# Analysis of a model currently used for assessing sustainable VIPIH

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Analysis of a model currently used for assessing sustainable VIPIH

© Journal of The Royal Society of New Zealand, Volume 29,"Number 2, June 1999, pp 175-184 Analysis of a model currently used for assessing sustainable V I P I H in yield in inrlia<mmi« indigenous fnrpsts forests Murray Efford* Problems are identified in the use of a variant of the Usher matrix model to underpin the harvest of beech (Nothofagus spp.) and rimu (Dacrydium cupressinum) on the West Coast of the South Island. The model has been suggested as a means of determining the harvest that is sustainable in the sense of maintaining the present forest structure, and uses as inputs the size structure and estimates of size-specific radial growth rate. The model contains a bias because the equations for transition coefficients in the projection matrix assume an inappropriate geometric model for stage duration. The effect is to overestimate population growth by about 22% over a 15 year felling cycle. Alternative formulae are given for a more realistic model of fixed stage duration. The mortality rates needed to maintain the initial size structure of the population may be inferred from recursive formulae that are derived here separately for the geometric and fixed models of stage duration. Using a model with fixed stage duration it is found that the method is unworkable, in the sense that no set of mortality rates can be found to keep observed stand structures constant. The estimate of sapling recruitment number used in the harvest calculations does not appear to be well-founded. Even assuming that the forest is in a steady state and natural mortality is known, the assumption that harvest mortality substitutes for natural mortality rather than adding to it appears to be unwarranted. I suggest that matrix models cannot be used to determine ecologically sustainable forest management without additional information on natural forest dynamics and the response to harvesting. Keywords Usher matrix population model; sustainable harvesting; Nothofagus; Dacrydium cupressinum; sizestructured models; stage duration INTRODUCTION The New Zealand state-owned enterprise Timberlands West Coast Ltd (TWC) is seeking to mill old-growth Nothofagus and Dacrydium cupressinum forest according to a sustainable and ecologically sensitive regime. Models are used to determine the size-specific harvest of trees that is consistent with retaining the current demographic profile of the forest. The adequacy of the models is important not only because the TWC initiative is controversial (over 12 000 people responded in November 1998 to a call for public submissions), but also because it is seen by some as an exemplar for sustainable indigenous forestry on land in private ownership (Williams 1998). This note investigates the particular modelling approach used by TWC (1998 a,b,c) to determine whether it provides an adequate basis for ecologically sustainable adaptive management of West Coast beech forests. *Landcare Research, Private Bag 1930, Dunedin. [email protected] 176 Journal of The Royal Society of New Zealand, Volume 29, 1999 THE TWC MODEL Timberlands (TWC 1998 a,b,c) use a size-structured single-species matrix projection model (Usher 1966): 1 0 0 0 R2 a2 b2 0 0 0 «3 0 0 Nt,2 0 The population is described by a vector whose elements Ntj are the stem density (ha"1) at time t in the rth size class, i = 1 ...n. Trees are grouped by diameter at breast height (DBH) into 10 100 mm size classes (0-100 mm, 100-200 mm, ..., 900-1000 mm) and one open-ended class for trees over 1000 mm DBH. The number of trees in each class changes from year to year as some die naturally or are felled, and others grow across the boundary into the next size class. Rates of mortality (M,) and of radial tree growth are assumed to depend only on the size of the tree. The model (TWC 1998 a,b,c) assumes that trees are evenly distributed within size classes. On these assumptions, the diagonal* (a,) and subdiagonal (b,) elements of the projection matrix are given by b,=^Y (2) where a,- is the size-specific annual survival rate (1 -M,), (7, is the size-specific growth rate (mm.yr 1 ) and S is the size class width (mm). Intuitively, a fraction Gj I S of the surviving trees in class / are close enough to the upper boundary of the size class to grow into the next size class within one year. The product bjNj is the outgrowth from class i and the ingrowth to class i+\. Tree diameter growth rates may be estimated independently from increment cores (Table 1). Mortality rates are unknown, although it appears from Wardle (1984) that mortality in the first size class is an order of magnitude greater than among mature trees. Saplings are recruited to the smallest size class once they grow to a minimum height (1.4 m). In the Usher model the number of recruits into the smallest size class increases directly with the number of mature, seed-producing trees; the ratio of recruits