A MODEL FOR OPTIMAL SALMON MANAGEMENT' 



Douglas E. Booth" 



ABSTRACT 



Considerable attention has been given in the literature recently to continuous time dy- 

 namic maximizing models for fisheries in general, but the time discreteness and inter- 

 dependency problems encountered in the case of most salmon fisheries have been largely 

 ignored. Hence, a discrete time profit maximizing model for a salmon fishery is devel- 

 oped in this paper, and it is shown that a correct salmon management policy takes the 

 form of an investment decision with respect to the level of escapement and that a man- 

 agement policy of maximum sustained yield may be incorrect from an economic stand- 

 point. It is hoped that continued research including construction of a working model will 

 provide some indication of the difference between the magnitude of spawner stocks se- 

 lected on the basis of maximum sustained yield and stocks selected on the basis of economic 

 optimality. 



Continuous time dynamic maximizing models 

 have been developed in the literature recently 

 to handle the problem of optimally managing 

 a fishery resource (Brown, 1969"; Quirk and 

 Smith, 1970*) . The continuous time approach to 

 analyzing management policy for a salmon fish- 

 ery tends to be unrealistic since the reproductive 

 process for salmon is periodic, and for certain 

 species reproduction involves rather complex 

 time interdependencies. In the simplest case 

 salmon spawned in a given time period will re- 

 turn to their spawning ground in some future 

 time period, while in more complex cases salmon 

 spawned in a given time period will return to 

 their spawning grounds in several diflferent runs 

 over a number of time periods; also, the level of 

 spawning activity in one time period may aflfect 

 the fertility of the spawning grounds in future 

 time periods. Such discreteness and time inter- 

 dependencies cannot be adequately characterized 

 in a continuous time mathematical model. 



' Contribution No. 361, College of Fisheries, Univer- 

 sity of Washington. 



" Fisheries Research Institute, 260 Fisheries Center, 

 University of Washington, Seattle, WA 98195. 



* Brown, G. W., Jr., 1969. An optimal program for 

 managing common property resource with congestion 

 externalities. Univ. Washington, Seattle. [Mimeo- 

 graph.] 



* For a static linear-programming model useful for 

 analyzing fishery management problems see Rothschild 

 and Balsiger, 1971. 



Hence, the purpose of this paper is to develop 

 a discrete time maximizing model based on cur- 

 rently accepted views of biological spawner-re- 

 turn relationships for salmon; the model is de- 

 veloped with the biological properties of the 

 Bristol Bay fishery foremost in mind (Mathews, 

 1967). It is shown that a correct fishery man- 

 agement policy takes the form of an investment 

 decision with respect to the level of escapement 

 and that a management policy of maximum sus- 

 tained yield may be incorrect from an economic 

 standpoint. In essence the fishery manager must 

 decide whether to invest in spawners which yield 

 a return of additional fish at future points in 

 time, or to catch and sell potential spawners 

 today. 



In the first section of the paper, the no- 

 tation and assumptions of the analysis are pre- 

 sented, and in the second section, a simple 

 first-order diflference equation model of a salm- 

 on fishery is developed and discussed. In the 

 third section, the model is extended to account 

 for the fact that salmon spawned in time per- 

 iod t will return to the spawning grounds in 

 time periods t + i, t + 5, and t + 6, and also 

 to account for the possibility that fish spawned 

 in time period t will deplete the spawning 

 grounds of food to such an extent that the 

 number of fish the spawning grounds can sup- 

 port in time period t + 1 will be reduced. De- 

 sirable refinements and applications of the model 



Manuscript accepted December 1971. 



FISHERY BULLETIN: VOL. 70, NO. 2, 1972. 



497 



