR.
At all the PGRs examined the numbers of H. iris sustainability harvested was theoretically
maximised with a slot type harvest system, which left the larger adults unharvested. In
contrast the greatest sustainable biomass yield was constantly suggested from harvesting
the largest adult H. iris, with limits above the currently regulated 125 mm minimum shell
length. The harvest levels may be underestimated, as the ideal length may be slightly more
or less than recorded, as the only relevant shell harvest lengths tested were the whole numbers
between 140 and 145 mm.
These consistent percentages of 46-49% (slot system) and 63-67% (minimum length) of number of adults in the harvestable class across all the population growth rates tested might be
expected to be found in a stable fished population, assumed to have been harvested with
a consistent system for many years. However, as only four population growth rates were
explored, the possibility exists that results outside this range were not found.
65
The maximum sustainable number of H. iris harvested annually was obtained via a slot
type harvest system when using this model, with the full analysis for the different population
growth rates in the appendices in Tables A.3 - A.6 and a visual representation in Figures 3.2a,
3.3 and 3.4a. The use of a slot type harvest system has the advantage of supplying more,
smaller H. iris. The use of a slot size limit was not recommended in an eggs-per-recruit
(EPR) study of red abalone H. rufescens (Leaf et al., 2008). Adult mortality rates used in
the Leaf et al. (2008) study were much higher than in H. iris, making juvenile survival more
important, and increasing the proportion of the adult population in the smaller classes. A
stable stage distribution was calculated using the ‘popbio’ package (Stubben and Milligan,
2007) in ‘R’ for a H. rufescens matrix model from Rogers-Bennett and Leaf (2006). Assuming
I can apply these figures to the work in Leaf et al. (2008), for an unharvested population
there are 6.7% of adults in the longer than 178 mm class, compared to around 36% of adults
in the 152-203 mm slot class. The much larger proportion of H. rufescens within the slot,
compared to the percentage above 178 mm, means that at similar harvest percentages a slot
harvest would remove many more H. rufescens adults and is thus likely to be less sustainable
due to the number harvested, rather than the type of harvest system.
One disadvantage of a slot type harvest system is that while it raises the number of sustainably
harvestable adults by between 5 and 9%, it results in a loss of biomass yield. The second
and third pairs of bars in Figure 3.3 show the poor biomass yield from the best slot systems
at the various population growth rates. The loss of potential biomass yield incurred by
implementing the best slot systems ranges from 19% (at a population growth rate of 2.5%),
to 35% (at a population growth rate of 15%).
The 9% gain in numbers occurs at the lowest investigated population growth rate (2.5%),
which is also the scenario whereby the potential biomass yield loss is minimised (19%).
Therefore if 2.5% were found in a practical situation to be a realistic population growth
rate the implementation of a slot system may be realistic. In a non-commercially harvested
population the gain in numbers taken may be more desirable than the biomass loss. Because
fecundity continues to increase with age older H. iris are important, although numbers do
decline due to natural mortality. For example individual adults above 146 mm (over 17.5
years of age) produce 50% more larvae than shorter adults. Therefore any poaching of larger
adults would have a very detrimental effect on the successful implementation of a slot type
system.
I next looked at harvest regulations that maximised relative biomass yields from the two
adult classes. This analysis yielded new results compared to the number maximising system,
due to the different average body weight of H. iris in each adult class. When maximum
sustainable yield across the two classes was considered in terms of body weight, different
shell harvest lengths were recommended, with the full analysis for the different population
growth rates again in the appendices in Table A.3 - A.6 and visual representation in Figures
3.2b, 3.3 and 3.4b.
The harvest system yielding the highest sustainable biomass identified in this study involved a
larger minimum length than is currently in use in the Kaikoura region (Ministry of Fisheries,
66
2011c). For all the population growth rates studied a larger shell harvest length had an
added advantage, as it reduced the effort per kg required to harvest the catch. However the
reduction in fisher workload is only around 2-5% (dependent on population growth rate - see
column seven, Table 3.1).
Sainsbury (1982a) found yield per recruit (YPR) was increased by decreasing the minimum
shell harvest length for H. iris using a YPR model, however YPR models do not incorporate
a requirement to maintain a spawning population for replacements. Breen (1992) found that
YPR models are not sufficient tools alone for Haliotis management decisions, as they do not
consider the sustainability of the fishery. Yield per recruit models are now primarily only
used where information on recruitment is limited (Gerber et al., 2003).
Under the current system of harvesting H. iris above 125 mm the proportion of the population
within that harvestable class (column three, Table 3.1) is less at higher population growth
rates. At the higher population growth rates the composition of the population is different,
with a lower percentage of the population in the harvestable (greater than 125 mm) class.
Table 3.1 also shows the effect of changing the population growth rate on recommended
sustainable harvest levels.
Although unharvested populations with a higher population growth rate are theoretically
more sustainable, this attribute may well disappear under a maximum sustainable harvest
system. The harvest system recommended to maximise biomass yield for a population growth
rate of 5% (the dotted lines in Figure 3.2b) moved away from the recommended maximum
much more slowly than the recommendations from a population growth rate of 15% ( the
dotted lines in Figure 3.4b), which fall away either side of the maximum with a much more
rapid slope. This means that any measurement or miscalculation that results in the harvesting of H. iris from outside the recommended minimum shell length will rapidly lower the
sustainable yield, and may result in over exploitation. Estimations of optimal shell harvest
length regulations are thus more important in H. iris populations with higher natural population growth rates, as is accuracy in shell measurement. The steeper plots in Figure 3.4
mean that a small deviation from the recommended harvest length regulations will have a
comparatively larger impact on the sustainable harvest, compared to the effects of a similar
deviation at lower population growth rates.
In common with the slot type systems, the greatest gains with a change to an alternative
minimum shell length system (from the current 125 mm) from this theoretical homogeneous
population are gained if the population growth rate is small. The 4% gain in biomass occurs
at the lowest investigated population growth rate (2.5%), which is also the scenario whereby
the potential numbers loss is minimised (1%). This is also the population growth rate where
the greatest change in minimum harvest length is needed (125 mm to 137 mm) to maximise
sustainable yield. This suggests that if 2.5% were found in a practical situation to be a
realistic population growth rate longer minimum shell lengths should be investigated further.
This deterministic matrix population model was also able to predict the relative sustainable
yields if restrictions on either shell harvest length or harvest rate were not enforceable. In
the first instance, if the desire was to pass a length restriction so that all H. iris above that
shell length were sustainably harvestable, then the shortest harvestable shell length would
be 146.05 mm, and would result in 3.0% of the population being taken annually, assuming a
67
population growth rate of 5%. This limits yield to 70% of the maximum sustainable number
and 83% of the maximum biomass. This result is shown in the dotted line in Figure 3.2b.
as the point of inflection at 22.8 years (equivalent to a harvest length above 146.05 mm).
Relative harvest declines steeply after this as the harvest becomes further removed from
the ideal to maximise harvest. The decrease in the size of the harvest class (H. iris older
than 22.8 years) cannot be compensated for by an increase in the recommended harvest rate
beyond 100%.
A slot system could not allow 100% harvest, as there would be no remaining adults to mature
into the larger classes. Unfortunately a slot smaller than 100-121 mm was not modelled,
however the loss of sustainable harvest under the 100-121 mm model (see Figure 3.3) hints
at a poor result from narrow slots targeting the smallest adults. The alternative option of
limiting the number in the harvest class (via much longer minimum harvest lengths) returns
sustainable yields that are much higher.
Similar results can be seen for a population growth rate of 15%. However the point of
inflection in the dotted line in Figure 3.4b. is at a younger age of 14.1 years, showing that all
the H. iris above 144 mm can be sustainably harvested. The effect of changing the population
growth rate from 5% to 15% in the matrix is large, lowering the length where 100% harvest
is sustainable from 146.05 mm to 144 mm.
Alternatively, if length restrictions were difficult to enforce so that all adults (above 100 mm)
were harvested, then the sustainable yield was 85-88% of the maximum number, or 8183% of the maximum biomass achievable. Further, this could only be sustained at very
low harvest rates that removed between 1.2 and 5.5% of all adults per annum. This would
entail well implemented and strict harvest regulations to limit harvest rates to this low level.
Both the drop in yield and the enforcement requirements mean that shell length restrictions
have been seldom if ever abandoned in favour of harvest systems that only limit the total
number of adults taken in modern Haliotis fisheries systems (Hahn, 1989; Cook and Gordon,
2010; Morales-Bojorquez et al., 2008; Johnson, 2004; Ministry of Fisheries, 2011c; Chick and
Mayfield, 2012).
The effect of sustainably fishing on the numbers of H. iris remaining in the population, compared to an unharvested population, was also examined (the fishdown level). Both the current
and recommended harvest regulations did not result in a fishdown level less than 0.340 in the
harvestable class. In New Zealand fisheries management soft and hard limits are considered
as important management tools (Ministry of Fisheries, 2011c), although measurements of
changes to population size and demographics in commercially harvested Haliotis populations
are difficult to obtain in the field (Chick and Mayfield, 2012; Ministry of Fisheries, 2011c).
