FISHERY BULLETIN: VOL 77. NO, 2 



Fisheries for various other pelagic, schooling 

 species, such as anchoveta, herring, and mackerel, 

 also appear to involve aggregative processes. Sev- 

 eral of these fisheries have in fact experienced 

 collapses which are qualitatively similar to those 

 predicted by our aggregation models. Other 

 mechanisms, however, may be involved in these 

 fisheries, including: predation (Clark 1974); com- 

 petitive exclusion (Murphy 1966); increased 

 catchability (Fox ); depensation in stock- 

 recruitment relationships (Clark 1976). In some 

 cases, stocks have failed to recover following a 

 collapse, even when fishing has been greatly cur- 

 tailed (Murphy 1977). Dynamic behavior of this 

 kind is not consistent with any of the traditional 

 models employed in fishery management. 



On the other hand, discontinuous behavior of 

 continuous nonlinear systems is a well-known 

 phenomenon in applied mathematics. Thus the 

 term "bifurcation" refers to such discontinuous 

 changes induced by continuous parameter shifts 

 in explicit mathematical models. More recently 

 the subject "catastrophe theory" has been de- 

 veloped as an abstract approach to these 

 phenomena (Thom 1975; Zeeman 1975; see also 

 the report in Science by Kolata (1977)). 



A discussion of catastrophe theory as it applies 

 in the fishery setting appears in Jones and Walters 

 (1976). Indeed these authors assert that ". . . the 

 tropical tuna fisheries have almost certainly 

 moved into a cusp region, . . . where small changes 

 in investment policy or failure to rapidly adjust 

 catch quotas could lead to fishery collapse." (Jones 

 and Walters 1976:2832). Since no specific biologi- 

 cal (or technological) catastrophe-inducing 

 mechanism has been suggested by Jones and Wal- 

 ters, their assertion stands only as a plausible 

 conjecture — a warning that possible nonlinear 

 system effects ought to be investigated more fully. 



In this paper we shall investigate in some detail 

 the interactions between the schooling behaviour 

 of tuna and the operation of the purse seine 

 fishery. Since current knowledge about the school- 

 ing strategy of tuna is limited, we shall construct a 

 variety of models in order to investigate the possi- 

 ble effects of and interactions with the fishery. In 

 particular, we shall discuss the following topics: 



^Similar collapses have not occurred in tuna stocks, perhaps 

 because of their relative diflfuseness. 



^Fox, W. W., Jr. 1974 An overview of production model- 

 ling. Unpubl. manuscr. Southwest Fisheries Center, Na- 

 tional Marine Fisheries Service, NOAA, P.O. Box 271, La JoUa. 

 CA 92038, 



1. yield-effort relationships, 



2. indices of stock abundance, 



3. fishery dynamics, 



4. management implications. 



The results turn out to be highly, perhaps sur- 

 prisingly, sensitive to the assumptions and 

 parameters of our models. Of particular impor- 

 tance is the way in which the size of surface tuna 

 schools depends upon the overall abundance of 

 tuna. If it is the case that school size ( as unaffected 

 by the fishery) is relatively independent of total 

 tuna abundance, then our models indicate the pos- 

 sibility (under certain additional conditions) of a 

 catastrophic collapse of the tuna fishery as the 

 intensity of fishing passes some critical level. That 

 such a prediction could arise from a potentially 

 biologically realistic tuna model was completely 

 unexpected at the beginning of the study, in spite 

 of the theoretical investigations mentioned above. 



Another significant result of our analysis is 

 that, under our model assumptions, the catch- 

 per-unit-effort(CPUE) statistic may constitute an 

 extremely unreliable index of stock abundance. 

 The bias may be in either direction depending on 

 the model adopted — CPUE may severely either 

 underestimate or overestimate the decline in 

 abundance as the fishery develops, while in other 

 cases CPUE may quite accurately represent 

 abundance. 



Following the description and analysis of our 

 various models, we shall present some simple 

 simulated development paths for the tuna purse 

 seine fishery, based upon the models. The first 

 simulation that we performed utilized our best 

 guesses as to realistic parameter values. In this 

 simulation the fishery experiences a catastrophic 

 collapse when effort is increased to 18,000 stan- 

 dardized vessel days per annum. The decline of the 

 tuna population itself occurs quite gradually, but 

 is not reflected by any significant decline in catch 

 or in CPUE, until the fishery is virtually de- 

 stroyed. In other words, the collapse of the fishery 

 involves not an abrupt change in the stock, but 

 rather an abrupt change in the input-output rela- 

 tionship. 



TUNA PURSE SEINE FISHERY 



The commercial fishery for tuna in the eastern 

 tropical Pacific Ocean began in the years following 

 World War I, the two main species taken being 

 yellowfin tuna and skipjack tuna, Katauivonus 



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