FISHERY REGULATION VIA OPTIMAL CONTROL THEORY^ 



William J. Palm^ 



ABSTRACT 



This paper attempts to show how control theory can be used to formulate a regulatory scheme for 

 fisheries. The regulatory mechanism considered is a limit imposed on fishing effort. It is shown that 

 static optimization methods, such as maximum equilibrium yield analysis, need to be supplemented 

 with dynamic methods, such as optimal control theory, which take into account the variable nature of a 

 fishery. The dynamic analysis is used to show that the size of a limit on effort should be a feedback 

 function of the variables in the state of the fishery. The concept of the Linear-Quadratic Optimal 

 Control Problem is introduced as a method for devising such a feedback scheme for fishery regulation. 

 A single- variable logistic model is used to introduce the basic concepts. A model with three variables 

 is then analyzed to show how the techniques are easily extended to the general multivariable case. 

 Details of the general method are given in an Appendix. 



The need for fishery regulation is apparent and 

 will become even more important with the es- 

 tablishment of resource management zones off 

 our coasts. Regulatory mechanisms include catch 

 quotas and limits on fishing effort (number of 

 boats permitted entry into the fishery, number of 

 hooks used, etc.). A mathematical model of the 

 fishery, which includes biological and perhaps 

 economic factors, is useful for determining the 

 best regulatory scheme. Some of the more familiar 

 examples of these models are given by Schaefer 

 (1954, 1968), Beverton and Holt (1957), Ricker 

 (1958), Larkin (1963, 1966), Pella and Tomlinson 

 (1969) and Fox (1970). The above models are said to 

 be dynamic because they utilize differential equa- 

 tions to describe how the fishery changes with 

 time. The inclusion of economic factors, multiple 

 species, and other biological variables, such as size 

 and age, results in multivariable models which are 

 quite complex. 



Much of the analysis of fisheries is based on the 

 concept of an equilibrium. Perhaps the best known 

 is the maximum equilibrium yield analysis. 

 However, equilibrium is an idealization and is 

 never actually encountered in reality because con- 

 tinually changing environmental infiuences act as 

 disturbances which displace the system from its 

 equilibrium condition. For unstable systems this is 

 disastrous because equilibrium is never regained. 



'Part of this work is a result of research sponsored by NOAA, 

 Office of Sea Grant, Department of Commerce under Grant 

 #9X-2()-6807C. 



Department of Mechanical P]ngineering and The Institute of 

 Knvironmental Biology, University of Rhode Island, Kingston, 

 HI 02881. 



and for stable systems with large time constants, 

 the return to equilibrium might take so long as to 

 negate the assumptions and usefulness of the 

 equilibrium-based analysis. Thus "static" or 

 equilibrium-based analysis should be supplement- 

 ed with dynamic methods which take into account 

 the variable nature of the fishery. A purpose of this 

 paper is to show that the above considerations in- 

 dicate that any regulatory scheme should contain 

 "feedback"; that is, the size of any quota or limit 

 should be a function of the state of the fishery. 

 Also, the concept of the Linear-Quadratic Optimal 

 Control Problem will be introduced as one way of 

 devising such a feedback scheme for fishery 

 regulation. 



The Linear-Quadratic Optimal Control Problem, 

 which has been widely applied in engineering, is 

 one method within the larger framework of op- 

 timal control theory. Other optimal control 

 methods have recently been applied to problems in 

 fishery management which are unlike the problem 

 treated here. Goh (1969, 1973) applied the so-called 

 "singular" control method to the problem of 

 maximizing yield with a single-species model. 

 Saila (in press) describes Goh's results in more 

 detail. Clark et al. (1973) analyze the problem of 

 optimal reduction of effort in an overexploited 

 fishery. They calculate the fishing mortality func- 

 tion which maximizes the total present value of all 

 profits and utilize a Beverton- Holt model for the 

 fishery. Clark (1973) has presented a similar 

 analysis for a logistic fishery model. The above 

 three analyses lead to control functions which have 

 been loosely described as a "bang-bang" control 



Manuscript accepted Januar\' 1974. 

 FISHERY BULLKTIN: VOL. 73, NO. 4, 1975. 



830 



