CLARK and MANGEL: AGGREGATION AND FISHERY DYNAMICS 



pelamis. Annual catches in the between-war 

 period rose to a total of about 70.000 short tons. 

 Following World War II "there was a great up- 

 surge in the fishery" (Schaefer 1967:89). which has 

 continued to the present time, see Figure 1. The 

 entire period has also seen a progressive expan- 

 sion of the fishery into the offshore waters, con- 

 comitant with progressive developments in 

 technology. Of particular significance is the 

 switchover from bait boats to purse seiners, which 

 occurred in the early 1960's and has resulted in 

 substantial continuing increases in the catch of 

 yellowfin tuna. Much of this increase has resulted 

 from the offshore fishery on porpoise-associated 

 tuna schools. 



The purse seine tuna fishery operates by locat- 

 ing schools of tuna at or near the surface of the sea. 

 The main types of schools encountered are: a) non- 

 porpoise associated schools (pure yellowfin tuna, 

 pure skipjack tuna, or mixed schools) and b) por- 

 poise schools (yellowfin tuna only). Schools of tuna 

 that are not associated with porpoise are some- 

 times associated instead with concentrations of 

 floating debris ("log schools"). Management of the 

 yellowfin tuna fishery has been complicated by the 

 controversial problem of limiting the incidental 

 kill of porpoise, but this question will not concern 

 us here. 



Figure L— Annual catches of yellowfin i YFl and skipjack iSK) 

 tuna in the eastern tropical Pacific Ocean, 1945-75. 



Schools of tuna are normally located by visual 

 search, often by noting the presence of flocks of sea 

 birds. After sighting and approaching a school, the 

 vessel attempts to capture tuna by setting its 

 purse seine net about the school. During a set on 

 porpoise schools, speedboats may be lowered into 

 the water to assist in concentrating the porpoise so 

 that the school can be encircled by the net. Of the 

 daylight hours spent on the fishing grounds, 

 perhaps 70% are spent in searching for schools and 

 30% on setting of nets. 



According to biological observations (Sharp 

 1978),only a portion of the total tuna population is 

 available to the fishery, as schooled fish, at any 

 given time. It appears that the magnitude of this 

 available portion may be related to environmental 

 conditions, particularly the depth in the ocean of 

 certain thermal isoclines. Furthermore, it seems 

 evident that there must exist a dynamics of school 

 formation and exchange. The fishery interacts 

 with this dynamic process by removing some of the 

 schools. To our knowledge, the implications of 

 such a dynamic availability phenomenon have not 

 been previously investigated in detail. 



Since present knowledge about the schooling 

 strategy of tuna is limited, we shall discuss a 

 coterie of submodels for the formation of schools. 

 The models have been chosen in an attempt to 

 "bracket" the possible range of schooling 

 strategies; a wide variety of alternative models 

 could obviously also be set up (see Appendix B). 



We next describe a submodel for the purse seine 

 fishery. In order to keep the length of this paper 

 within bounds we discuss only a single fishery 

 submodel, in which vessels search at random for 

 randomly distributed surface schools. Finally we 

 introduce our submodel of tuna population 

 dynamics, which will be the standard Schaefer 

 model. In the main body of the paper we employ 

 the continuous-time version of the Schaefer 

 model, but a discrete-time version will be dis- 

 cussed in Appendix A. 



In Appendix B we describe several more de- 

 tailed models pertaining to the schooling strategy 

 of tuna, using techniques known from chemical 

 kinetics. This approach yields as special cases the 

 two submodels described in the text proper and 

 also gives rise to a number of interesting new 

 details. 



Although the background of our schooling and 

 fishery models is stochastic, we concern ourselves 

 only with expected values, so that the analysis 

 remains essentially deterministic. (Explicit 



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