Many Australian abalone fisheries use similar systems, incorporating lower limit or target
reference points, although they are not based on a percentage of unharvested biomass (Chick
and Mayfield, 2012; Hesp et al., 2008) (as is done in New Zealand), thereby making the
Australian regulatory measurements difficult to compare to what happens in New Zealand’s
H. iris commercial harvest areas.
68
The slot type system decreased the catchability compared to the harvest of the largest sized
H. iris. This is because a lower percentage of this theoretical adult population was in the
slot type harvestable classes. The lowering of the recommended shell harvest lengths (in
both columns eight and twelve) as population growth rate increases seems tied to this, as the
catchability figures (columns 10 and 14) remain relatively constant at different population
growth rates. Thus, the slot type harvest systems consistently recommend harvesting within
the smallest 46-49% of the Kaikoura adult population, whilst the harvest systems to maximise biomass yield target the largest 63-67% of the population (with a smaller harvest rate),
despite varying the annual population growth rate from 2.5% to 15% in this study. H. iris
at Stewart Island in zone PAU5B (Ministry of Fisheries, 2011c) have similar body weights to
these H. iris analysed at Kaikoura, with the current figures of biomass harvestable: biomass
spawning ratios of 1120:1487 and 1120:1528 published by Ministry of Fisheries (2011c) (assuming ”biomass harvestable” is the biomass of adults in the harvestable class and ”biomass
spawning” estimates the total adult weight). I used the similarities between H. iris around
Kaikoura to those at Stewart Island (Ministry of Fisheries, 2011c) to convert these figures to
catchabilities (count based) of 60.7% and 58.2%, using a calculation based on average body
weight. These results are lower than my estimations of 63-67%. Current harvest practices
in PAU5B of increasing the minimum harvest length are predicted to decrease the biomass
harvestable: biomass spawning ratio (Ministry of Fisheries, 2011c), which will tend to make
that field data from PAU5B further from my estimations of an ideal system. This finding
again raises the possibility that recent increases in the minimum harvest length at Stewart
Island to 135 mm (Ministry of Fisheries, 2011c) may be too long to maximise biomass yield.
Robustness
Refining the model
While it was possible to reparameterise the matrix A to reflect different points of division
between the harvested and unharvested adult classes, an examination of the lambda measurement for each of these matrices showed a small level of inconsistency. This implied the
population growth rate changed as I shifted the division between small (Y) and large (M)
adults, which is not biologically realistic. These small changes in lambda could affect any
ideal harvest length regulations gleaned from the model, and so I negated them by artificially
changing the egg settlement rate SE . By raising the SE associated with lower lambda values,
and vis versa, the changes in SE required to maintain a level population growth rate were
less than 5% (columns three, five, seven and nine of Table A.2). The small amount of change
required to cause this alteration points to the stability and fit of the matrix models. The
fact that lambda was correct to four decimal places after implementing SEi implies that no
unfair advantage was accorded any specific shell length harvest regulation. These two points
further strengthen the robustness of this length-based H. iris population model.
69
Summary
Here I examined the demographic changes that could occur due to changes in shell harvest
length regulations in a theoretical midrange (maximum length 146.2 mm) population of
Haliotis iris. This analysis was used to identify the regulations that theoretically maximise
the annual sustainable yield of H. iris both in terms of numbers harvested, and annual
biomass yield. As the population growth rate of a healthy population of any Haliotis species
could not be identified, I conducted this investigation over a range of values. I also examined
some of the effects of any regulation changes on the work of the harvesters.
Catchability (what percentage of the population is in the harvested class) remained relatively
constant across the different population growth rates, but was quite different for the different harvest systems, whereas the dropdown (population drop due to fishing) varied across
different population growth rates, but was consistent at the same population growth rate for
the two different harvest systems. The recommended percentage of the population within
the harvested class (assuming tight adherence to harvest rates) was not affected by lack of
knowledge about the population growth rate.
For a slot system, to maximise the number that can be taken this model recommends 46-49%
of the smallest adults should be in the harvestable class, whereas for a weight maximisation
model the recommendation is that the harvestable class should encompass the largest 63-65%
of adults. However the actual percentage of adults within these classes that can be taken
annually (the harvest rate) varies widely, depending on both the chosen harvest system, and
the population growth rate.
Longer minimum shell lengths than the current 125 mm were recommended to sustainably
maximise biomass yield from this population, for all population growth rates between 2.5%
and 15%. Slot systems targeting only the smallest H. iris in their first one to two years
of emergence lowered yield to 10-36% of that possible under other systems, and was not
recommended.
70
Chapter 4
General Discussion, conclusions and
recommendations
The aims of my thesis were to design and use an appropriate population model for Haliotis iris
to explore harvest length regulations. My first task was to clarify and possibly expand upon
the current knowledge about matrix population modelling, and specifically models of Haliotis
populations. Secondly, I calculated which harvest systems for a specific H. iris population
returned the maximum sustainable yield per annum, and then looked at the broader picture
around that harvest information.
General discussion
The first part of this Chapter draws together the background material in Chapter 1 and
the analysis in Chapters 2 and 3 and looks at how these increase the base of knowledge
already known about this species, both biologically and in relation to fishing practices. Due
to the large variations that exist between H. iris populations (see Chapter 1) the analysis is
based on H. iris measured at Kaikoura over two years (1967-69) by Poore (1972a,b,c, 1973).
According to a classification system for H. iris designed by Naylor et al. (2006) using von
Bertalanffy growth parameters, Poore’s data was from a midrange population. However I
am not suggesting that any specific results of the matrix analysis made in Chapter 2, or
shell harvest length recommendations made in Chapter 3, can be extended to any H. iris
populations without further analysis.
Relevance of the matrix analysis
Density dependence
If juvenile density is limiting population growth then the protection (or possibly provision) of
cryptic habitat sites for juveniles should increase the population size. Links between cryptic
formations and population dynamics have been found (Aguirre and McNaught, 2012). This
71
is supported by the work of Nash et al. (1995) who found good settlement rates on artificial
collectors.
Eigen analysis
The ratios of small and large adults I calculated for this theoretical H. iris fished population
in Chapter 3 were close to the ratios calculated for a fished population of H. iris measured at
Stewart Island (Ministry of Fisheries, 2011c), however the unfished stable stage distribution I
generated in Chapter 2 did not agree with population counts at Kaikoura in 1967-68 (Poore,
1972c). If my results are correct then the mismatch at Kaikoura may be due to a loss of
larger H. iris from the Kaikoura population prior to the analysis in 1967. Alternatively if
my calculations are incorrect there is possibly a larger difference in population parameters
between H. iris at Kaikoura in 1967-68, and those at Stewart Island in 2011 than I have
allowed for.
Because of the composition of the matrix with the census conducted immediately post harvest,
the number available to harvest in the following year would be affected by both intervening
natural mortality and the growth of individual H. iris into and out of the class being harvested. If the parameters of the population did not change, and the population growth rate
was zero, then mortality (natural loss from a class) would equal growth into the class. However as the population growth rate is positive and the stable stage distribution is constant
(as explained in section 3), net growth into each of the adult classes would occur. As this
growth was assumed to occur throughout the year, the numbers alive in each class at the
beginning of the harvest period would be more than at census time, and so harvest levels may
be underestimated. Doubleday (1975) found that the timing of harvest, spawning and seasonal variation in vital rates affected the yield in matrix population modelling. However due
to the consistent stable stage distribution, although the total number of harvestable H. iris
may change, any advantage in a comparative analysis will be unchanged, and it follows that
the harvest length recommendations will be unaffected.
Elasticity
Elasticity analysis was completed on several alternative matrices with the divisions between
the small and large adult classes changed in each matrix to reflect the shell lengths that led
to the largest maximum sustainable yields. These results are discussed below.
The highlighted (or grey) entries in Table 2.4 are of the classes of H. iris that were harvested
under different scenarios, assuming a population growth rate of 5%. The H. iris adults in the
slot harvest of (100-143 mm) had the lowest elasticity readings of the three harvest options
with e(SY 143 ) < e(SM 135 ) < e(SM 125 ) and e(FY 143 ) < e(FM 135 ) < e(FM 125 ). This trend is
consistent at the different population growth rates (Figure 2.4) where e(FY ) < e(FM ) and
e(SY ) < e(SM ). This means that changes in the numbers of harvested small adults, (in this
case below 143 mm) under a slot harvest system were predicted to have a lesser effect on the
current population growth rate compared to the other two harvest systems. A slot harvest
was thus a likely candidate as a harvest system for maximising sustainable yield, when yield
is measured in numbers rather than weight.
A comparison at the PGR of 5% of the elasticities of the matrix divided where L = 125 mm
72
(the current harvest system) versus the matrix divided where L = 135 mm (the harvest
system maximising biomass yield) shows support for the advantages that can be gained by
harvesting under this alternative system. e(FM 135 ) < e(FM 125 ) and e(SM 135 ) < e(SM 125 ),
implying that harvesting above 135 mm as opposed to harvesting above 125 mm will decrease
the impact of the harvest on the current population growth rate λ, making it a better prospect
to have a minimal impact on this population.
As the elasticities relating to the juvenile parameters were small this implied that the harvest
regulations suggested here are more robust to changes in juvenile survival and growth, and
that this robustness does not change if the harvest length regulations are altered. However
events such as irregular spawnings and sedimentation can have large effects on juvenile numbers (Phillips and Shima, 2006), perhaps comparable to the effect of harvesting on adult
numbers. This means that these events may have a degree of importance to the maintenance
of the population simply because they have such a large effect on juvenile numbers.