to mature trees is a size-specific coefficient of regeneration. Timberlands' proposal adapts the Usher model by setting recruitment to a constant number: i=2 This avoids the problem of estimating each Rh although it is still necessary to estimate k (see Problem 4 below). With k held constant, the total recruitment is decoupled from the number of large trees. This has a stabilising effect on the overall dynamics of the model because per capita recruitment becomes 'compensatory' or 'density-dependent' (i?, a iVf1). The Timberlands papers (1998) do not identify this effect, but I note here that it is biologically * Here I assume that trees in the 0-100 mm class do not reproduce, and hence R\ = ax (cf Manly 1990 p. 106). Efford—TWC model for sustainable forestry plausible, at least over a range of densities near equilibrium and in comparison to a model with regeneration linearly related to adult tree number. The TWC procedure for estimating the sustainable yield entails finding the mortality rate in each size class that will maintain the present stand composition, and treating that mortality as a harvestable surplus. The permissible harvest of trees 300-1000 mm DBH is set initially at 50% of the calculated surplus to allow for uncertainty in the model inputs. 177 ° °5 oo • Problem 1. Bias arising from 10 20 30 40 50 assumption Of even distribution Stage duration (years) within size classes. The assumption of even distribution Fig. 1 Comparison of two models for stage duration within size classes yields a geometric i n a n Usher matrix model for varying mean stage distribution of residence time within each duration (growth rate"1) and within-stage mortality size class. (Residence time is frequently rate - Contours of percentage bias in outgrowth referred to as the 'stage duration' in the e s t i m a t e d from a m o d e l w i t h geometric stage duration literature). Growth is then equivalent to a ( E q n ? r e l a t i v ; c t o a ™ r e realistic model with fixed ,. .. , • , . , * ... . . stage duration (Eqn 3). The bias in outgrowth increases lottery m which every tree within a size ° , iU ^ , .. , , _ .. , , . , , . . with both stage duration and annual mortality, and class has an equal chance of progressing e x c e e d s , 0 % for a] , c o m b i n a t j o n s o f d u r a t i o n > 2 0 in a particular year. years and annual mortality >om However, a geometric distribution of stage durations is inconsistent with the underlying plant biology. Trees grow incrementally and, as far as TWC's proposal was concerned, deterministically, through each size class. It is therefore more appropriate to treat the duration of each stage as a fixed value equal to the time a tree takes to grow from the lower boundary to the upper boundary of the size class. The proportion of surviving trees that grow into the next size class in a particular year is then given by a more complicated expression that takes into account the uneven within-class distribution resulting from within-class mortality (Caswell 1989). At equilibrium (stationary size distribution and finite population growth rate A. = 1) the formulae are: where T, = S/G, and o, < 1. If the within-class size distribution is not stationary, then neither equation is reliable: a problem which may be solved by recasting the model in terms of age (e.g., Caswell 1989 p.85). The geometric and fixed models of stage duration give different projection matrices and hence different dynamic properties for the models. For particular values of mortality, growth rate and size class width, equation (3) yields slower outgrowth than equation (2) (Fig. 1). Slower outgrowth also exposes trees to prolonged high mortality in the juvenile size classes. 178 Journal of The Royal Society of New Zealand, Volume 29, 1999 The additional early risk seems unlikely to be balanced by delayed entry to the largest size classes where there may (or may not) be a senescent increase in mortality rate*. Population growth and harvestable yield will therefore be lower in a model with transition coefficients formulated more correctly than in the TWC model. The extent of this bias depends on the schedules of size-specific mortality and growth. It is most clearly measured by X, the asymptotic growth rate of the Usher model**. In order to examine the effect on X it was necessary to estimate the separate size-specific recruitment coefficients Rh and to provide plausible estimates of mortality. Approximate values of the Rs were obtained by distributing k (as estimated by TWC) in proportion to basal area (DBH2) over the existing stand structure (cf. Caswell 1989 p. 94). Mortality in the 0-100 mm and 100-200 mm classes was set as a fixed ratio of mortality in later classes (10 : 1 and VlO : 1 respectively; values chosen to reflect the observed decline in mortality with size (Wardle 1984)). All mortality rates were scaled to give*** X = 1 when transition coefficients were calculated with equations (3). This provided a benchmark for population growth under the improved model with fixed stage duration. The