Elsevier

Fisheries Research

Volume 243, November 2021, 106047
Fisheries Research

Fisheries management under incomplete information by optimal stochastic control and Hidden Markov Model filter

https://doi.org/10.1016/j.fishres.2021.106047Get rights and content

Abstract

This paper deals with mathematical modelling of strategies in harvesting of fish populations. Since too aggressive harvesting may have severe consequences, a cautious control of the process is called for. The determination of optimal harvesting based on the biomass size is made even more challenging since there is considerable uncertainty in the estimation of the stock size, the so-called incomplete information problem.

Fishery management is in this paper established as an optimal control problem for a model based on nonlinear stochastic differential equations, with economic performance as objective. The so-called certainty equivalence principle, by which the estimate is used for the purposes of optimal feedback control as if it were the certain value of the state variable, is adopted. Markov controls are identified by solving the stationary Hamilton-Jacobi-Bellman equation. State estimates are obtained by a Hidden Markov Model filter, where the forward Kolmogorov equation governs the temporal evolution of estimates with the uncertain quantities. Fishery profit over time as economic performance is computed by Monte Carlo simulations, in order to compare the performances of the strategies considering control rules and precautionary approach.

Two control rules in harvest policy – harvest control rule and effort control rule – are compared, with respect to their robustness given the uncertainty. The effort control rule produces a significantly higher cumulated profits, and is less sensitive to the uncertainty in stock assessment present in the system. Moreover, a precautionary approach taking the uncertainty associated with biomass estimates into account, can be achieved via quantile estimators. This approach produces an economic gain by making a more appropriately cautious decision, and leads to sustainable harvesting of fish resources.

Introduction

There is an growing demand for consumption of natural resources, such as renewable resources like fish. Overfishing can lead to economic collapse of fisheries and even biological collapse of the stock. The collapse of Northwest Atlantic cod stock is an example, where mortality due to fishing is a factor in the failure of the stock to recover (Rice and Rivard, 2003). Optimal control is a useful tool in the analysis of many real-world dynamical systems. In establishing a control model, the type of information available to the decision-maker at each instant time plays an important role. A standard assumption is that variables are observed completely. However, this is rarely the case in real-world situations. Fishery resource management is subject to a variety of unobservable processes, which is a challenge for fisheries science, when aiming to maintain the stock for sustained future harvesting.

Fish population dynamics is reasonably well described as a stochastic process. The problem of maximizing expected sustainable yields is studied in Ewald and Wang (2010), using an approach called the mean-variance analysis of sustainable yields. Fisheries management can be posed as a Markov control problem: how to choose the catch rate. Such an optimization problem can be solved by the stationary Hamilton-Jacobi-Bellman equation, as presented in Thygesen (2016). This is treated as a stochastic optimal control problem under complete information, where the state of the system can be directly measured. In comparison, its counterpart under incomplete information is substantially more difficult. The sources of random variation in the fishery are mentioned in Roughgarden and Smith (1996). Several sources of uncertainty are important factors in the problem of overfishing. One could be variability in fish dynamics, which comes from the environmental variations resulting in fluctuations of parameters. This kind of problem might be serious for more complicated models. One such mathematical model is studied in Yoshioka et al. (2019) where the aim is to find the optimal harvesting policy. Here, the incompleteness of information is due to uncertainties involved in the body growth rate of the fish resource. Another source of uncertainty could be inaccurate stock size estimates, which means that the sampling error leads to an incorrect catch quota. Using dynamic programming as a tool, it is claimed in Sethi et al. (2005) that uncertainty in measurements has the greatest potential to affect fishery policy.