In general the elasticities were supported by the biological life history of the H. iris, reflecting
an animal with low egg settlement and juvenile survival, high adult survival, a variable
adolescence, indeterminate and irregular fecundity that increases throughout adulthood, and
a slow growth rate.
Robustness
Matrix population models have been described as data hungry models, and the usefulness of
any model is restricted by the accuracy of the data, the appropriateness of the data, and the
level of confidence that exists in its outputs. The suitability of the model can be estimated
in part from how realistic the outcomes of the simulation are. One indication of this is that a
fishing rate of 100% is sustainable, but only above long minimum shell harvest lengths, which
did not achieve maximum yields (tables A.3-A.6). Comparisons with other matrix models
for Haliotis species is difficult as changes in the number and length of classes has large effects
on the sensitivities and elasticities of the analysis (Carslake et al., 2009).
Maximising harvest systems
The slot shell length harvest
One aim of this study was to discover if a change in the harvest rules to a slot type size
limit (for example minimum shell harvest length 100 mm, maximum 135 mm), which left
the larger more fecundant females (in this case those above 135 mm) to reproduce, would
be advantageous. Slot systems were traditionally set under tikanga (rules) traditionally
implemented by some indigenous Māori kaumātua (senior people in the kin group) (Gibson, P.
on behalf of Ngāti Konohi, 2006). I investigated if any of a range of slot harvest systems
could lead to a long term increase in the harvest of H. iris, measured as individuals, and/or as
harvestable biomass. Over the wide range of population growth rates examined a slot system
was found to consistently maximise the sustainable yield from this model, as regards the
number of H. iris harvested annually. That being said, a slot size limit did not significantly
increase eggs-per-recruit (EPR) yield in a red abalone study (Leaf et al. 2008), although I
was unable to ascertain if class numbers were considered in that analysis.
73
Additionally, implementation of a slot size regulation for H. iris may help undo humaninduced genetic changes, if removal of the larger, faster growing animals has genetically
changed the species in favour of slow growing, early maturing phenotypes. Fishery-induced
genetic changes have been reported to reduce yield in some finfish species (Biro and Post,
2008; Enberg et al., 2012), in a process known as fisheries-induced evolution. Its effect
on abalone species is unknown, as there appears to be little research on fisheries-induced
evolution in aquatic species other than finfish (Jorgensen et al., 2007; Enberg et al., 2012).
However fisheries-induced evolution has been found to cause genetic change in characters
with high levels of phenotypic plasticity (Perez-Rodriguez et al., 2012), such as growth in
H. iris.
If the H. iris phenotype altered in the future so that individuals spend on average more time
or reproductive effort outside the harvested class, this would require a recalculation of the
population matrix, due to changes in the growth and fecundity rates. The corresponding
ideal harvest system would also change, as it is dependent on the matrix parameterisation.
Fisheries-induced genetic changes in H. iris populations that minimise time in the harvested
cohort, and human induced regulatory changes to maximise sustainable harvest, could evolve
into a ‘Red Queen’ type race between evolutionary changes in H. iris, and catch up harvest
regulatory changes (Kerfoot and Weider, 2004). This seems more likely than the suggested
evolutionary tug-of-war between natural and harvest selection seen in pike (Edeline et al.,
2007), due to the long life cycle and dynamic harvest regulations of H. iris. However a
reversal in trends does not happen in all species (Allendorf et al., 2008), and it may not be
possible to induce change towards a pre-harvest phenotype (assuming one is needed), with
the use of a slot size regulation in H. iris harvests. Additionally, due to the long life of H. iris
any foreseeable harvest system should be little affected by genetic change, and as the longterm effects of fisheries-induced evolution on population growth rates have been questioned
(Kuparinen and Hutchings, 2012), I feel that maximising harvest returns is a more immediate
consideration than reversing the effects of undiagnosed fisheries-induced evolution.
The maximum sustainable number of H. iris harvested annually was obtained via a slot
type harvest system when using this model. The use of a slot type harvest system has
the advantage of supplying more, smaller H. iris. With the current regulations specifying a
maximum recreational harvest of ten H. iris per person per day this means that an increase in
the sustainable numbers that can be harvested could satisfy a larger number of recreational
fishers. However, maintaining the long-lived, larger and more fecund H. iris outside the
slot in the population would be a trial for fisheries officers combating poaching.One possible
solution to this is suggested from an examination of New Zealand culture, which includes the
national adoption of many living taonga (treasures) such as the kiwi and kereru (Channel
Three News. March 26th, 2010). If larger breeding H. iris could be afforded a similar taonga
status then poaching of these more fecund adults left under a slot type harvest system would
be less likely to occur, although such a mindset change may take some time to implement.
A slot size limit was traditionally used by some Māori tribes, has been examined with overseas
Haliotis species (Leaf et al., 2008), and is currently being trialed in the Marlborough Sounds
area with blue cod, from 1st April 2011 (Ministry of Primary Industries, 2012b, 2013c), so
the suggestion of its use in New Zealand’s H. iris management is not without precedent.
I anticipated numerical support for a slot type system may be possible as the high fecundity
levels in larger animals results in higher reproductive success, making them more important
(in the short term) to maintaining population growth. Removal of the same number of small
74
adults results in the removal of less biomass, and as biomass is proportional to fecundity,
a smaller proportion of potential offspring are lost, therefore a higher harvest rate can be
sustainable.
Sustainable semi-regulated systems
Semi-regulated systems are a possible solution to a lack of enforceable restrictions on either
the number of H. iris harvested or the shell length at which they are harvested. At all
population growth rates above 5% a maximum harvest length regulation existed that allowed
100% of adults longer than the specified shell length to be harvested. Therefore setting a long
shell length is one method of restraining harvest without setting harvest rate limits, although
it does limit yield to around 80% of that achievable in a more regulated harvest system.
The narrow slot harvest of small adults from 100-121 mm gave a very low sustainable harvest, although harvest rates were higher than in wider slots. This low yield was somewhat
unexpected, as this harvest would not remove any of the older very fecund adults after they
were through the slot. Examination of the elasticities in equation 2.43 shows (at a division
of 100-125 mm) small adult stasis is the second most important element, after large adult
survival. Because juvenile survival is so low compared to the adult survival, all adults are
important to maintaining the population in this model. This is supported by the equalities
in the importance of fecundity of both small and large adults in equations 2.42.
At all the population growth rates examined maximum yield could not be achieved with a
harvest regulation that allowed 100% harvest of the selected cohort. The loss of yield when
compared to the best harvest system (at all population growth rates) dropped yield to 8182% of the maximum biomass. Interestingly, on examination of Tables A.3-A.6, incorporating
population growth rates from 2.5% to 15%, none of the minimum shell length harvest systems
reduced yield below 80% of the maximum biomass yield, except those where the recommended
harvest rate is 100%. The smallness of this drop away from the maximum achievable bioweight
yield even at extreme misfits in the model may account for less interest in determining ideal
harvest length, as there have been few investigations in this area of harvest management
(Johnson, 2004; Ministry of Fisheries, 2011c; Chick and Mayfield, 2012).
An alternative method of restraining harvest was restricting the annual harvest to taking a
small number of any length from within the entire adult population. This would be difficult
to enforce by means of either a closed season, or the use of reserves. At a population growth
rate of 5% the number taken is only 2% of the whole adult population. While the figures are
slightly more generous at a population growth rate of 15%, the harvest rate of 5.5% would still
be difficult to practically enforce, and limits yield to around 83% of the best achievable. If
there were difficulties in enforcing any initial minimum or maximum shell length regulations,
then this alternative seems little better, as limiting catch to a small percentage would also
be difficult to enforce practically.
It has been estimated that around 26% of the total H. iris taken annually is illegally harvested
(Haas, 2009), and many of these are undersized (Ministry of Primary Industries, 2012a,d;
Ministry of Fisheries, 2011c). Because large numbers of small H. iris are taken I suspect that
the total harvest take, at least close to larger towns and cities, is probably following the above
model, where H. iris of any length are harvested. This would mean that the total catch is
limited to around 83% of the maximum achievable. If this is so, it will affect all harvesters.
Therefore, not only are poachers removing 26% of the harvest, their poor size selections are
75
possibly decreasing the legal harvest (at least in some areas) to around (83 − (83 × 26%)) =
60% of what it could be without poaching.
Biomass maximisation
A change in the minimum shell length harvest regulation could lead to a sustainable increase
in the biomass of H. iris harvested annually. Using a range of population growth rates from
2.5% to 15% per year in this model I found in all instances that an increase from the current
125 mm minimum shell length was needed to maximise the sustainable biomass yield.
Maximum sustainable weight is an important component of any H. iris population analysis.
The financial returns and quota allowances from commercially harvested H. iris are based
on biomass (Statistics New Zealand, 2010; Ministry of Primary Industries, 2013b), and data
from analyses of H. iris commercial fisheries has also been calculated in biomass (Ministry
of Fisheries, 2011c). Restrictions on shell length have been an important regulatory tool
used in abalone fisheries for many years, both in New Zealand (Johnson, 2004; Ministry of
Fisheries, 2011c), and overseas (Rogers-Bennett and Leaf, 2006; Chick and Mayfield, 2012).
