Lesson 8

This weeks schedule
Mon, Wed: Lectures on species interactions and competition.
Wed lab: 24 species, dwarf shrubs.
Fri: Tilman and Grimes papers?
Mon, 13 Feb: Lecture Exam 1.
The exam is designed as a review of the material.
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Open book, open notes. 70 questions covering lessons 2 through 9. Work on the
exam independently.
Questions will be short answer.
Covers the lecture and the journal papers discussed in class.
Review all the lectures before taking the exam. Make sure you understand all the
material discussed in class.
I will not accept any answers that are clearly clipped from the lecture slides, the
textbook ,or the journal papers. I will award points on the basis of whether or not you
demonstrate that you understand the question and the answer.
The exam will likely take 3-4 hours (possibly more) to complete. You can use the
lecture period on Monday to work on the exam.
I will email the exam to everyone on Fri 10 Feb by 5 pm and you can email the
answers back.
The exam is due Tues, Feb 14 at 9 am in class (Happy Valentines Day!). I will deduct
5 points if the exam is late that day, and I will not accept it beyond 5 pm, Feb 14.
Lesson 8: Species interactions:
competition and amensalism
• Simple interactions
• Competition
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Measuring competition
Experimental evidence of competition
Models of competition and resource limitation
Limiting resources and plant strategies
• Amensalism
– Allelopathy
– Interactions between trophic levels (e.g.,
herbivory)
Introduction
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Individuals and populations respond much differently when
grown with other species.
Plant ecologists have long recognized that studying plants
in relationship to other species is critical to understanding
ecosystems.
In this lesson, we will first look at a simple classification
scheme of pairwise interactions.
We will then examine two of these: competition and
amensalism, with most of the focus being on competition
between plant species
Simple interactions
Pairwise interactions
• Competition (-. -): Plants are competing for the same
resource decreasing the to total fitness and/ or
growth of both species.
• Amensalism (-, 0): One plant has a negative effect on
the other, but the other has no effect on the first.
• Commensalism (0, 0): Plants are apparently
indifferent to each other.
• Mutualism (+, +): Plants have mutual benefit to each
other
• Parasitism (also herbivory, predation, pathogenicity)
(+, -): One plant benefits, the other is affected
negatively.
Using pattern to infer
interactions
Positive association: A nonrandom
clumped distribution, such as in
(a), denotes a positive association
between species (e.g., mutualism,
or parasitism).
Negative association: If the species
show negative association with
each other, as in (b), this
indicative of a negative spatial
association (e.g., allelopathy).
No association: Two species that
show totally random dispersal
patterns in relation to each other,
generally have no interaction,
whereas a nonrandom pattern is
indicative of an interaction
(positive or negative).
• However, these patterns are not
necessarily indicative of a
relationship. For example, both
species may be associated with
some environmental factor, such
as water availability, and may have
no real interaction with each other.
Competition
• Since plants require the same basic resources
(carbon, water, nutrients) in roughly the same
proportion, and they do this through the same
basic mechanism (photosynthesis, root uptake),
then it stands to reason that they ought to
compete for access to these resources.
• This has been a central focus of studies since the
inception of plant ecology (e.g., de Candolle 1820,
Clements 1929, Tilman 1982, Keddy 1989, Connell
1990).
Three types of competitive interactions
1. Direct interference competition: Species actually confront each
other (e.g., strangler figs, allelopathy in plants).
2. Exploitation competition: Species have a negative impact on each
other through competition for resources (e.g. competition for
light, water, or nutrients).
3. Apparent competition: Species have a net negative impact on one
another, but this is indirectly mediated through a third species.
(e.g., if two species are affected by a herbivore, increasing plant
species A may increase the herbivore population, with a greater
net negative effect on species B. Thus, increasing A may lead to a
decline in B with no direct interaction between them.
Ways to study competition
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Analysis of the results of competition:
– Studies of pattern and diversity of species.
– Example: How does the the presence of Species A affect the pattern
and abundance of Species B?
– Use plant community studies.
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Analysis of the mechanisms of competition:
– Studies of resource acquisition and use.
– Example: How does the presence of Species A affect the growth or
water uptake of Species B?
– Use plant physiology and autecological studies.
Niche (Hutchison 1957)
“the multidimensional description of a species with all aspects
of its biotic and abiotic environmental requirements”.
