A review of harvest theory and applications of optimal control theory in fisheries management

Abstract

Management. Can. 1. Fish. Aqrmat. Sci. 44(Suppl. 2): 75-83.Optimal control theory is a useful tool for analysis of general mathematical models of fisheries and harvest strategies. It provides a framework for management decisions, but is too general for direct application to many specific management problems. A review of applications of optimal control theory to fisheries management is presented, with a general audience in mind. Advantages and shortcomings of optimal control theory in the context of fisheries management are discussed with some specific examples.La theorie du contr6le optimal constitue un outil utile pour I'analysede modeles mathematiques g6neraux des peches et des strategies d'exploitation. Elle fournit une base pour les dkcisions relatives a la gestion mais elle est trop generale pour 6tre appliquee directement a d e nornbreux probl&mes d e gestion determines. L'armteur presente, a I'intention du public, un examen des applications de cette theorie a la gestion haliermtique. II illustre a I'aide d'exemples ses avantages et ses inconvenients dans le contexte de la gestion haiieutique.In vector notation, (1) becomes wherex(t) is an n x 1 vector, representing the state of the system (e.g. species density), u(t) is an r x 1 input or control vector (e.g. harvest quota or harvest rate), and f is a vector-valued function. With time delays T, we have and for discrete systems, where k is the kth time interval.The optimal control problem is stated as follows: Given the state eq. (I), a set of boundary conditions on the state variables at the initial and terminal times, and a set of constraints on the state and control variables, determine the admissible controls u(t) so that a performance index (cost function) is minimized (or maximized). The boundiary conditions are given by x(to) and SEX($)] where to, 9, and S denote, respectively, the initial time, the final time, and the target set. Note that to is always fixed, but may be part sf the optimal control problem; e.g. one may strive to minimize rf:The performance index is expressed a% a scalar quantity where 6 and F are scalar-valued functions.The main theoretical approaches to the optimal-control-Can. J . Fish. Aquar. Sci., Vo%. 44, I987 7% Can. J. Fish. Aquat. Sci. Downloaded from www.nrcresearchpress.com by McMaster University on 11/25/14 For personal use only. Can. J . Fbh. A q u t . Sei., V01.44, 1987 Can. J. Fish. Aquat. Sci. Downloaded from www.nrcresearchpress.com by McMaster University on 11/25/14 For personal use only. (May 1974). When the ecosystem model is not large (e.g . m < 5), and when the parameters of F in (8) are not known, the Routh-Hunwitz stability criterion may be used to establish Can, 9. Fish. Aquat. Sci., VoI. 44, 1987 Can. J. Fish. Aquat. Sci. Downloaded from www.nrcresearchpress.com by McMaster University on 11/25/14For personal use only.

The Importance of Functional Form in Optimal Control Solutions of Problems in Population Dynamics

“…This approach has been used to analyze a wide range of questions, including life history evolution (Schaffer 1983) and the optimal timing of biological processes (Williams and Nichols 1984). One of the most fruitful areas of ecology to which this approach has been applied is optimal foraging theory (Mangel and Clark 1986, Clark and Mangel 2000, understood in a broad sense to include such problems as optimal choice of habitat patch as a function of intrinsic and extrinsic conditions (Clark andLevy 1988, Houston et al 1988), optimal choice between mating behavior and foraging Mc-Namara 1986, McNamara et al 1987), optimal size of foraging groups (Clark 1987), and optimal harvest of renewable resources by humans (Williams 1982, Walters 1986, Cohen 1987, Johnson et al 1997.…”

Review of Dynamic Optimization Methods in Renewable Natural Resource Management

“…The reader's attention is directed to for a mathematically sophisticated review and documentation of much of this work, especially as concerns fisheries management. Several interesting applications of dynamic optimization are documented in Cohen [1987] and Vincent et al [1987].…”

“…It is not my purpose here to reproduce this theory, or to discuss in detail the solutions for specific resource models that have been analyzed. Rather, my intent is to provide the reader with comparative information about optimization methods that have been applied to biological systems, to illustrate applications to selected problems, and to forward some general suggestions about approaches to different kinds of renewable resource problems (see Jaquette [1972b], Walters and Hilborn [1978], Goh [1980], Shoemaker [1981], Clark [1985], Reed [1986], and Cohen [1987] for surveys that are organized along somewhat different lines). I focus below on six approaches to dynamic optimization, namely variational mathematics, dynamic programming, linear programming, nonlinear programming, simulation-optimization, and classical procedures.…”

A viability analysis for a stock/price model

“…When fishing activities are included, the model becomes the Schaefer [33] model. This model is widely used as the underlying framework by optimal control theory since the latter has been introduced in fisheries sciences (see, for instance, Clark [10], Clark and Kirkwood [9], Goh [17], Charles [6], McKelvey [29] and Cohen [13]). …”