Steps to increase the minimum shell harvest length are being taken in many commercial
H. iris catchments throughout New Zealand (table 2.1), which reflect the findings of this
model. However, at a population growth rate of 15% this model recommends a minimum
harvest length of only 129 mm, which has already been exceeded by voluntary increases
in harvest length in some commercial catchment areas (Ministry of Fisheries, 2011c; Paua
Industry Council Ltd, 2012).
Financial and fisher considerations
Annual counts from the field of changes in population size and demographics provide an
important check on the health of the fishery, and are often used as an aid to setting annual
harvest levels overseas (Chick and Mayfield, 2012). However the level of this type of assessment varies across New Zealand’s H. iris commercial catchment zones (Ministry of Fisheries,
2011c).
A reduction in fisher workload is achieved by increasing the minimum harvest shell length
from 125 mm to the lengths recommended by this model. The longer H. iris will enable fishers
to full the biomass quota faster by harvesting fewer adults, with a reduction in workload of
around 2-5% (column seven, Table 3.1) depending on the population growth rate. As the
annual commercial harvest is over 700 tonnes per annum (Statistics New Zealand, 2010), the
total reduction in the time that fishers take may be important. An average haul for one
vessel of 300 kg per diver per day, over 500 diver days, (Fu, 2010) means an annual saving
of between 10 and 25 diver days per boat for the same gross return. This assumes that the
300 kg per diver per day can be increased by harvesting larger H. iris, but is not decreased
by the time taken to identify these larger more scarce adults. The probably exists that the
decreases in catchability shown in Table 3.1 could adversely affect the achievable kg per diver
per day.
A further point to consider in determining the possible implications of this suggestion is
how financial returns to the fishers affect these recommendations. Reed and Clarke (1990)
examined harvest decisions and asset valuations, and found that the optimal harvesting size
76
did not depend on price. This implies that if an optimal harvest size does exist, then this
may be price independent, and although some smaller abalone are worth more per kilogram
overseas, the commercial wild harvested H. iris are not suitable for this market (JLJ group,
2010).
Different population growth rates
Unfortunately Haliotis species are one of many groups where population growth rates are
unknown, and difficult to calculate. A fuller discussion of this difficulty is included under
the subheading ’Egg numbers’, in section 2. Following on from this, the opportunity was
taken to explore a wide range of possible population growth rates (PGR) for the theoretical
H. iris population analysed in this thesis. PGR from 1% to 16% were trialed to find their
effect on the elasticity measurements in Chapter 2, and four PGR between 2.5% and 15%
were examined in relation to the suggested harvest regulations in Chapter 3.
The one main effect of the changes in PGR was that large adult survival remained the most
important, but its elasticity decreased with an increase in the PGR, and at the same time
all the other population parameters increased their elasticities. This is possibly due to the
changing structure of the population. A further interesting point in the elasticity analysis of
the different PGR is that parameters with a high temporal variability will often have a low
elasticity (Pfister, 1998). Therefore, a comparative analysis of the left and right hand sides
of the graphs in Figure 2.4 could possibly be used to shed some light on which PGR is most
feasible biologically. Population parameters with high temporal variability should have a low
elasticity, and vice versa, at the point in the graphs were the PGR is accurate. However, as
the proportions of the population in the classes changes as PGR increases, I considered the
possibility that the effect of numbers in each class would swamp any comparative influence of
temporal variability, and therefore this type of graphical analysis lent no insight into which
PGR was most biologically feasible.
The sustainable harvest rates in Table 3.1 increased at higher PGR, with the increases in
harvest rate being consistent with the increase in PGR. At higher PGR maximum sustainable
yields were indicated at a shorter shell harvest length regulation, perhaps due to the changing
demography of the population. At the lowest examined population growth rate of 2.5% none
of the investigated harvest length regulations supported a 100% harvest rate, probably due to
a requirement to maintain in the population the largest H. iris with their very high fecundity.
There was a large change in the length at which 100% harvest becomes sustainable as the
PGR changed, which was influenced by two factors. Firstly, the structure of the population is
different at different population growth rates. With a higher population growth rate there are
more younger (and thus smaller) H. iris, so the average shell length is less. This is reflected
in the drop in elasticity of survival of the largest adults as they become less important to
maintaining the population as their relative numbers decrease. Secondly, at higher population
growth rates less of the population is needed to sustain population growth, so 100% harvest
rates become sustainable at a younger age.
77
Conclusions
Improving the analysis of harvested Haliotis species is an ongoing process, with the sometimes
contradictory aims of both maximising short term productivity and protecting the species
from over-harvesting. The purpose of this study was threefold: firstly, to parametrise and
analyse a matrix for the species Haliotis iris using a specific theoretical population; secondly,
to use that matrix to predict if a change in the harvest regulations could increase the sustainable yield; and thirdly, to look at some of the effects of any suggested changes in the harvest
regulations, on both the harvested H. iris population, and on the people who harvest them.
The parameters used in the formation of the Lefkovitch length-based matrix model were
based primarily on an analysis of H. iris at Kaikoura in 1967-69 by Poore (1972a,b,c, 1973).
Population growth rates were chosen based on both scientific research and Ministry of Primary
Industries published data, with the egg settlement rates calculated to ensure the desired
population growth rates were generated. This variability in population growth rates resulted
in a wide range of results, however several interesting trends remained constant.
The matrix analysis in Chapter 2 provided some interesting insights into both H. iris populations, and matrix analysis in general. My calculation of a population growth rate of 16%
for H. iris at Stewart Island, based on Ministry of Primary Industry data, provides a starting
point for the analysis of healthy Haliotis populations. The most important elasticity measures calculated related to survival, with large adult survival the most important parameter
in population growth. I found that the choice to change either egg settlement or juvenile
survival had little effect on the elasticity readings, and that population growth rate was the
most important parameter in this matrix analysis.
I also found that the matrix analysis was robust as distribution error in the matrix model
was largely removed. The construction of a usable matrix with only three classes was made
possible by more accurate calculation of average class fecundities. These fecundities were
calculated using the integration of a spine function based on the length dependent fecundity
power function. This generated nearly identical population growth rates from the different
matrix formations. This increase in accuracy allowed me to work with a manageable number
of biologically relevant matrix elasticities which were then separated from elasticity measures
influenced by the matrix construction. And finally, the new terms of promotion and relegation
were introduced to better describe movement between the matrix classes.
The current harvest regulation, taking H. iris with minimum shell lengths of 125 mm, was
found to be consistently too short for this theoretical population with a maximum shell
length of 146.2 mm, despite varying population growth rates (PGR) between 2.5% and 15%.
I suggest that a longer minimum length could increase the sustainable biomass yield for this
population by 4-8%. As maximum shell lengths may vary from 79 to 163 mm (Naylor et al.,
2006) throughout New Zealand, I assume similar gains could be possible in several other populations. Assuming 4-8% is an average reduction in yield (as this is a midrange population)
due to non optimising harvest systems, an increase of this level in exports worth $55 million
annually will earn another $2.7 − 4.4 million per year. The current 125 mm regulation also
decreased the average body weight of individual harvested H. iris and thereby increased the
work per kilogram harvested by between two and five per cent. At a PGR of 2.5% the recommended minimum shell harvest length was 137 mm, although at the higher PGR of 15% the
minimum harvest length recommendation was only 129 mm. In the commercial catchment
PAU5B at Stewart Island the minimum length has recently been increased to 135 mm, and
78
although my model was not designed for H. iris in PAU5B, these results suggested that this
length may be too long to either maximise the yield, or best increase the biomass.
Large increases in the fecundity of H. iris at longer shell lengths were measured by Poore
(1973), and this has resulted in a slot type harvest system returning the highest sustainable
yield (numbers of H. iris) in this simulation. A slot type system harvests from the small adult
H. iris cohort and leaves the larger surviving adults to freely reproduce. Again, the specific
harvest shell length recommendation depends on the chosen PGR: ideal slot harvest lengths
suggested varied from 100-141 mm through to 100-144 mm. These were found to increase
the number of H. iris available to harvest over the current system of harvesting H. iris with
shell lengths longer than 125 mm, although the increase in numbers harvested was small,
in the order of one to four per cent. However adopting the best slot harvest system lowers
the biomass yield to 75-80% of the maximum achievable. This means that a slot harvest
should be difficult to recommend in a commercially harvested fishery, due to their emphasis
on biomass yield.
A level of harvest, or specific harvest rate, whilst proposed by this analysis (and shown in
columns two and three of Tables A.3 through A.6), is not considered important in this final
analysis. Firstly, this is because harvest rate varied proportionally to the selected population growth rate, and was possibly influenced by the chosen census time. Also, maximum
sustainable yield is now used more often as only one step towards setting an upper limit
in the tiered management approaches (Pikitich, 2012) used in many Haliotis assessments
(Gorfine et al., 2001; State of California. Dept of Fish and Game., 2010; Mayfield et al.,
2011). Therefore harvest rate recommendations are not made here, and I suggest they are
perhaps better calculated by other methods, more responsive to the large levels of temporal
and spatial variability within H. iris populations.