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Although intellectually appealing, the actual niche of a plant
is very difficult to define because so many different factors
influence the occurrence of species.
If two plants have similar niches, the more likely they are to
compete for resources.
Gauses (1934) competitive
exclusion principle
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“one niche, one species”
Gause concluded that in
order for species to
coexist in nature they
must evolve ecological
differences (i.e., occupy
different niches).
Exponential vs. logistic population
growth
Intraspecific (within species) competition:
the Verhulst-Pearl Equation
dN/dt = rN(K-N)/K
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dN/dt is the rate of growth of a species, or the slope of the line.
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In a one species system, the quantity (K-N)/K is the intraspecific
competition component because as N approaches K, the change
in the population size, dN/dt, approaches 0, but when N is small,
dN/dt approaches 1, the maximum rate of population increase.
In other words, individuals of the same species are limiting the
population size as the population approaches the carrying
capacity.
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Two species system: Lotka-Volterra
equations
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The Lotka-Volterra equations describe the relationship between two
species using the same resource.
Assume a two species system, where the sum of individuals of
species 1 and 2 add up to the carrying capacity:
N1 + N2 = K1,
where is competition coefficient for species 2 on species 1, i.e., is the inhibiting (competitive) effect on species 1 on species 2.
For a two species system, we can introduce the negative effect of the
second species into the Verhulst-Pearl equation by substituting (K1 N2) for N1 on the right side of the equation dN1/dt = r1N1(K1-N1)/K1).
For species 1:
dN1/dt = r1N1(K1 - N1 - N2)/K1.
For species 2:
dN2/dt = r2N2(K2 - N2 - N1)/K2,
where is the competition coefficient for species 1 on species 2.
Zero Net Growth Isocline diagrams (ZNGIs)
(Tilman 1982)
ZNGI diagrams devised by Tilman can help visualize outcomes of
pairwise competitive interactions.
(a)
Isocline for species 1:
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The line is the isocline for species 1 and represents various combinations of
species 1 and 2 that result in the joint carrying capacity, K1.
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N1 + N2 = K1, so when N1 = 0, N2 = K1/, and when N2 = 0, N1 = K1.
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To reach equilibrium, Species 1 will increase left of the line and decrease right
of the line.
Pairwise competitive interactions: Outcome when one
species always inhibits the growth of the second species
Line 2 is the isocline for species 2:
• Note: that N1 + N2 = K2, so when N1 = 0, N2 = K2, and when N2 = 0, N1 = K2/.
The line represents various combinations of species 1 and 2 that
result in the joint carrying capacity, K2 .
• Species 2 increases below the line and decreases above the line.
Situation when Species 1 always inhibits growth of Species 2
(Species 2 isocline is always below Species 1 isocline ):
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At point A, Species 1 and Species 2 increase.
At point B, Species 1 and Species 2 decrease.
At point C, Species 1 increases, Species 2 decreases.
Species 2 will continue to decline once Line 1 is reached and
Species 1 will increase, until K1 . This is the stable equilibrium point.
Situation where the isoclines cross
Interesting patterns can occur depending on the relationship of the isoclines to each other:
• When the isoclines cross with K1 exceeding K2/. , each species limits the other more than it
does itself), population trajectories are such that stable equilibrium points exist at both
species carrying capacities, K1 and K2.
• If K1 is less than K2/. , (i.e., if each species limits itself more than it limits the other
species), then there is a stable equilibrium at the intersection of the isoclines and both
species can coexist. The most obvious way for this to happen is through niche separation.
Conclusion: It is very difficult for species to coexist at equilibrium, unless each species
limits itself more than it limits the other.
However, natural populations may not come into equilibrium very often, or other interactions
may limit the full competitive interaction between species.
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Tilman 1982
Experimental Evidence of
Competition
• Replacement series experiments (De Wit 1960)
• Target-neighborhood experiments (Fonteyn and
Mahall 1981)
• Root vs. shoot competition experiments (McGraw
1985)
Replacement series experiments
(De Wit 1960)
• The ratio of seeds planted for two species,
A and B, is compared to the ratio of some
measure, such as biomass, of the
resulting crop.