With these considerations in mind, a single species biomass model is formulated in this paper as a stochastic surplus production model, and incomplete information in the form of uncertain assessments of the stock is selected as the key uncertainty. The certainty equivalence principle (Sethi, 2019, Simon, 1956) is adopted. It emphasizes the fact that the estimate is used for the purposes of optimal feedback control as if it were the certain value of the state variable. Here a Hidden Markov Model (HMM) filter (Thygesen, 2016) is chosen as an approach to estimate the states in the nonlinear exploited fisheries models. Based on stochastic differential equations, there are several estimation methods, each having pros and cons. A HMM filter has the advantage of supplying an accurately estimated distribution. This is a full posterior distribution, and therefore it can be applied to most problems. In particular, the method is superior for nonlinear filter problems. The optimal control strategy is identified by applying dynamic programming, which reduces to solving the stationary Hamilton-Jacobi-Bellman (HJB) equation just mentioned.

Certainty equivalence means that the full information control law is applied with the certain state replaced by its estimate, from different choices of estimators. In this way, the control is influenced by the uncertainty on estimate and further on to the system performance. The relative performance of different control rules to meet the fishery objective are affected differently. The fishery profit will in this paper be computed from specified control rules: harvest control rule (HCR) controlling the catch rate or effort control rule (ECR) regulating fishing mortality rate. Each control rule corresponds to a control rule function. In practice, the most important yield indicator of HCR is the landed catch, also called landings. ECR can be conducted via limiting inputs such as number of participants, and fishing time restriction, i.e. number of days at sea (Committee on Fish Stock Assessment Methods, 1998). Moreover, the so-called precautionary approach1 has emphasis on attempting to react with caution to uncertainties, when deciding the optimal fishing. It strives to be more cautious when information is less certain (Hammill and Stenson, 2007). From a mathematical point of view, precautionary approach can be achieved by estimators, which are considered in this paper as well. Mathematical questions like how uncertainty on estimates will affect the optimal harvesting, and how precautionary an approach is optimal are of importance. Specifically, this paper will address:

  • which control rule function is more robust with respect to uncertainty;

  • how various estimators can mitigate the loss due to incomplete information by optimally incorporating the uncertainty into the control mechanism.

The rest of the paper is organized as follows. In Section 2, the mathematical model for fishery management as a stochastic control problem under incomplete information is formulated. The process noise and the measurement noise, as well as the objective as the expectation of cumulated profit will be considered. In Section 3, based on a nonlinear stochastic differential equation the Hidden Markov Model filter with control will be introduced. After this, the control rules and the precautionary approach will be addressed from a mathematical view. The results will then be presented in Section 4, and the conclusions of this paper will be summarized. Finally in Section 5, some complications are discussed, and future perspectives of the research are suggested.

Section snippets

Model

To formulate fishery management as a stochastic optimal control problem, the dimensionless exploited stock model and the economic performance objective will be introduced. For the incomplete information problem, a measurement equation will also be introduced.

Methods

In fisheries management, the aim is to find a sustainable strategy. This corresponds to control in the infinite horizon T→ ∞. The problem posed is to decide the catch rate in order to get continued economic gain under Markov control. The stationary Hamilton-Jacobi-Bellman equation is used to identify the optimal Markov strategy Ct = μ*(Xt) analytically as an exact result with complete information, as presented in Thygesen (2016). In this section a method to solve the stochastic optimal control

Harvest control rule vs. Effort control rule

To explore the influence of uncertainty in stock size assessment, various levels of the measurement noise intensity σY have been considered. Here the case σX = 1 and κ = 0.25 is shown. Other choices of the parameter value exhibit similar behavior of the system. Using Monte Carlo simulations, the harvest control rule (HCR) and the effort control rule (ECR) with a median estimator (M) are first examined, to get an overview. The effect of other estimators will be examined in Section 4.2. The ratio

Discussion

Results similar to some of those presented in the previous section have also been mentioned in the literature, even if other mathematical approaches were applied. For instance, it is emphasized in the paper by Sethi et al. (2005) that imprecise stock estimation affects fishery policy significantly. A higher measurement error lowers the level at which the fishery is shut down. Fig. 2 has revealed a very similar causal relationship. As the measurement noise level increases, the evaluated optimal

Declaration of interests

None.

Acknowledgements

This paper presents some results developed from a M.Sc. Thesis at DTU Compute. The author wishes to thank the thesis supervisor professor Uffe H. Thygesen. The author thanks also two anonymous reviewers for their helpful comments.

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