Whilst the effect of population growth rate on suggested shell harvest length regulations was
large (table 3.1) it may be possible to base shell harvest length regulations on a more consistent population parameter, namely the harvest class:spawning class ratio, which remained
relatively consistent across population growth rates ranging from 2.5% per annum to 15%
per annum.
The drop in biomass from an unharvested population when harvested via the suggested best
two harvest systems, (one maximising number and the other maximising bioweight) is comparable at the same population growth rate, shown in columns 11 and 14 of Table 3.1. Therefore
the biomass necessary to maintain the population is consistent across the two harvest systems
and bears little relationship to the number of individuals harvested. This is possibly linked
to the relationship between egg production and body weight which is proportional, whereas
the relationship between egg production and the total number of adults is influenced by the
composition of the adult population.
Limitations
Topics including implementation times, non-commercial harvests, different size slots, the
Allee effect, social and environmental factors and data variability are also briefly examined
as factors possibly affecting any recommendations based on this model.
79
Implementation times
Assuming current fish stocks do not reflect the matrix generated stable stage distribution,
and assuming all parameters are exact, there is still the problem of what happens to the
population and the sustainable yield, as it adjusts toward that stable stage distribution.
This is called transient dynamics, that is, what happens to the population due to a change
in harvest regulations during the transition phase (Ezard et al., 2010). The length of the
stabilisation process can vary (Rogers-Bennett and Leaf, 2006), and is influenced by both the
initial population structure (Buhnerkempe et al., 2011), and the chosen density dependent
effect (Bardos et al., 2006). If regular changes to harvest regulations are occurring, then
transient effects may be more important than long term results.
Cultural, recreational and poaching harvest
A rate to reflect the level of cultural, recreational and/or poaching harvest could be incorporated into the matrix in a similar way to the commercial harvest terms, and could be
applied to any combination of the three length-based classes, possibly giving a more accurate
analysis of the effect on yield than is currently contained in section 4 in relation to poaching.
In Chapters 2 and 3 of this thesis the cultural, recreational and poaching harvest were not
included as the levels are difficult to quantify and vary within and between different quota
management areas around New Zealand (Ministry of Fisheries, 2011c). This means that the
recommended harvest rates should be considered as representing TAC (total allowable catch)
rather than TACC (total allowable commercial catch).
120 mm to 130 mm slot
I recently became aware of a push to investigate a specific slot measuring 120 to 130 mm.
This size best allows two whole H. iris per can, and could result in a 20% price premium on
the international market (Pickering, 2012). Unfortunately I was unable at this late stage to
incorporate this into the model.
Allee effect
This study assumed that the average rather than the absolute number of adults H. iris present
in the population at spawning was the only factor affecting egg settlement rate. As H. iris
are broadcast spawners requiring aggregation for successful fertilisation, the absolute number
of fertile adults may have an effect on the egg settlement (Lundquist and Botsford, 2011)
via the Allee effect. If so, harvesting a smaller number of large H. iris (via a minimum shell
harvest length) versus a larger number of smaller animals (harvested in a slot type system)
may have compounding effects on the egg settlement that are not considered here. Strong
Allee effects have been observed in different Haliotis species (Shepherd and Brown, 1993),
but the less detectable weak Allee effect has been more difficult to quantify (Lundquist and
Botsford, 2011). Although patches of Haliotis have returned to pre-harvest aggregation levels
within 10 weeks of harvest (Officer et al., 2001), and higher levels of aggregation in H. iris
are associated with spawning (Hepburn, pers. comm. 2012), a weak Allee effect cannot be
discounted as affecting the final analysis.
80
Social and environmental factors that can influence harvest levels
Changes in human attitudes to both legal and illegal harvesting will have profound effects on
the stability of H. iris populations. In South Africa a large Haliotis commercial harvest was
destroyed after changing attitudes increased poaching (Edwards and Plagányi, 2008; Plagányi
et al., 2011), and in New Zealand both recreational and illegal harvest levels of H. iris are
consistently increasing (Johnson, 2004; Ministry of Fisheries, 2011c). One effect of this is
to increase the difficulty in making accurate population predictions, as the shell length and
number taken in non commercial harvests are seldom recorded (Ministry of Fisheries, 2011c).
Increased illegal harvesting will have the added disadvantages of reducing the legal harvest
as it removes both legal and illegally sized H. iris (Powley, 2003), and thereby possibly sets
up suboptimal population ratios. This is because poachers are unconcerned with the long
term maximisation of the harvest.
There are many human and environmental threats to Haliotis populations (Neuman et al.,
2010). Climate change is one that could have major effects in New Zealand (Mullan et al.,
2008). Coastal aquatic environments are undergoing, or predicted to undergo, several changes
due to increases in global temperatures. These changes could include rising seawater temperatures, increased carbon dioxide levels that will reduce seawater pH, and increased storms
causing more runoff and shallow sedimentation (Mullan et al., 2008; United States Environmental Protection Agency, 2013). These changes are predicted to have a largely detrimental
effect on many colder climate Haliotis populations (Vilchis et al., 2005; Phillips and Shima,
2006; Kroeker et al., 2010; Mayfield et al., 2012; Won et al., 2012).
A single-species population model will miss key biotic factors from species interactions that
can affect stage-structured population models (Fujiwara et al., 2011). One specific interaction
affecting H. iris is the interaction with New Zealand kina, the endemic species of sea urchin
Evechinus chloroticus. Interspecies interactions with E. chloroticus may decrease H. iris
populations (Naylor and Gerring, 2001; Aguirre and McNaught, 2011) due to the creation
of barrens (Estes et al., 2005). An increase in barrens has been observed in Tasmania,
where it was linked with warmer sea currents and increasing numbers of the sea urchin
Centrostephanus rodgersii (Johnson et al., 2011).
Data variability
Biological parameters (xi ) are often written with both an average value (µi ) and a calculated
estimation of the spread of data about that average value (σi ). All the biological parameters
can then be expressed as xi = (µi , σi ). Inclusion of these values into mathematical calculations
enables a measure of the probability (or likelihood) that a result is significant. However in
all instances I have simply used xi = µi , with σi = 0.
One reason for using xi = µi is that the matrix analysis is based on the equation Nt+1 = ANt
for all t, and when the population is at equilibrium Nt+1 = Nt . But if there is temporal
variability in the parameters (xi ) that make up the matrix A, then this equilibrium will not
be achieved.
Some limitations of the biological data this theoretical population was based on were discussed
in section 2, and alternative egg settlement, juvenile survival, and population growth rates
were explored in Chapter 3. However as well as the effects of this data variability on the
accuracy of the parametrisation, variability can also effect the usefulness of the analysis.
81
Highly variable survival, recruitment and growth patterns can significantly reduce the response of exploited populations below expectations (O’Neill et al., 1981; Troynikov and
Gorfine, 1998) and several complex problems have been found by researchers when incorporating variability into population models. Firstly, temporal variability can lead to lower
population growth rates (Tuljapurkar and Orzack, 1980); secondly, stochastic recruitment can
lead to lower estimations of biomass in an unfitted population, this worsens as stochasticity
increases (Cordue, 2001); Breen (1992) found that large annual variations in egg settlement
rates may require a more cautious shell length limit; and thirdly, the effects of temporal
and spatial variation on demographic variability may be quite different (Salguero-Gomez and
de Kroon, 2010).
In reality some abalone in a population will never reach the average maximum shell length
(L∞ ), and the closer the harvest length is to L∞ the higher the percentage of H. iris within
this non-harvestable group will be, thereby unnecessarily decreasing the catch rate and perhaps becoming overly conservative. These unharvestable smaller H. iris could become an
important perennial source of recruits (Troynikov and Gorfine, 1998).
The advantage of including variability in the parameter estimations is that these identified
problems could be considered. Therefore any specific recommendations from this analysis are
not immediately suitable for direct application to a specific population.
Recommendations (areas of future work)
This thesis was designed to improve population modelling in relation to H. iris, with the
aims of both finding out more about the species, and learning more about alternative harvest
systems that may increase the sustainable yield. The following areas of research would also
prove useful.
1. The gathering of more information about New Zealand’s valuable Haliotis iris populations
is an ongoing need to aid in both selecting appropriate models and planning future regulatory
changes. Genetic tagging and the gathering of high resolution field data including shell length
analysis is aiding in this task.
Genetic tagging of seeded Haliotis is becoming more popular (Roodt-Wilding, 2007; News
and Events, Anatomy, 2012) and more information about survival and growth rates will be
gained as tagged H. iris grow through the population. However, any information gained will
be specific to the environment investigated. Both fishery independent (Pickering, 2012) and
fishery dependent data (Paua Industry Council Ltd, 2012) are being gathered, which will aid
in optimising regulations.
It is difficult to know if current fish stocks are above or below the ideal fishdown value,
as the original size of the unharvested population is unknown, and even currently gathered
information on population sizes has a large degree of uncertainty (Ministry of Fisheries,
2011c). This allows the possibility that stock sizes are probably not ideal to maximise yield.
This model suggests fishdown values ranging from 34 to 45%, depending on the population
growth rate. Clearly more information is needed in this area.
