Elsevier

Journal of Theoretical Biology

Volume 360, 7 November 2014, Pages 46-53
Journal of Theoretical Biology

A stochastic Pella Tomlinson model and its maximum sustainable yield

https://doi.org/10.1016/j.jtbi.2014.06.012Get rights and content

Highlights

  • Pella–Tomlinson and Fox surplus production models with environmental noise are considered.

  • The maximum level of noise that a stock can tolerate before extinction is derived.

  • New formulas for the maximum sustainable yield (MSY) that account for the random noise are provided.

  • Models with a large asymmetry parameter p are most sensitive to environmental noise.

Abstract

This paper investigates the biological reference points, such as the maximum sustainable yield (MSY), for the Pella Tomlinson and the Fox surplus production models (SPM) in the presence of a multiplicative environmental noise. These models are used in fisheries stock assessment as a firsthand tool for the elaboration of harvesting strategies. We derive conditions on the environmental noise distribution that insure that the biomass process for an SPM has a stationary distribution, so that extinction is avoided. Explicit results about the stationary behavior of the biomass distribution are provided for a particular specification of the noise. The consideration of random noise in the MSY calculations leads to more conservative harvesting target than deterministic models. The derivations account for a possible noise autocorrelation that represents the occurrence of spells of good and bad years. The impact of the noise is found to be more severe on Pella Tomlinson model for which the asymmetry parameter p is large while it is less important for Fox model.

Introduction

Surplus-production models (SPM) are dynamic models especially useful to study the evolution of the biomass of a fishery and to estimate the maximum sustainable yield (MSY), which is the maximum catch that does not jeopardize the long term survival of the population. They are the simplest models available that can assess fish stocks and determine a harvest rate because they regroup recruitment, mortality and growth into a single production function. Furthermore, they require very few data in comparison with more complex models such as age-structured models; only time series of stock abundance indices and the corresponding catches are needed. Accordingly, they can be used when the data is limited (Chaloupka and Balazs, 2007, Hilborn and Walters, 1992). Even if criticized for being less realistic than age-structured models, SPM are still helpful since they can sometimes provide results as useful as those obtained with more complex models at a lower cost (Ludwig and Walters, 1985, Laloe, 1995, Chavance et al., 2002). These models are also widely used for fishery management (Panhwar et al., 2012, Prager, 2002). Although this paper focuses on discrete-time models, similar results could be obtained with continuous time as illustrated in Ewald and Wang (2010).

To each SPM is associated a production function f(·) which gives the stock production at time t+1 as a function of the stock biomass at time t. The stock biomass difference between time t+1 and t is the difference between the production of the stock biomass and the catches at time t:Bt+1Bt=f(Bt)Ct,where Bt is the stock biomass at time t, Ct is the catches at time t. Each parameter of the population, except for the catches, is included in the production function. In this paper we focus on the Pella and Tomlinson (1969) model whose production function is given byf(Bt)=p+1prBt(1(BtK)p),where r(0,1) characterizes the population growth, K the carrying capacity (the maximum population size for growth to be positive) and parameter p>0 allows to introduce an asymmetry in the production curve.

Two interesting special cases are p=1, that gives Schaefer׳s model (Schaefer, 1954) with a symmetric production function f(Bt)=2rBt(1Bt/K), and the limiting case when p tends to 0 corresponds to the Fox model (Fox, 1970) whose production function is f(Bt)=rBtlog(Bt/K).

In a deterministic model, the MSY is the maximum over B of f(B); it is the maximum surplus production. The management strategy is to keep the stock biomass equal to the value that maximizes the surplus production and to harvest the latter. For a Pella and Tomlinson model, the biomass which maximizes the production function is BMSY=K/(p+1)1/p, the MSY is rK/(p+1)1/p and the optimal fishing rate is FMSY=MSY/BMSY=r. Letting p go to 0, the corresponding characteristics for Fox model are BMSY=K/e, MSY=rK/e, and FMSY=r, where e=2.718 is Euler׳s number. Notice that in each case, the deterministic optimal fishing rate is equal to r; this is a consequence of the parametrization of (2).

Deterministic surplus production models have been criticized because of their overly optimistic estimation of MSY (Larkin, 1977, Boerema and Gulland, 1973, Lewis, 1981, Amundsen and Bjorndal, 1999). Doubleday (1976) advised to set harvesting quotas as a fraction (two third) of the deterministic MSY to reduce the risk of stock collapse and suggests that stochasticity should be taken into account. To pursue this objective, we will incorporate a positive multiplicative random noise εt (a process error in the terminology of Punt, 2003) with variance σ2 (Gore and Paranje, 2001) to reflect the natural variability. This leads to the following stochastic model:Bt+1=[Bt+p+1prBt(1(BtK)p)ϕBt]εt,where ϕ is the fishing rate so that, in (1), Ct=ϕBt. The deterministic equilibrium assumption (Bt+1=Bt) then becomes a stochastic equilibrium assumption, namely that the distribution of Bt+1 is the same as the distribution of Bt.