projection matrix was then recalculated for the TWC model with geometric stage duration using values of a,- and bj from equations (2) and the same underlying growth and mortality rates. For the resulting projection matrix X was 1.0134 per year, which corresponds to an increase of 22.2% in the abundance of each size class over a 15-year felling cycle as proposed by TWC. This is an approximate indication of the bias in estimates of population trend when transition coefficients are estimated with equations (2) relative to equations (3). The size of the bias depends somewhat on the distribution of mortality over size classes. Calculations were repeated for different mortalities in the first two size classes (0-100 mm and 100-200 mm) expressed as a ratio of mortality in later classes. When the ratios of initial to later mortality were 20 : 1 and V20 : 1, the overall bias was slightly increased (24.4% over 15 years). Conversely, relatively low initial mortalities (5 : 1 and V5 : 1 respectively) lead to a slight reduction in the bias (19.9% over 15 years). The estimate of bias therefore appears to be robust over the likely range of variation in mortality schedules. Problem 2. Fitting mortality rates to maintain a stable population structure The TWC model used a numerical algorithm to determine the annual mortality rate in each size class that will maintain the current stand composition. It is slightly more efficient to use an explicit recursive formula (Appendix 1; equation A4), but the end result is much the same (unpublished results). The problem arises when we try to repeat the calculations using the more biologically accurate fixed stage durations. The required mortality rates a,- (0 < a,- < 1, i - 1, 2,... n) are then real roots of the i equations T N: N: , n given a population vector, stage durations Tt and constant recruitment k\ (Appendix 1; ingrowth k, ,i>l, is estimated recursively). Using the stand structure of Maruia red beech as an example, some size classes / have no real solution to (4) in the range 0-1. In other words, * Variation in the rates of size-specific mortality is relevant here only if it is determined strictly by size rather than age. ** The asymptotic growth rate of the TWC model is 1.0 because k is fixed. *** A. is the dominant eigenvalue of the projection matrix. Efford—TWC model for sustainable forestry 179 there is no set of mortality rates that can maintain a stable tree population when combined with the given recruitment (k, = 57) and growth rates (Table 1). Intuitively, this happens when the annual outgrowth from one size class is inadequate to replace the outgrowth from the next class even when no trees die. Timberlands (TWC 1998a) suggest that the Maruia red beech have a visibly non-equilibrium size structure, and might argue that this is a special case. However, the problem is pervasive: there is also no solution for the main species in the Grey Working Circle (Nothofagus truncata), for which size-class frequencies (TWC 1998c) show a 'classic' reverse-J distribution (using k, = 33.1 calculated from TWC 1998c table 5.1.2, and values of G, in TWC 1998c table 5.4). Greater subdivision (e.g., by compartment) is likely to exacerbate the problem, as statistical sampling errors will then compound the lack of fit to stable size structures. The finding that many, perhaps most, existing stands cannot be reconciled as equilibrium stands with the TWC recruitment and growth estimates is not surprising. We might predict that this would be the case from the acknowledged role of occasional, widespread mortality in rejuvenating Nothofagus forests (e.g., Wardle 1984) and the slow approach of matrix models such as (1) to a stationary size structure (MGE unpublished results). Timberlands (TWC 1998a) assert that "... mortality tends to follow a pattern of minor fluctuation, punctuated by periodic large mortality events." Each Nothofagus stand we encounter is likely to be changing spontaneously, and we do not yet know how to diagnose its trend or predict its ultimate steady state. Timberlands' model attempts to estimate natural mortality by seeking size-specific mortality rates that are consistent with a steady state. When the population is not in a steady state this is likely to fail, as shown above. Nor can we be sure if estimates are obtainable that the population is actually in a steady state, and that the 'estimated' rates are therefore meaningful. With the TWC methodology, poor estimates of natural mortality must translate directly into wrong estimates of sustainable yield. Table 1 Parameter values for size-structured matrix model of Nothofagus fusca in the Maruia Working Circle. Diameter growth rates (G) and observed stand composition from TWC 1998b, Tables 5.1, 5.2; figures in parentheses show linear extrapolation of growth rate to older classes. Recruitment coefficients (R) were varied in proportion to basal area so that the total number of recruits kt = 57 for the observed stand composition; mortality rates M were adjusted to give A, = 1.0 for the Usher matrix