2. Specific information on the fecundity and reproductive success of H. iris of different lengths
and ages is another area needing further study, including the possibility that maternal effects
influence reproductive success. Genetically tagged H. iris may enable some future analysis
82
of how fecundity changes over the life of an adult H. iris, and information on the relative
contributions of small and large adults to the next generation can be made as the tagged
H. iris move up through the length classes. The method of calculating fecundity for each class
used in this analysis allows a very accurate consideration of relative fecundity, as evidenced
by the consistency in population growth rates across matrices generated with different length
divisions.
3. The mathematical and statistical analysis of Haliotis populations is an ongoing area of
research. The use of a zero population growth rate in environmental based studies, and the
omission of any stock-recruitment relationship in fishery targeted analysis are two areas of
concern. The method of determination of a population growth rate, and the effects of varying
the population growth rate explored in this thesis offer two methods that are possibly useful
in overcoming these problems.
4. Juvenile cryptic habitat is critical in both maintaining and rebuilding H. iris stocks.
Protection of this habitat from future damage is important. The possibly exists that global
warming will increase storms, and this and other human activities will increase sedimentation
(Phillips and Shima, 2006). Monitoring and protection of cryptic habitats is necessary because
sediment can severely effect juvenile survival (Hepburn, pers. comm. 2012).
5. Regulatory changes are often being considered in several different areas of H. iris management, as well as different Haliotis species overseas. In New Zealand these involve firstly
customary management considerations, including mātaitai reserves and taiāpure. Secondly,
in many commercially harvested areas the annual quota in each local area is set regularly,
often involving input from recreational and commercial fishers, as well as other interested
parties. Finally, the allocation of commercial quota to specific areas, and the recommended
minimum harvest length within each area are also regular considerations (Paua Industry
Council Ltd, 2010, 2012, 2013). These decisions are based on several different assessments,
including Ministry of Primary Industries publications (Ministry of Fisheries, 2011c). I hope
this thesis provides some information that may prove useful in the ongoing task of improving
the assessment and management of Haliotis populations.
83
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100
Appendix A
Appendix: Harvest tables
Tables containing figures used in the matrix calculations are included here. The number of
significant figures does not reflect the degree of accuracy in the calculations, but is instead
included to enable further analysis if so desired.
101
.
Table A.1: The fixed harvest parameters used to calculate maximum
sustainable yield at the different MHL (minimum or maximum shell
harvest lengths). Note that the number of decimal places included
does not indicate the degree of accuracy, this is instead a display of
derived numbers used in further calculations (average age entering Y
class = 4.34 years).
Shell
length
MHL
(mm)
121.00
122.00
123.00
124.00
125.00
126.00
127.00
128.00
129.00
130.00
131.00
132.00
133.00
134.00
135.00
136.00
137.00
138.00
139.00
140.00
141.00
142.00
143.00
144.00
144.50
145.00
145.25
145.50
145.75
145.85
145.95
146.00
146.05
146.10
146.125
146.15
146.175
146.20
Years in Y
class
TY
Egg numbers
Average weight,
EM
EY
W eightY
W eightM
1.953
2.083
2.219
2.361
2.510
2.665
2.829
3.001
3.183
3.376
3.581
3.801
4.036
4.290
4.565
4.867
5.199
5.570
5.989
6.470
7.037
7.725
8.601
9.808
10.639
11.761
12.514
13.498
14.921
15.731
16.815
17.534
18.460
19.767
20.693
22.000
24.233
Inf
2718000
2736000
2753000
2773000
2793000
2812000
2830000
2851000
2872000
2891000
2913000
2936000
2959000
2982000
3006000
3029000
3055000
3081000
3107000
3134000
3162000
3191000
3221000
3252000
3268000
3285000
3294000
3303000
3312000
3316000
3320000
3322000
3325000
3327000
3328000
3329000
3331000
3334000
636400
660200
684200
713100
742100
771100
800100
833800
867400
900900
938800
981000
1023000
1068000
1117000
1165000
1221000
1282000
1345000
1416000
1493000
1581000
1680000
1800000
1870000
1956000
2005000
2063000
2136000
2171000
2213000
2238000
2268000
2305000
2328000
2357000
2398000
2420000
0.5109
0.5201
0.5291
0.5398
0.5503
0.5606
0.5707
0.5822
0.5934
0.6044
0.6167
0.6300
0.6429
0.6567
0.6713
0.6854
0.7012
0.7184
0.7357
0.7548
0.7752
0.7978
0.8230
0.8527
0.8700
0.8906
0.9024
0.9162
0.9334
0.9417
0.9516
0.9575
0.9645
0.9731
0.9785
0.9852
0.9949
1.000
1.082
1.086
1.090
1.095
1.100
1.105
1.109
1.114
1.118
1.123
1.128
1.133
1.138
1.143
1.148
1.153
1.158
1.164
1.169
1.175
1.180
1.186
1.192
1.198
1.201
1.204
1.206
1.207
1.209
1.210
1.211
1.211
1.211
1.212
1.212
1.212
1.213
1.213
102
.
Table A.2: The variable harvest parameters used to calculate maximum sustainable yield at the different MHL (minimum or maximum
shell harvest lengths) for the four trialled population growth rates.
Note that the number of decimal places included does not indicate
the degree of accuracy, this is instead a display of derived numbers
used in further calculations. The ratio leaving the Y class each year
is GY .
Shell
length
MHL
(mm)
121.00
122.00
123.00
124.00
125.00
126.00
127.00
128.00
129.00
130.00
131.00
132.00
133.00
134.00
135.00
136.00
137.00
138.00
139.00
140.00
141.00
142.00
143.00
144.00
144.50
145.00
145.25
145.50
145.75
145.85
145.95
146.00
146.05
146.10
146.125
146.15
146.175
146.20
Population growth
rate of 0.025
GY
SE ×10−7
Population growth
rate of 0.05
GY
SE ×10−7
Population growth
rate of 0.10
GY
SE ×10−7
Population growth
rate of 0.15
GY
SE ×10−7
0.4910
0.4576
0.4269
0.3987
0.3726
0.3484
0.3259
0.3048
0.2850
0.2664
0.2488
0.2321
0.2162
0.2011
0.1866
0.1726
0.1591
0.1459
0.1331
0.1204
0.1078
0.09500
0.08179
0.06761
0.05981
0.05114
0.04627
0.04079
0.03427
0.03115
0.02751
0.02538
0.02293
0.01994
0.01809
0.01582
0.01265
0
0.4851
0.4513
0.4204
0.3919
0.3655
0.3411
0.3183
0.2971
0.2771
0.2583
0.2406
0.2238
0.2078
0.1926
0.1780
0.1640
0.1505
0.1373
0.1245
0.1118
0.09927
0.08661
0.07357
0.05968
0.05211
0.04375
0.03908
0.03388
0.02777
0.02489
0.02156
0.01963
0.01743
0.01479
0.01318
0.01124
0.00860
0
0.4738
0.4393
0.4078
0.3788
0.3520
0.3271
0.3040
0.2824
0.2621
0.2431
0.2252
0.2082
0.1921
0.1768
0.1622
0.1482
0.1346
0.1216
0.1089
0.09643
0.08415
0.07188
0.05941
0.04635
0.03936
0.03181
0.02768
0.02317
0.01804
0.01568
0.01304
0.01155
0.00989
0.00797
0.00685
0.00554
0.00386
0
0.4630
0.4279
0.3959
0.3664
0.3392
0.3140
0.2905
0.2687
0.2482
0.2290
0.2110
0.1939
0.1778
0.1625
0.1479
0.1339
0.1206
0.1077
0.09526
0.08316
0.07131
0.05961
0.04789
0.03588
0.02961
0.02300
0.01948
0.01573
0.01160
0.00978
0.00779
0.00671
0.00554
0.00423
0.00350
0.00268
0.00170
0
4.488
4.490
4.494
4.492
4.493
4.494
4.498
4.497
4.499
4.502
4.503
4.500
4.501
4.500
4.497
4.499
4.497
4.491
4.489
4.483
4.476
4.467
4.456
4.439
4.428
4.412
4.402
4.389
4.370
4.361
4.349
4.341
4.331
4.317
4.308
4.296
4.277
4.441
6.199
6.213
6.229
6.238
6.248
6.262
6.278
6.287
6.300
6.316
6.326
6.332
6.343
6.350
6.354
6.365
6.368
6.365
6.365
6.358
6.348
6.330
6.304
6.262
6.233
6.189
6.161
6.126
6.073
6.047
6.013
5.991
5.964
5.928
5.905
5.873
5.826
5.927
103
10.21
10.26
10.32
10.37
10.42
10.47
10.53
10.57
10.62
10.68
10.72
10.76
10.80
10.83
10.85
10.88
10.90
10.89
10.89
10.86
10.82
10.75
10.64
10.48
10.37
10.22
10.12
10.00
9.840
9.758
9.654
9.590
9.513
9.414
9.350
9.268
9.148
9.143
15.043
15.168
15.299
15.405
15.518
15.636
15.762
15.862
15.970
16.085
16.177
16.246
16.324
16.380
16.418
16.469
16.475
16.441
16.402
16.312
16.184
15.991
15.726
15.344
15.092
14.760
14.559
14.315
13.993
13.836
13.642
13.525
13.386
13.212
13.102
12.963
12.764
12.686
.