Suppose that a stock experiences an environmental shock that results in an important drop of its biomass. It then grows according its average intrinsic annual growth rate, in the terminology of Reed (1978). For model (3) this growth rate is γ=(p+1)r/pϕ. If p is large, it is rϕ and the stock will take much longer to recover from this unexpected drop than when p is small. This suggests that, in the Pella Tomlinson SPM, the sensitivity to environmental shock increases with p. For the model of Fox, the near zero growth rate is rlog(Bt/K); it goes to as Bt goes to 0 and is much larger than for a Pella Tomlinson model. Thus the Fox model might not be as affected by a random environment as it postulates a larger growth when the stock is small. An objective of this work is to investigate this question formally and to characterize the sensitivity to environmental shocks of these models in the stochastic framework given in (3).

Thus our first objective will be to investigate conditions leading to a stochastic equilibrium for the Pella Tomlinson process, insuring the long term survival of a stock. Reed (1978) showed that a constraint on the noise distribution is necessary for the non-extinction of the population in models such as (3). An upper bound to the noise variance, specific to that model, that insures that a stochastic equilibrium exists and that extinction is avoided is proposed. Reed (1978) also compared the long-run average yield of different harvest policies in a stochastic framework. He found that the deterministic approach was overestimating the available resources. His work has been complemented by Bousquet et al. (2008) who focused specifically on the Schaefer model. The second objective of that work is to devise harvesting strategies for the Pella Tomlinson model that take into account the environmental noise.

This paper generalizes the results from Bousquet et al. (2008) to the Pella Tomlinson model (3), thereby a similar framework will be used: first we study the stochastic process defined by (3) and derive conditions on the environmental noise to insure the existence of a stationary distribution. Second, we exhibit a distribution for the innovations εt that leads to a closed form expression for the stationary distribution and derive a bound on the innovation variance, which is a sufficient condition to the existence of the process. Then we investigate the impact of the innovation׳s variance on the optimal reference points by deriving stochastic versions of MSY and FMSY. These calculations are presented in a general context where, besides the innovation variances, the effect of an autocorrelation between innovations is also investigated. This autocorrelation could be associated with spells of good and bad years for the stock. The limiting Fox models is also considered and results, specific for that model, are presented. As an illustration we use the results of the analysis presented in Chaloupka and Balazs (2007) about an Hawaiian stock of green sea turtle to evaluate the impact of an environmental noise on a an harvesting strategy constructed using model (3).

Section snippets

Description and stationarity

This section focuses on the study of the stochastic Pella Tomlinson model (3). We exhibit a particular model for the innovations that leads to a closed form for the stationary distribution of (3). We study the tolerance to environmental noise of this process. For given ϕ and r, we derive an upper bound for the innovation variance that insures the stationarity of the model, viz. the long term survival of the stock.

Without loss of generality, we can assume that, in (3), E(εt)=1. Indeed, if E(εt)=μ

Derivation of biological reference points for stationary SPM processes

When Bt is a stationary process, the optimal harvest rate is the value of ϕ that maximizes ϕE(Bt). This fishing rate is optimal on the long run; it maximizes the average harvest over several years of fishing. Besides its expectation, which is assumed to be equal to 1, the only assumption on the environmental noise εt in this section is that its variance σ2 is small enough for the process Bt to have a non-trivial stationary distribution. The objective is to derive an o(σ2) approximation to the

Illustration

To get a better understanding of the impact of a random environment on the calculations of biological reference points, consider the Hawaiian green sea turtle stock of Chaloupka and Balazs (2007) studied in Section 2.1, with p=2.97,r=0.035 and σ^2=0.019. Assume a deterministic Pella Tomlinson model leads to the following reference points: MSY(0)=41.0t,E[BMSY](0)=1171.1t,FMSY(0)=0.035.Theorem 3.1 gives the following approximations for the optimal reference point that account for a random

Discussion

This paper has studied the impact of a random environment on surplus production models. Two types of impact have been studied in this work. The maximum innovation variance that a stationary process can tolerate before becoming extinct has been derived. Then the first order contributions of that variance to the biological reference points have been calculated. We have found that Pella and Tomlinson models with large p, e.g. where the maximum surplus production is more that 50% of the carrying

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