calculated using equations (3) (see text for further detail). Size class mm 0-100 100-200 200-300 300^00 400-500 500-600 600-700 700-800 800-900 900-1000 >1000 Observed stems ha"1 G mm.year' R M 742 61 29 25 21 22 15 11 7 4 6 2.56 2.55 2.85 3.11 3.19 2.88 2.62 2.32 2.04 (1.75) (1.47) 0.029 0.079 0.155 0.257 0.383 0.536 0.713 0.916 1.144 1.398 0.0748 0.0236 0.0075 0.0075 0.0075 0.0075 0.0075 0.0075 0.0075 0.0075 0.0075 180 Journal of The Royal Society of New Zealand, Volume 29, 1999 Problem 3. Is logging mortality additive? Suppose that natural mortality is known. Is TWC correct to claim that a harvest of up to this size is ecologically sustainable in the sense of not altering forest structure? Putting aside salvage harvest, which will usually be a small fraction of the total (e.g., TWC 1998b 5.2.2), the claim is based on the idea that harvesting 'pre-empts' natural mortality. Pre-emption is not defined in the context of the model, but it appears to mean either that harvesting is restricted to trees that are about to die anyway, or that for every tree harvested another tree that would have died in the same year actually survives*. To evaluate this we need to think of the agents of natural mortality: for established trees these are most likely weather (wind throw, snowfall, drought), geomorphology (slips, earthquakes) and pathogens (insects and fungi, probably in combination). Local tree density may moderate mortality from these agencies, but it is implausible that low local density provides absolute protection, or that the death of individual mature trees may be anticipated on a 15-year logging cycle. There are even concerns that logging may elevate mortality from pinhole borer {Platypus spp.) and wind throw among unlogged neighbouring trees, rather than decrease it. The argument becomes even less plausible when we consider the effect of logging on natural mortality of trees distant from logging coupes. For geometrical reasons these form the majority of the natural mortality in any year, but it is barely conceivable that their deaths can be prevented by logging. We must conclude that logging imposes mortality that is largely additional to natural mortality in any one year. Timberlands plan initially to harvest at the rate of half their estimate of natural mortality to provide a 'safety margin'. Clearly this will result in slower change to the forest than if the harvest equalled estimated natural mortality. However, the planned harvest is no more likely to be fully compensated by a reduction in natural mortality: a smaller response is required, but the stimulus is also smaller. The consequential effects of reduced competition for light and nutrients will no doubt increase growth rates among neighbouring survivors. How these responses feed through into population processes is a more subtle question that requires, at the least, some functions that relate tree performance to local density. Such density dependence is missing from the TWC model, which therefore cannot predict how forest structure will change under harvesting. Problem 4. Recruitment The estimation of recruitment is a problem area acknowledged by TWC, and which they set as a research priority (TWC 1998a). Some comments may be helpful. The seedling class (height <1.4 m) is omitted from the matrix model, although it might well be included. All that is required is to add an initial extra row and column to the projection matrix and to cast S (class width) and G (growth rate) in terms of height when calculating a,- and Z>, with equations (3). Here I use the subscript 0 for the seedling stage. Recruitment (as used by TWC) is the ingrowth to the sapling class (0-100 mm DBH). It is estimated by TWC from the number of seedlings No, the number of saplings JVJ, and the duration of the seedling stage To = SQ I GQ, where SQ and GQ are defined in terms of height: k From the numerical examples in the Maruia Plan, OQ appears to have been estimated from the ratio of saplings to seedlings: * There is a trivial sense in which harvest mortality must pre-empt natural mortality, as a tree cannot die twice. Efford—TWC model for sustainable forestry N, , N, 181 1000- a 0 = —l- => h = —L We can compare this estimate with one from an appropriate application of (3) to describe b§. E q u a t i o n (5) identifies a range of recruitment values equally consistent with the inventory data but differing in the unknown level of seedling mortality (Fig. 2). No particular value of Arj is implied by the data unless we have independent information on survival rate. This result raises two points. Firstly, the way TWC (1998 a,b,c) obtains its estimate of recruitment numbers is arbitrary other estimates are equally compatible with the existing data (Fig. 2). Secondly, using the current number ' ,,. • , , . of seedlings as an input merely begs the 00 D1 a2 Annual 04 °3 mortality of seedlings F »g- 2 Estimated recruitment to sapling class as a function of mortality in seedling class. Curve shows values consistent with inventory and growth data presented by TWC for Maruia red beech (Wo= 23634; r =U "> \7f' ° \ R e c r u i t m e n t m u s ' l i e outside the hatched area to replace the outgrowth from the sapling . .