• Input ratio = (seeds sown of A)/(seeds
sown of B)
• Output ratio = (biomass A)/(biomass B)
Application of replacement series to study
weed competition (Fischer et al. 2000)
Kochia scoparia (Kochia)
sanangelo.tamu.edu/ agronomy/newsltr/kochia_ko
Hordeum distychum (Barley)
Triticum aestivum (Wheat)
http://www.hops.co.uk/sectiontwo/Images/Barley.jpg
www.oznet.ksu.edu
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Kochia is a weed infecting cereal crops, severely reducing yields and has developed resistance to
herbicides. Alternatives are needed for integrated management of the weed.
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Replacement series experiments with barley and wheat were conducted under a variety of
temperature, soil moisture, and light conditions to determine what environmental conditions
would render Kochia susceptible to competition by small grained crops.
Fisher, et al. 2000. Interference between spring cereals and Kochia related to environment and photosynthetic pathways.
Agron. J. 92: 173-181.
Fisher et al. (2002) Experiments
Experiment
PAR
μmol m-1s-1
Day/Night
Temperature
(˚C)
Day length
(h)
Leaf
Temperature
(˚C)
1. Early May
500
15/11
15
16
2. June
550
21/17
16
26
3. July
550
23/19
16
31
4. Moisture
stress
550
23/19
16
30
5. Light/Shade
550/250
22/18
16
28
• Seeds of Barley and Wheat were planted in separate experiments with
the following ratios to Kochia: 100:0, 75:25, 50:50, 25:75, and 0:100, 4
replicates each. (Fisher et al. 2000.)
Fischer et al. (2002) Results
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Experiment 1
Experiment 2
(Fisher et al. 2000.)
In the first two experiments, Barley
suppressed Kochia more than
wheat did because of its larger
canopy, despite its lower
photosynthetic rates.
Fischer et al. (2002) Results
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Fig. 2 Relative yields and relative yield totals (RYT) (open
triangles) of wheat (solid circles) and kochia (solid triangles)
grown at 15/11 and 22/18°C day/night temperature regimes
in replacement-series experiments. Error bars represent ±
standard errors of the mean
(Fisher et al. 2000.)
Under high radiation conditions and warm
temperatures, growth and photosynthesis
were greater for kochia than wheat.
Warm temperatures also increased dark
respiration and reduced water use
efficiency under low radiation conditions,
however, thus limiting kochia's
competitiveness under a closed canopy.
Water stress did not affect competition.
Fischer et al. (2002) Results
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Fig. 4 (a) Net CO2 assimilation rates and (b) photosynthetic water use
efficiency of barley (open circles), wheat (solid circles), and kochia
(solid triangles), as affected by levels of photosynthetically active
radiation (PAR), when grown under moisture stress at 23/19°C
day/night.
(Fisher et al. 2000.)
Net photosynthetic rates of kochia were
greater at photosynthetically active
radiation (PAR) values > 400 μmol m-2 s-1.
Growth and CO2 exchange rates varied
among four different kochia accessions,
but growth of all accessions was reduced
by shade.
Results suggest that a leafy, cold-tolerant
crop or cultivar, grown early in the season
to produce necessary ground cover,
should provide opportunity to suppress
kochia.
Replacement series (De Wit 1960)
(a)
If the output ratio is equal to the ratio of the input for all seed mixes
(diagonal line in (a) then there is no competition. If for all input ratios, the
output ratio (biomass of A/biomass of B) is consistently less than the
input ratio (seeds of A/seeds of B) then B will eliminate A, and vice versa.
(b) If the output ratios vary with differing
input ratios, there can be two
(
outcomes.
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If the slope is >45˚, then competition will eliminate one of the species
depending on the input ratio.
If the slope is <45˚, then there is a stable equilibrium seed ratio.
Other experimental seeding designs for competition
experiments
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(a) Partial additive: Can be used to test the
effect of varying the abundance of seeds of
species 2 against a fixed abundance of
seeds of species 1. This tests the effects of
species 2 on species 1, but not vice versa.
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(b) Replacement series of DeWit with
mixtures varying from total dominance of
species 1 to total dominance of species 2.
This allows testing the effects on either
species on the other.
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(c and d) Additive series are more complex
and allow one to test the interaction of a full
range of input ratios of seeds.
Competition experiments: Targetneighborhood experiments (Fonteyn
and Mahall 1981)
The study site near Cottonwood Springs in Joshua Tree National Park,
California. The formation is Colorado Desert on a bajada of the Eagle
Mountains, 20 km south of the transition to Mojave Desert. Light gray
shrubs in the foreground are the dominant perennial of the system,
Ambrosia dumosa (Asteraceae).
