Control of structured populations by harvest

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Abstract

It has long been recognized that demographic structure within a population can significantly affect the likely outcomes of harvest. Many studies have focussed on equilibrium dynamics and maximization of the value of the harvest taken. However, in some cases the management objective is to maintain the population at a abundance that is significantly below the carrying capacity. Achieving such an objective by harvest can be complicated by the presence of significant structure (age or stage) in the target population. In such cases, optimal harvest strategies must account for differences among age- or stage-classes of individuals in their relative contribution to the demography of the population. In addition, structured populations are also characterized by transient non-linear dynamics following perturbation, such that even under an equilibrium harvest, the population may exhibit significant momentum, increasing or decreasing before cessation of growth. Using simple linear time-invariant models, we show that if harvest levels are set dynamically (e.g., annually) then transient effects can be as or more important than equilibrium outcomes. We show that appropriate harvest rates can be complicated by uncertainty about the demographic structure of the population, or limited control over the structure of the harvest taken.

Introduction

Age- or stage-structure has long been included in the modelling of harvested populations, particularly fish and forests (Getz and Haight, 1989). Early models were linear and deterministic Beddington and Taylor, 1973, Doubleday, 1975, Rorres and Fair, 1975, progressing later to models that included density dependence Reed, 1980, Getz, 1980, Jensen, 1996, Jensen, 2000, seasonal effects (Getz, 1980), environmental effects on vital rates Milner-Gulland, 1994, Pascual et al., 1997, Xie et al., 1999, and spatial structure (Brooks and Lebreton, 2001). Optimization of harvest is usually carried out with the purpose of maximizing sustainable yield, and so equilibrium dynamics are a common focus.

As more complicated models have been developed, it has become common to simulate and observe dynamics under different scenarios as an alternative to formal optimization. Other characteristics of harvest strategies have been considered, such as variance in yield Milner-Gulland, 1994, Sæther et al., 2001, harvest structure Ginsberg and Milner-Gulland, 1994, Jensen, 1996, population structure Milner-Gulland, 1994, Jensen, 1996, Jensen, 2000, Xie et al., 1999, abundance Ginsberg and Milner-Gulland, 1994, Milner-Gulland, 1994, Pascual et al., 1997, Xie et al., 1999, Sæther et al., 2001, Mayaka et al., 2004, and transient dynamics (Jensen, 1996).

Only a small number of studies have attempted to optimize harvest when the initial demographic structure of the population is markedly different from the equilibrium structure. In such cases, transient dynamics are of particular importance (Fox and Gurevitch, 2000). When sequential harvest decisions are made over time, transient dynamics may have a large influence on observed population change between decisions, and so must be accounted for during the decision-making process. Harley and Manson (1981) were the first to discuss how a structured population might be most efficiently brought to equilibrium in a finite number of steps. A small number of other studies Stocker, 1983, Milner-Gulland, 1997, Hauser et al., 2005 have used stochastic dynamic programming to determine the optimal harvest decision for any possible initial population structure and abundance. However, these studies have divided the target population into only a small number of classes, and have generally assumed complete knowledge of the structure of the population at the time of the harvest. While modelling a small number of classes may have been adequate for these studies, incorporating more classes is likely to be limited by data constraints and the computational method.

Even in the absence of dynamic decision-making, population momentum can be used as a measure of the long-term effect of transient dynamics. It was first noted by Keyfitz (1971), but has only recently been discussed in the context of population management Caswell, 2001, Koons et al., 2005a, Koons et al., 2005b. While a management action such as harvest can reset the asymptotic growth rate of a population to replacement only, transient dynamics may cause the population to grow or decline before it reaches equilibrium.

In this paper, we will explore the harvest of a structured population with the purpose of control. Structured models have rarely been used to optimize harvest for control (but see Brooks and Lebreton, 2001). We assume that the desired level of control maintains the population at a steady abundance well below carrying capacity, so that a linear (density independent) matrix model is appropriate. This model has been considered frequently in the past Beddington and Taylor, 1973, Doubleday, 1975, Rorres and Fair, 1975 but usually with the purpose of maximizing yield under equilibrium conditions. Instead we will consider the structure of the harvest taken, the structure and abundance of the population, and the possible effects of population momentum. We discuss our results in the context of uncertainty, outlining potential challenges in meeting the control objective (sensuHunter and Runge, 2004).

Section snippets

The model

For the unharvested population, we use the matrix modelNt+1=ANt,where Nt is a vector giving the number of individuals in each stage at time t and A is the population projection matrix (Caswell, 2001). We will assume that Ais primitive (aperiodic), so that it has a real positive eigenvalue λ which is greater in magnitude than all other eigenvalues. This eigenvalue gives the asymptotic growth rate and we will assume that the population is growing so that λ>1.

The associated left and right

Equilibrium condition: constant harvest

When the population is in equilibrium, it is neither growing nor declining. When our objective is to maintain the population at a particular abundance, the equilibrium condition indicates what this abundance can be. We find the equilibrium condition for the model with constant harvest by removing the time dependence in Eq. (3):Neq=ANeqYeq.Rearrangement of this equation gives the steady stage-structured population stateNeq=(AI)1Yeq.Since λ1 then the inverse (AI)1 exists, but we must also

Equilibrium condition: proportional harvest

We obtain the equilibrium condition under proportional harvest by removing the time dependence in Eq. (4):Neq=(IHeq)ANeq.Note that this is an eigenvalue equation in the form of (2). We wish to find harvest Heq such that (IHeq)A has a dominant eigenvalue of 1. That is, we set the long-term growth rate of the population to 1, replacement only. Then the steady population structure is given by the corresponding right eigenvector Neq.

If the initial population is given by vector N0, then we can

Transient dynamics and population momentum

We have seen in our example that given an initial population of 1000 individuals, different harvest strategies can produce different short-term and long-term results even if all harvest strategies involve the repetition of a harvest that satisfies the equilibrium condition. Population momentum is one measure of the long-term outcome of managing using the equilibrium condition. It is defined asM=limt|Nt||N0|,where |N| denotes the total number of individuals in N(Caswell, 2001). This is the

Partial control and observation

Let us briefly return to the unstructured model (10). To compare this model with our example involving three stage-classes, we set N0=1000 and λ=1.309. From the equilibrium condition (11) we might choose to remove 300 individuals from the population, with the intention of allowing the population to grow slightly over the next time step (N1=1009). If the population actually has the life history characteristics given in (12), then we might observe the population to decline over the next time step

Conclusions

Harvest management is never a simple task. The challenges of harvest management are compounded when structured populations are considered. We have shown that when populations are structured (based on age or some other demographic variable), the harvest required to achieve an equilibrium objective depends on initial population size, structure and reproductive value at the time of harvest.

Although derivation of the equilibrium harvest vector for simple structured models is straightforward, there

Acknowledgements

We would like to thank the members of the Adaptive Management Conference Series (in particular J.D. Nichols, F. Johnson, M. Runge, C. Fonnesbeck, and B.K. Williams), Hugh Possingham (University of Queensland), David Koons (Auburn University) and two anonymous reviewers for comments at various stages during the development of this paper. We also thank the New York Department of Environmental Conservation and the University of Queensland Graduate School, for providing partial financial support of

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