-., •*_.,. • ° ..• - ° „ class if the size structure is stationary. The upper question of recruitment to the seedling ^ ^ rf^ h a t c h e d ^ ^ = N/r;) c o r r e s p o n d s c ass ^ ' to zero mortality in the sapling class; recruitment Recruitment and early mortality are a b o v e t h i s threshold must be balanced by positive rapid processes compared with those mortality of saplings, acting in later stages of the life cycle, and self-thinning and compensatory growth introduce substantial early density dependence. The duration of the seedling stage is particularly variable, providing the 'advance crop' of potential recruits to the sapling stage (Wardle 1984). Constant recruitment number is a plausible interim solution near equilibrium. Unless density dependence is introduced elsewhere in the model, the equilibrium frequency of all succeeding size classes varies in proportion to recruitment number. The assumption that recruitment is constant and non-limiting at low densities of mature stems therefore needs to be used with caution. More particularly, a valid estimate of k\ should be obtained, presumably by tagging saplings. DISCUSSION This paper has been critical of the matrix modelling used by TWC (1998 a,b,c) to underpin their assertion that the particular harvest levels they have chosen and are now implementing will not cause appreciable change in the structure of existing West Coast beech forests. Formulae used by TWC for the transition coefficients of the model imply a geometric distribution of stage durations which is inconsistent with the growth biology of the trees. More appropriate formulae that assume a fixed stage duration are provided here, following Caswell (1989). Calculations with plausible parameter values show that population growth is substantially overestimated by the TWC transition coefficients. For example, a stable forest 182 Journal of The Royal Society of New Zealand, Volume 29, 1999 would be misinterpreted as growing at a rate of 22% per 15 years, the duration of the proposed felling cycle. This difference is similar in size to the harvestable increment as calculated by TWC (1998 a,b,c). Further logical errors were identified in procedures used by TWC to infer the harvestable yield from size composition and growth data. Their procedure is based on determining the mortality rates that must prevail to maintain the forest at its present composition, and treating this as a harvestable surplus. Under the more biologically realistic model with fixed stage durations, the desired mortality rates may be found as the roots of a polynomial. However, for forests with certain initial size compositions there is no set of size-specific mortality rates that can maintain the present forest structure. This applies to the main species of beech in both the Maruia and Grey Working Circles. The proposed rule for determining harvest is therefore inoperable. A more fundamental logical problem arises from the implicit belief that harvesting pre-empts natural mortality, and is therefore not additive. This cannot strictly be true, as the harvest in any year will be focussed on a small fraction of the total area, whereas natural mortality will be dispersed or at least unpredictable. Other problems could be raised regarding interspecific competition, estimation of growth rates, and the interaction between harvesting and time-varying natural mortality. However, sufficient evidence has been presented to confirm that the 'sustainability' of the proposed harvest cannot be said to have a demonstrated scientific basis, particularly when evaluated against the accompanying claim that the existing structure of the forest will be maintained. The industry's response is essentially that they will 'learn by doing' in the spirit of adaptive management. Their plans include monitoring of forest change and improved estimation of model parameters. However, this is 'adaptive management' only in a weak sense: the models on which the work plan is based are used uncritically and no hypotheses are tested (cf. Walters 1986). Milling of old-growth forest trees between 100 and 400 years old is irreversible within human lifetimes. It is therefore extremely unclear how the philosophy of adaptive management can validly be applied. Since harvesting experiments will not be conducted on 400-year time scales, long-term effects must be inferred non-adaptively from suitably constructed and fitted models. The present TWC models appear to be inadequate for this purpose. More robust modelling would allow informed public discussion of the acceptability or otherwise of the changes in forest structure that are likely to follow from harvesting. The bias identified as Problem 1 is a hazard endemic to size-structured models that rely on indirect estimation of transition coefficients. For example, Vanclay (1995 p. 13) noted that "Simple stand projection and matrix approaches may