Ambrosia dumosa (Burro weed)
http://www.jaeger.ws/history/099/06.JPG
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Left: Effect of Ambrosia dumosa on
Larrea tridentata 100 m2 plots
(clockwise from upper left):(1) control,
(2) removing all Larrea and all
Ambrosia except the Larrea target, (3)
removing all the Ambrosia, (4)
removing all the Larrea except the
target.
• Right: Experiment examining control
of Larrea on Ambrosia.
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Examined the effect of the removal
experiments on stem xylem pressure
of target species .
Fonteyn, P.J. and B.E. Mahall. 1981. An Experimental Analysis of Structure in a Desert Plant Community. Journal of Ecology
Vol. 69, no. 3, pp. 883-896.
Competition experiments: Targetneighborhood experiments
(Fonteyn and Mahall 1981)
(a) Larrea showed some
reduction in water stress when
other plants were removed.
This increased somewhat as
the summer progressed.
(b) Ambrosia showed a much
stronger response to removal,
particularly of Larrea.
Fonteyn and Mahall. 1981.
Competition experiments:
Examination of root and shoot
competition (McGraw 1985) (1)
No competition
Shoot competition
McGraw. 1985.
Root competition
Full competition
Manipulated the aboveground
and belowground space with
partitions to separate or
enclose roots and/or shoots of
two ecotypes of Dryas (F =
Fellfield ecotype, S = Snowbed
ecotype).
Competition experiments:
Examination of root and shoot
competition (McGraw 1985) (2)
Generally, the snowbed ecotype
responded positively to shoot
competition (dashed lines)
competition; whereas the
fellfield ecotype responded
negatively.
McGraw. 1985.
Competition for light and soil moisture
(Shirley 1945)
Objective: determine the relative importance of
competition for light and soil moisture to
pine seedlings.
Picea glauca, Pinus strobus, P. resinosa and P.
banksiana are the overstory species in north
central Minnesota, but they do not reproduce in
their own shade. Usually hardwood seedlings
will occur beneath the trees.
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(a) Effect of competition for light. Pine seedlings
were grown beneath different layers of screens to
achieve different levels of sunlight (a). Growth was
not satisfactory below about 65% light.
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(b) Combined effect of shade and root competition
for water.
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The overstory had three treatments. (b), (uncut, 1/3
removed, and clear cut).
The understory was also varied (control, all
understory plants removed, weeded and trenched to
sever plant roots).
The results were complex and appeared to depend on
initial site moisture. In moist areas, opening the
canopy, weeding, and trenching improved
seedling growth. In dry areas opening the canopy
decreased seedling survival, but not seedling
growth.
Resource-ratio hypothesis
(Huston and Smith 1987, Tilman 1988)
“Differences in the relative supply rates of limiting
resources should lead to differences in the
composition of plant communities.”
Nutrient flux gradients (Huston and De Angelis 1994)
(a)
High nutrient flux: Plants can
coexist because each has access to
only a small portion of the total
available resource. Species with
similar resource requirements, but
restricted rooting zones (as in a) can
coexist because each can access
only a small portion of the of the
total resources available.
(b)
Low nutrient flux: Plants deplete
nutrients over a much broader area.
If soil resource depletion zones
extend into the rooting zones of
neighboring individuals, then
competitive effects become
important.
Models of competition along resource gradients:
root vs. shoot competition (Wilson and Tilman 1991)
Root vs. Shoot competition
• Wilson and Tilman examined the
survivorship of roots vs. shoots
in little bluestem, Schizachyrium
scoparium along a nutrient
gradient.
Wilson & Tilman 1991, cited in Barbour et al. 1999.
• When N availability was low, root
competition was relatively high,
and when N availability was high,
shoot competition became more
important.
Resource competition: Effect of competition between species
for a single resource, R. Tilman model (1982)
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Curves are the population
growth rates for species A
and B.
ma and mb are the mortality
rates for species A and B.
The intersection of the curves
with the m lines represent the
minimum amount of the
resource R needed to sustain
the population.
Best competitor is the one
with the lower R* for the
limiting resource.
Tilmans resource-ratio model (1982): How 2 species can
coexist competing for the same resources (1)
Lines A and B are Zero Net Growth Isoclines (ZNGIs) of species A and B for
resources R1 and R2.
