AGGREGATION AND FISHERY DYNAMICS: A THEORETICAL STUDY 

 OF SCHOOLING AND THE PURSE SEINE TUNA FISHERIES' 



Colin W. Clark^ and Marc Mangel^ 



ABSTRACT 



This paper describes mathematical models of exploited fish stocks under the assumption that a certain 

 portion of the stock becomes available through a dynamic aggregation process. The surface tuna fishery 

 is used throughout as an example. The effects of aggregation on yield-effort relationships, indices of 

 abundance, and fishery dynamics are discussed. The predictions of the theory are notably different 

 from those obtained from general-production fishery models, particularly in cases where the available 

 substock has a finite saturation level. Possible effects include fishery "catastrophes" and lack of 

 significant correlation between catch-per-unit-effort statistics and stock abundance. Various man- 

 agement implications of the models are also discussed. 



The relationship between fishing effort, catch 

 rate, and stock abundance is of fundamental im- 

 portance to the management of commercial 

 fisheries. To a first approximation, it is usually 

 assumed that catch per unit effort (ClE ) is propor- 

 tional to stock abundance (P), with a fixed con- 

 stant of proportionality (catchability coefficient). 



C = qEP, 



(1) 



where C denotes catch per unit time and E denotes 

 fishing effort. By combining this relationship with 

 an appropriate model of population dynamics, one 

 obtains a dynamic fishery model which can then be 

 used as a basis for management policy (Schaefer 

 1957). 



The form of Equation (1 ) is predicated on certain 

 underlying assumptions pertaining to the fishing 

 process, particularly a) that fishing consists of a 

 random search for fish and b) that all fish in the 

 stock are equally likely to be captured. More pre- 

 cisely, by introducing an explicit stochastic model 

 of the fishery based upon such assumptions, one 

 can deduce Equation ( 1 ) for the expected catch rate 

 C. But such models can also be employed to inves- 

 tigate the consequences of alternative, and possi- 

 bly more realistic, assumptions. For example, 



'Research performed under contract to NOAA, National 

 Marine Fisheries Service, Contract No. 03-6-208-35341. 



^Department of Mathematics, University of British Columbia, 

 Vancouver, B.C., Canada V6T 1W5. 



^Center for Naval Analyses, 1401 Wilson Boulevard, Ar- 

 lington, VA 22209. 



Manuscript accepted October 1978 

 FISHERY BULLETIN VOL 77, NO 2, 1979 



Stochastic models of purse seine fisheries, incor- 

 porating detailed descriptions of the operation of 

 fishing vessels, have been discussed by Neyman 

 (1949), Pella (1969), and Pella and Psaropulos 

 ( 1975). On the other hand, the effects of concentra- 

 tion offish and of fishing effort have been studied 

 by Calkins (19611, Gulland (1956), and others. 



In this paper we discuss fishery models in which 

 the assumption of equal availability of all portions 

 of the stock is relaxed. Specifically, we are con- 

 cerned with fisheries that exploit aggregations of 

 fish; these aggregations are assumed to constitute 

 a dynamically changing substock of the entire 

 population. Although a general class of such mod- 

 els could be developed, we shall restrict the discus- 

 sion here to the case of the tuna purse seine 

 fisheries, in which aggregation apparently occurs 

 through the process of surface school formation. 

 Several alternative models of the interchange pro- 

 cess between surface and subsurface tuna sub- 

 populations will be presented, and the effects of 

 the surface fishery will be investigated for each 

 model. Evidence arising from studies carried out 

 at the Inter-American Tropical Tuna Commission 

 (Sharp 1978), and at the Southwest Fisheries 

 Center, National Marine Fisheries Service, shows 

 that yellowfin tuna, Thunnus albacares, captured 

 in surface schools in the eastern tropical Pacific 

 Ocean do in fact spend part of their time below the 

 surface. Little seems to be known, however, about 

 the dynamics of the interchange process; our 

 analysis of alternative models indicates that such 

 knowledge could become crucial to the manage- 

 ment of the fishery. 



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