Table A.3: Exploring different maximum (slot) and minimum shell harvest
lengths (MHL). Harvest rate is expressed here as the number of adults that can
be taken annually (note that as MHL increases the proportion of adults in the
large adult class decreases). The maximum yield compares each MHL option
with the highest maximum sustainable yield, which is given a value of one. The
last column shows the effects of this management system on H. iris, comparing
harvested vs non-harvested populations. The current harvest regulations (above
125 mm) and the best recommendations are highlighted.
Note that the class ”Large harvest” above 146.2 mm is considered empty, and for
all 40 trials a density independent matrix generated λ = 1.025.
Shell
length
MHL
(mm)
121.00
122.00
123.00
124.00
125.00
126.00
127.00
128.00
129.00
130.00
131.00
132.00
133.00
134.00
135.00
136.00
137.00
138.00
139.00
140.00
141.00
142.00
143.00
144.00
144.50
145.00
145.25
145.50
145.75
145.85
145.95
146.00
146.05
146.10
146.125
146.15
146.175
146.20
Harvest rate (per
numbers in that
class)
Small slot Large
harvest
harvest
0.1052
0.01383
0.09985
0.01395
0.09491
0.01407
0.09029
0.01419
0.08597
0.01432
0.08190
0.01446
0.07806
0.01461
0.07443
0.01477
0.07095
0.01494
0.06768
0.01512
0.06451
0.01531
0.06146
0.01552
0.05853
0.01575
0.05570
0.01600
0.05293
0.01629
0.05025
0.01660
0.04758
0.01697
0.04495
0.01740
0.04236
0.01789
0.03973
0.01849
0.03707
0.01925
0.03431
0.02025
0.03140
0.02164
0.02819
0.02387
0.02641
0.02563
0.02438
0.02838
0.02324
0.03049
0.02195
0.03366
0.02038
0.03926
0.01962
0.04313
0.01876
0.04931
0.01824
0.05423
0.01764
0.06178
0.01692
0.07575
0.01649
0.08905
0.01593
0.11570
0.01518
0.21170
0.01302
NA
Maximum
(numbers
H. iris)
Small slot
harvest
0.5092
0.5466
0.5818
0.6148
0.6460
0.6756
0.7039
0.7307
0.7564
0.7811
0.8046
0.8270
0.8486
0.8692
0.8887
0.9078
0.9254
0.9417
0.9571
0.9707
0.9827
0.9923
0.9987
1.0000
0.9977
0.9916
0.9866
0.9793
0.9678
0.9614
0.9529
0.9474
0.9406
0.9316
0.9257
0.9179
0.9063
0.8901
yield
of
Large
harvest
0.9121
0.9141
0.9160
0.9176
0.9191
0.9204
0.9215
0.9222
0.9227
0.9230
0.9229
0.9222
0.9212
0.9197
0.9176
0.9149
0.9113
0.9067
0.9011
0.8940
0.8852
0.8740
0.8596
0.8399
0.8269
0.8103
0.7999
0.7871
0.7706
0.7620
0.7515
0.7451
0.7376
0.7280
0.7219
0.7143
0.7034
NA
104
Maximum
yield
(biomass of H. iris)
Small slot
harvest
0.2464
0.2693
0.2916
0.3144
0.3368
0.3588
0.3806
0.4030
0.4253
0.4473
0.4701
0.4936
0.5169
0.5408
0.5653
0.5895
0.6148
0.6409
0.6671
0.6942
0.7217
0.7500
0.7787
0.8079
0.8223
0.8367
0.8436
0.8501
0.8559
0.8578
0.8591
0.8595
0.8595
0.8589
0.8581
0.8568
0.8543
0.8821
Large
harvest
0.9350
0.9408
0.9464
0.9522
0.9579
0.9631
0.9681
0.9731
0.9778
0.9820
0.9861
0.9899
0.9931
0.9959
0.9981
0.9994
1.0000
0.9997
0.9980
0.9949
0.9898
0.9820
0.9705
0.9531
0.9407
0.9243
0.9137
0.9004
0.8827
0.8734
0.8620
0.8549
0.8465
0.8359
0.8291
0.8204
0.8081
NA
Change from unharvested population (biomass)
Small slot Large
harvest
harvest
0.4941
0.4510
0.4910
0.4510
0.4881
0.4509
0.4854
0.4509
0.4828
0.4509
0.4805
0.4510
0.4783
0.4509
0.4761
0.4510
0.4742
0.4511
0.4723
0.4511
0.4705
0.4513
0.4688
0.4514
0.4672
0.4514
0.4656
0.4517
0.4641
0.4517
0.4626
0.4520
0.4614
0.4522
0.4600
0.4523
0.4586
0.4527
0.4574
0.4530
0.4562
0.4534
0.4550
0.4538
0.4539
0.4544
0.4529
0.4550
0.4523
0.4555
0.4521
0.4559
0.4516
0.4563
0.4513
0.4567
0.4513
0.4572
0.4514
0.4575
0.4511
0.4578
0.4513
0.4580
0.4514
0.4583
0.4514
0.4585
0.4510
0.4587
0.4513
0.4589
0.4512
0.4591
0.4512
NA
.
Shell
length
MHL
(mm)
121.00
122.00
123.00
124.00
125.00
126.00
127.00
128.00
129.00
130.00
131.00
132.00
133.00
134.00
135.00
136.00
137.00
138.00
139.00
140.00
141.00
142.00
143.00
144.00
144.50
145.00
145.25
145.50
145.75
145.85
145.95
146.00
146.05
146.10
146.125
146.15
146.175
146.20
Harvest rate (per
numbers in that
class)
Small slot Large
harvest
harvest
0.1812
0.02618
0.1727
0.02643
0.1647
0.02670
0.1572
0.02698
0.1501
0.02727
0.1433
0.02758
0.1369
0.02791
0.1308
0.02827
0.1249
0.02866
0.1193
0.02907
0.1139
0.02952
0.1086
0.03002
0.1035
0.03056
0.09853
0.03116
0.09364
0.03185
0.08888
0.03262
0.08414
0.03353
0.07942
0.03459
0.07471
0.03588
0.06995
0.03749
0.06510
0.03957
0.06007
0.04244
0.05474
0.04673
0.04889
0.05420
0.04564
0.06000
0.04198
0.07232
0.03992
0.08237
0.03760
0.1001
0.03487
0.1418
0.03357
0.1821
0.03207
0.2846
0.03121
0.4386
0.03022
1.000
0.02903
1.000
0.02831
1.000
0.02744
1.000
0.02626
1.000
0.0225
NA
Maximum
(numbers
H. iris)
Small slot
harvest
0.5208
0.5591
0.5951
0.6288
0.6607
0.6909
0.7197
0.7469
0.7730
0.7980
0.8216
0.8440
0.8655
0.8858
0.9049
0.9233
0.9401
0.9552
0.9692
0.9810
0.9907
0.9973
1.0000
0.9965
0.9913
0.9818
0.9748
0.9654
0.9517
0.9443
0.9350
0.9292
0.9222
0.9133
0.9076
0.9004
0.8903
0.8556
yield
of
Large
harvest
0.9213
0.9233
0.9250
0.9264
0.9276
0.9286
0.9293
0.9296
0.9295
0.9291
0.9282
0.9266
0.9246
0.9219
0.9183
0.9141
0.9087
0.9021
0.8942
0.8845
0.8727
0.8580
0.8395
0.8152
0.7995
0.7802
0.7683
0.7542
0.7366
0.7278
0.7174
0.7113
0.7040
0.6807
0.6542
0.6086
0.5214
NA
105
Maximum
yield
Small slot
harvest
0.2523
0.2758
0.2987
0.3219
0.3448
0.3673
0.3895
0.4124
0.4351
0.4575
0.4805
0.5043
0.5278
0.5518
0.5762
0.6002
0.6252
0.6509
0.6763
0.7023
0.7284
0.7546
0.7806
0.8059
0.8180
0.8293
0.8344
0.8389
0.8425
0.8434
0.8439
0.8439
0.8436
0.8429
0.8423
0.8414
0.8401
0.8115
Large
harvest
0.9455
0.9513
0.9567
0.9625
0.9678
0.9728
0.9774
0.9820
0.9860
0.9895
0.9928
0.9957
0.9978
0.9993
1.0000
0.9996
0.9982
0.9957
0.9915
0.9854
0.9768
0.9651
0.9489
0.9260
0.9105
0.8909
0.8786
0.8637
0.8447
0.8351
0.8237
0.8170
0.8089
0.7824
0.7521
0.6998
0.5997
NA
Small slot Large
harvest
harvest
0.4922
0.4182
0.4867
0.4182
0.4816
0.4182
0.4769
0.4182
0.4725
0.4183
0.4684
0.4184
0.4645
0.4185
0.4608
0.4186
0.4574
0.4187
0.4541
0.4190
0.4509
0.4192
0.4480
0.4194
0.4452
0.4197
0.4425
0.4200
0.4399
0.4204
0.4374
0.4208
0.4351
0.4212
0.4328
0.4217
0.4306
0.4223
0.4285
0.4230
0.4265
0.4238
0.4246
0.4248
0.4229
0.4259
0.4213
0.4274
0.4205
0.4319
0.4199
0.4292
0.4196
0.4301
0.4194
0.4307
0.4192
0.4316
0.4192
0.4321
0.4192
0.4326
0.4193
0.4329
0.4193
0.4408
0.4194
0.5078
0.4195
0.5510
0.4196
0.6063
0.4198
0.6869
0.4204
NA
.