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In the left figure, A can survive on lower levels of both resources, and will draw either
resource to a level that B cannot survive (Area 2).
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In the right figure, B is the superior competitor, and will draw either resource to levels
that A cannot tolerate (Area 6).
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Who can exist in areas 1, 3 and 5?
Tilmans resource-ratio model (1982): How 2 species can
coexist competing for the same resources (2)
In this case, the ZNGIs cross. A is a superior
competitor for R1 and B is the superior
competitor for R2.
A B
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R2
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1
2
4
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3
1
R1
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Think of R1 and R2 as light and water.
The black dot is the two-species equilibrium
point, where both species can coexist.
Species A will outcompete and replace B in Area
2.
Species B will coucompete and replace A in Area
3.
The outcome is less certain in Area 4 and
depends on the consumption rate of each
resource by each species.
The black dot is the point (amount of both
resources) where both species can coexist.
Which resource is most limiting Species A?
Tilmans resource-ratio model (1982): How 2 species can
coexist competing for the same resources (3)
CA and CB are resource consumption vectors for
each species.
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The slope of each vector is the ratio of
consumption of resource R2 divided by the
consumption of resource R1.
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In this situation, Species A consumes more of
resource 2 (the resource that is most limiting to
itself) than resource 1 (slope of CA>1). So in areas
2 and 3, it will out compete Species B.
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B consumes more of resource 1 (slope of CB<1),
and will out compete Species A in areas 6 and 5.
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In area 4, the both species can coexist as they
draw down the resources to the equilibrium point.
Tilmans resource-ratio model (1982): Where 2 species will not
coexist competing for the same resources (3)
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In this situation, Species A consumes more of
resource 1 (slope of CA<1). This resource is
most limiting to Species B.
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Similarly, Species B consumes more of
resource 2 (slope of CB>1). This resource is
most limiting to Species A.
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So in areas 2 and 3, A still out competes B, and
in areas 6 and 5 B still out competes Species A.
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But in area 4, the equilibrium point is unstable
because each species uses more of the
resource that limits the other species, so either
species could dominate at this point depending
on the initial conditions.
Productivity vs. species richness (Tilman
and Pacala 1993)
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Habitats intermediate
in resources (and
productivity) tend to
support the most
species.
Extremely poor soils
are likely to be
dominated by only a
few species that can
compete for a single
limiting resource.
Extremely rich soils
support high biomass
production and are
dominated by the few
species that compete
the most effectively for
light.
Implications of Resource-ratio hypothesis
(Tilman 1988)
Differences in the relative supply rates of limiting resources should
lead to differences in the composition of plant communities:
– Species allocation patterns: Species with allocation patterns focusing
on shoots are assumed to be relatively effective competitors for light,
and those allocating more heavily to roots are assumed to be good
competitor for below-ground resources (water, nutrients).
– Landscape implications: Various habitats within landscapes differ in
their level of key resources, and hence will favor either root or shoot
specialists depending on the local resource supply.
– Succession implications: Resource supply ratios also vary
systematically through successional series to first favor root
specialists (because soil nutrition is more limiting than light in primary
succession) and then shoot specialists because light is more limiting
in later stages of succession.
Reconciling the theories of Grime and Tilman
Grime focuses on plant strategies and adaptation to
certain environmental conditions, the role of environment
in relation to plants distributions, and how these determine
patterns of succession and competition between species.
Tilman focuses more on the interactions between plants
and the role of competition for resources.
Grime 1977
Tilman says that Grime's theories do not adequately
incorporate the importance of non-heterogeneous supplies
of nutrients and how these supplies are partitioned over
long time scales, and are inconsistent regarding the
importance of disturbance in nutrient-limited habitats and
need to reconsider the carbon economy of shade-tolerant
plants.
Tilman 1985
Reconciling the theories of Grime and Tilman: Craine (2005)
Craine, J.M. 2005. Reconciling plant strategy theories of
Grime and Tilman. J. Ecol. 93: 1041-1052.
•
Reconciling the approaches of Grime and Tilman leads to six scenarios for
competition for nutrients and light, with the outcome of each depending on the ability
of plants to preempt supplies.
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Under uniform supplies, pulses or patches, light competition requires leaf area dominance.
Nutrient competition requires root length dominance.
Craine has published extensively with Tilman so is hardly unbiased in his
reconciliation, which is strongly focused on competition for resources.