allow some stems to move n classes in n projection intervals, and may thus overestimate yields". Usher (1966) envisaged that the a, and b, would be obtained by remeasurement of the entire stand to determine the actual growth from one class to the next over a felling cycle. It is to be expected that field estimates from a stationary stand will approximate those obtained from (3), although technical problems remain (Kohyama & Takada 1998). Some divergence is to be expected because trees of the same size differ in growth rate. Caswell (1989) described modifications of (3) for the more general case of stochastic, non-geometric, stage durations, but we cannot be sure that these capture the likely correlations between individual growth rate and mortality etc. Individualbased models may be required (Liu & Ashton 1995; Pacala & Deutschman 1995; Vanclay 1995). ACKNOWLEDGMENTS I am grateful to Graeme Hall, Bill Lee, Bryan Manly, Sue Maturin, Jake Overton, the editor and two anonymous referees for their suggestions and comments on a draft. Part of this work arose out of a contract to the Royal Forest and Bird Protection Society N.Z. Inc. Efford—TWC model for sustainable forestry 183 REFERENCES Caswell, H. 1989: Matrix population models: construction, analysis and interpretation. Sunderland Mass., Sinauer. xiv+328 pp. Kohyama, T.; Takada, T. 1998: Recruitment rates in forest plots: Gf estimates using growth rates and size distributions. Journal of Ecology 86: 633-639. Liu, J. G.; Ashton, P. S. 1995: Individual-based simulation models for forest succession and management. Forest Ecology and Management 73: 157-175. Manly, B. F. J. 1990: Stage-structured populations: sampling, analysis and simulation. London, Chapman and Hall, xii+187 pp. Mathsoft, 1997: S-PLUS 4 User's Guide. Seattle, Mathsoft. xiv+620 pp. Pacala, S. W.; Deutschman, D. H. 1995: Details that matter: the spatial distribution of individual trees maintains forest ecosystem function. Oikos 74: 357-365. Timberlands West Coast Ltd 1998a: West Coast Beech Plan - Overview. Available: http:// www.timberlands.co.nz [1998, Nov. 8]. Timberlands West Coast Ltd 1998b: West Coast Beech Plan - Maruia. Available: http:// www.timberlands.co.nz [1998, Nov. 8]. Timberlands West Coast Ltd 1998c: West Coast Beech Plan - Grey. Available: http:// www.timberlands.co.nz [1998, Nov. 8]. Usher, M. B. 1966: A matrix approach to the management of renewable resources, with special reference to selection forests. Journal of Applied Ecology 3: 355-367. Vanclay, J. K. 1995: Growth models for tropical forests: a synthesis of models and methods. Forest Science 41: 7-42. Walters, C. 1986: Adaptive management of renewable resources. New York, Macmillan. 374 p. Wardle, J. A. 1984: The New Zealand beeches: ecology, utilisation, and management. Christchurch, New Zealand Forest Service. 447 p. Williams, J. M. 1998: Timberlands West Coast Ltd. Beech management prescriptions. Responses to advice from the Independent Review Panel. PCE Progress report No. 2. Wellington: Office of the Parliamentary Commissioner for the Environment. R99002 Received 3 February 1999; accepted 18 March 1999 APPENDIX 1 Calculation of mortality rates to maintain present stand composition. TWC adaptation of Usher model (constant recruitment number). For each size class we need to know the mortality that will cause the sum of outgrowth and mortality to exactly balance ingrowth, or recruitment in the case of the first class. This can be expressed in general terms as 'ingrowth = mortality + outgrowth', or k, = Niil-od+NitTib,, (Al) where k, represents ingrowth (or recruitment to size class /)>Nt is the current number of trees in the ;-th size class, o ; is their survival rate (1 - M, where M is mortality), and bt is a coefficient of outgrowth. We would like to rearrange this equation to obtain an expression for a, or equivalently for M. This will depend on the model used for stage durations, which determines whether (2) or (3) is used for the bt. In each case we assume that the within-class size distribution is at equilibrium, and that initial recruitment (k\) is a given. Outgrowth from one size class becomes ingrowth to the next, so we can recursively express each kt in terms of the numbers, growth rate and estimated survival rate in the previous class /-/: kj = Nl_]C,_xbi.ui = 2...n (A2) a. Geometric distribution of stage durations Substituting from (2) for b, in A2: (A3) And similarly by substituting for b, in (Al) and rearranging: 184 Journal of The Royal Society of New Zealand, Volume 29, 1999 b. Fixed stage durations Substituting from (3) for b, in A2: '~lTi J - ' ki=Ni_xai_x i = 2...n (A5) i—\ Similarly, by substituting for b, in (Al) and rearranging: T N, K I N, . n K I Equation (A6) is a polynomial in o, that may or may not have a real root in the range 0 < o, < 1. The required value of a, may be found by solving (A6) numerically (e.g., function uniroot() for a real root in [0,1), Mathsoft 1997). Although (A6) always has a root at o = 1 this is not a valid value in (2) and should be discarded.