Shell
length
MHL
(mm)
121.00
122.00
123.00
124.00
125.00
126.00
127.00
128.00
129.00
130.00
131.00
132.00
133.00
134.00
135.00
136.00
137.00
138.00
139.00
140.00
141.00
142.00
143.00
144.00
144.50
145.00
145.25
145.50
145.75
145.85
145.95
146.00
146.05
146.10
146.125
146.15
146.175
146.20
Harvest rate (per
numbers in that
class)
Small slot Large
harvest
harvest
0.2892
0.04866
0.2770
0.04923
0.2655
0.04986
0.2544
0.05055
0.2438
0.05123
0.2336
0.05203
0.2238
0.05283
0.2143
0.05368
0.2051
0.05465
0.1962
0.05573
0.1875
0.05687
0.1790
0.05818
0.1706
0.05961
0.1624
0.06132
0.1542
0.06326
0.1462
0.06548
0.1381
0.06822
0.1300
0.07164
0.1219
0.07591
0.1136
0.08167
0.1052
0.08982
0.09638
0.1025
0.08713
0.1254
0.07713
0.1814
0.07164
0.2601
0.06560
0.6053
0.06227
1.000
0.05862
1.000
0.05447
1.000
0.05259
1.000
0.05043
1.000
0.04925
1.000
0.04792
1.000
0.04635
1.000
0.04549
1.000
0.04443
1.000
0.04314
1.000
0.04016
NA
Maximum
(numbers
H. iris)
Small slot
harvest
0.5424
0.5822
0.6195
0.6543
0.6871
0.7182
0.7477
0.7753
0.8017
0.8268
0.8503
0.8723
0.8932
0.9126
0.9304
0.9472
0.9620
0.9745
0.9854
0.9935
0.9987
1.0000
0.9964
0.9855
0.9761
0.9624
0.9534
0.9422
0.9275
0.9202
0.9115
0.9065
0.9007
0.8938
0.8897
0.8849
0.8787
0.8655
yield
of
Large
harvest
0.9369
0.9384
0.9397
0.9404
0.9408
0.9408
0.9405
0.9395
0.9380
0.9360
0.9332
0.9296
0.9252
0.9199
0.9135
0.9060
0.8970
0.8864
0.8740
0.8594
0.8422
0.8217
0.7968
0.7660
0.7473
0.7255
0.7106
0.6752
0.6042
0.5592
0.4984
0.4590
0.4105
0.3477
0.3075
0.2571
0.1872
NA
106
Maximum
yield
Small slot
harvest
0.2631
0.2875
0.3112
0.3354
0.359
0.3823
0.4051
0.4286
0.4517
0.4745
0.4979
0.5218
0.5453
0.5691
0.5931
0.6164
0.6404
0.6647
0.6883
0.7120
0.7351
0.7575
0.7787
0.7979
0.8063
0.8138
0.8170
0.8197
0.8221
0.8228
0.8237
0.8242
0.8248
0.8258
0.8266
0.8278
0.8301
0.8219
Large
harvest
0.9626
0.9679
0.9729
0.9781
0.9827
0.9867
0.9902
0.9935
0.9961
0.9980
0.9993
1.0000
0.9996
0.9983
0.9958
0.9918
0.9865
0.9795
0.9702
0.9585
0.9438
0.9252
0.9017
0.8711
0.8520
0.8295
0.8135
0.7741
0.6937
0.6424
0.5729
0.5278
0.4722
0.4001
0.3539
0.2960
0.2156
NA
Small slot Large
harvest
harvest
0.4906
0.3733
0.4815
0.3735
0.4730
0.3736
0.4652
0.3737
0.4579
0.3740
0.4511
0.3740
0.4447
0.3743
0.4387
0.3748
0.4331
0.3751
0.4276
0.3753
0.4225
0.3760
0.4178
0.3765
0.4131
0.3773
0.4089
0.3778
0.4047
0.3786
0.4008
0.3798
0.3970
0.3808
0.3936
0.3819
0.3902
0.3834
0.3871
0.3851
0.3843
0.3869
0.3817
0.3890
0.3795
0.3915
0.3775
0.3945
0.3768
0.3963
0.3764
0.3983
0.3763
0.4276
0.3763
0.4932
0.3763
0.5785
0.3762
0.6218
0.3765
0.6740
0.3765
0.7053
0.3766
0.7417
0.3770
0.7863
0.3769
0.8136
0.3772
0.8466
0.3772
0.8907
0.3778
NA
.
Shell
length
MHL
(mm)
121.00
122.00
123.00
124.00
125.00
126.00
127.00
128.00
129.00
130.00
131.00
132.00
133.00
134.00
135.00
136.00
137.00
138.00
139.00
140.00
141.00
142.00
143.00
144.00
144.50
145.00
145.25
145.50
145.75
145.85
145.95
146.00
146.05
146.10
146.125
146.15
146.175
146.20
Harvest rate (per
numbers in that
class)
Small slot Large
harvest
harvest
0.3654
0.06959
0.3512
0.07062
0.3376
0.07175
0.3244
0.07288
0.3116
0.07410
0.2992
0.07551
0.2872
0.07692
0.2754
0.07852
0.2639
0.08031
0.2527
0.08228
0.2415
0.08444
0.2305
0.08698
0.2196
0.08989
0.2088
0.09328
0.1980
0.09732
0.1873
0.1023
0.1765
0.1085
0.1656
0.1168
0.1547
0.1281
0.1435
0.1446
0.1321
0.1717
0.1204
0.2243
0.1081
0.3733
0.09520
1.000
0.08827
1.000
0.08087
1.000
0.07693
1.000
0.07272
1.000
0.06807
1.000
0.06606
1.000
0.06384
1.000
0.06265
1.000
0.06137
1.000
0.05995
1.000
0.05916
1.000
0.05828
1.000
0.05725
1.000
0.05549
NA
Maximum
(numbers
H. iris)
Small slot
harvest
0.5614
0.6023
0.6406
0.6762
0.7097
0.7412
0.7710
0.7988
0.8251
0.8499
0.8730
0.8942
0.9142
0.9323
0.9486
0.9636
0.9761
0.9861
0.9940
0.9987
1.0000
0.9969
0.9886
0.9730
0.9615
0.9464
0.9372
0.9264
0.9135
0.9075
0.9008
0.8971
0.8931
0.8886
0.8861
0.8834
0.8802
0.8754
yield
of
Large
harvest
0.9484
0.9492
0.9497
0.9496
0.9490
0.9479
0.9463
0.9439
0.9408
0.9370
0.9322
0.9264
0.9197
0.9117
0.9024
0.8917
0.8793
0.8650
0.8487
0.8299
0.8083
0.7834
0.7545
0.7176
0.6755
0.6018
0.5482
0.4785
0.3848
0.3371
0.2802
0.2469
0.2090
0.1642
0.1380
0.1076
0.0698
NA
107
Maximum
yield
Small slot
harvest
0.2726
0.2977
0.3221
0.3469
0.3711
0.3949
0.4182
0.4420
0.4653
0.4882
0.5116
0.5354
0.5585
0.5819
0.6052
0.6276
0.6504
0.6732
0.6950
0.7164
0.7367
0.7559
0.7733
0.7885
0.7950
0.8010
0.8038
0.8067
0.8103
0.8122
0.8146
0.8164
0.8186
0.8218
0.8240
0.8271
0.8323
0.8319
Large
harvest
0.9752
0.9800
0.9842
0.9885
0.9921
0.9950
0.9973
0.9991
1.0000
0.9999
0.9992
0.9975
0.9945
0.9902
0.9846
0.9770
0.9679
0.9567
0.9429
0.9264
0.9066
0.8830
0.8545
0.8168
0.7709
0.6886
0.6282
0.5491
0.4422
0.3876
0.3223
0.2842
0.2406
0.1892
0.1590
0.1239
0.0804
NA
Small slot Large
harvest
harvest
0.4885
0.3422
0.4766
0.3422
0.4656
0.3422
0.4554
0.3426
0.4459
0.3430
0.4370
0.3431
0.4287
0.3437
0.4209
0.3443
0.4135
0.3448
0.4066
0.3455
0.4001
0.3465
0.3939
0.3474
0.3881
0.3484
0.3826
0.3497
0.3775
0.3512
0.3726
0.3527
0.3680
0.3547
0.3638
0.3567
0.3599
0.3590
0.3564
0.3617
0.3533
0.3646
0.3506
0.3680
0.3485
0.3718
0.3469
0.4074
0.3464
0.4741
0.3463
0.5570
0.3463
0.6074
0.3465
0.6668
0.3469
0.7402
0.3470
0.7755
0.3472
0.8163
0.3473
0.8395
0.3474
0.8656
0.3476
0.8956
0.3477
0.9130
0.3478
0.9328
0.3478
0.9570
0.3480
NA

Exploring harvest regulations of the New Zealand paua abalone

Transcription

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