Examples of situations where plants use
environmental tolerance to avoid competition
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Serpentine soils,
– Low in essential nutrients, extreme pH, high in toxic elements
(e.g., Ni, Cr)
– Support unusual plants, often highly endemic floras
– Experimental evidence (e.g., Kruckeberg 1954) indicate that
although serpentine plant species often can grow better in
nonserpentine soils if grown without other species, they are
poor competitors when grown with other species.
•
Saline soils
– Halophytes can grow in soil with > 0.2-0.25% salt.
– Many have special structure whereby they secrete excess
salts.
– Examples include mangroves, coastal salt marsh species,
beach plants, desert herbs.
Amensalism
Interaction which depresses one plant population while the
other species remains unaffected.
• For example, the strongly negative effect that a large
species such as a tree might have on some small ground cover species.
Allelopathy
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A negative biochemical influence of higher plants upon
another species (usually inhibition of germinaiton or
growth) that is caused by the release of metabolic
substances under natural conditions.
Examples: several lichens, alders, Artemisia (sagebrush),
Larrea (creosote bush).
Allelopathy: Salvia leucophylla-grassland interface, Santa
Barbara, CA (Muller 1966)
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•
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Light bands around soft
chapperal (Salvia) are devoid
of plants.
Salvia emits volatile oils
(cineole and camphor).
Could this be due to seed
predators around the shrubs?
Allelopathy: Ceratiola ericoides (Williamson 1990)
•
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Florida chamise.
Halos around individual
plants.
Too small to harbor rodents
or other herbivores.
Other ways plants change their environment:
Effect of overstory and understory plants on soil properties
(Tappeiner and Alm 1975)
• Pines create very acidic soils that are toxic to many species of plants and soil organisms, including worms, and
many bacteria. Fungi tend to dominate the microflora in these soils, whereas bacteria dominate the more neutral
soils beneath deciduous forests.
• The above table shows the difference in some key soil properties of pine and birch forests. The pine forest have
lower pH, lower bulk density, lower soil nutrients, and slower litter and nutrient turnover times.
• There is some variation due to understory species, but this effect is relatively minor.
From Tappeiner and A.A. Alm. Undergrowth vegetation effects on the nutrient content of litterfall and soils in red pine and birch stands in northern Minnesota.
Ecology 56: 1193-1200.
Effect of canopy water throughfall on soil chemistry
Dramatic changes occur in
the chemistry of rainfall as it
passes through an oak
(Quercus petraea) forest
overstory.
• Most nutrients increased
because they are leached
out of the tree leaves.
Carlisle et al. 1966, cited in Barbour et al. 1999.
• N is somewhat reduced
beneath the trees
because of direct
absorption of N into the
tree leaves.
Summary
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Major types of competition: (1) interference competition (species directly interfere
with each other, e.g. allelopathy), (2) exploitation competition (mediated by
exploitation for a shared resource, most plant competition is of this type), (3)
apparent competition (mediated through a third species such as an herbivore).
•
Regular or clumped distribution patterns can be used to infer competition in some
cases.
•
Gauses competitive exclusion principle for animals and the Verhulst-Pearl equations
can be applied to plants in modeling situations, but in the real world, plants often
coexist because natural populations may not come into equilibrium very often, or
other interactions may limit the full competitive interaction between species.
•
Spatial and temporal variation in resource availability allows for the coexistence of
several species. This can be inferred using differences in dispersal abilities, or
differences in above- and below-ground allocation.
•
Tilman focused on resource competition as the basis for most competitive
interactions. His resource-ratio models are based on species relative abilities to
compete for resources.
•
Grimes models predict the strongest competition in high resource environments.
Plants able to convert resources to high growth rates are the best competitors in
these situations.
•
Allelopathy is an example of an amensal (0,-) interaction (or interference competition).
Many plants release allelochemicals that are inhibitory to the growth of other species.
Literature for Lesson 8
Craine, J.M. 2005. Reconciling plant strategy theories of Grime and Tilman. J. Ecol. 93:
1041-1052. http://www.blackwell-synergy.com/doi/full/10.1111/j.13652745.2005.01043.x?cookieSet=1#h8
Fonteyn, P.J. and B.E. Mahall. 1978. Competition among desert perennials. Nature 275:
544-545.
Grace, J.B. 1991. A clarification of the debate between Grime and Tilman. Functional
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