CLARK and MANGEL AGGREGATION AND FISHERY DYNAMICS 



Qil) = Q* -{Q* -Qo)e-"'^'/^*. (6) 



As will be seen in the sequel, the characteristics 

 of our purse seine fishery model are severely 

 inflenced by the choice of the schooling submodel 

 A or B. Which of these submodels more accurately 

 reflects the actual schooling strategy of tuna is a 

 question we are not qualified to answer. It may be 

 the case that neither extreme (school size Q* 

 strictly proportional to tuna abundance N in sub- 

 model A, and Q* strictly independent ofN in sub- 

 model B) is realistic. For example, school size may 

 saturate for large A'', but exhibit density depen- 

 dence at low N, giving rise to a combination of 

 models A and B. Submodels involving more gen- 

 eral links between Q* and N could easily be con- 

 structed, but we will not attempt to work through 

 the details here. A more general class of schooling 

 submodels is discussed in detail in Appendix B. 



Let us remark here that models A and B assume 

 in effect a uniformly distributed "background" 

 tuna population. The models discussed in Appen- 

 dix B assume instead that the background popula- 

 tion consists of "core" schools; according to Sharp 

 (1978) the latter assumption is more realistic. In 

 certain cases the core-school models reduce to the 

 models A and B described above. 



MODEL OF 

 THE PURSE SEINE FISHERY 



We shall use a simple Poisson model to describe 

 the process whereby the fishing fleet searches for 

 schools of tuna. The h3TDotheses underlying this 

 model are well known (see, e.g., Ludwig 1974) and 

 will not be specified here. Let us note, however, 

 that our model pertains to a single type of school 

 (e.g., porpoise school, log school); a more refined 

 model might allow for a random intermingling of 

 school types. A nonrandom distribution of school 

 types, on the other hand, would lead to the as yet 

 unsolved problem of attributing allocation of ef- 

 fort by fishing vessels. 



The probability that the fishing fleet locates 

 exactly /f school attractors with the expenditure of 

 t days of searching effort, is given by 





where X = (a/A)K 



a = area searched per day 



A = total area of fishing ground 



K = number of school attractors. 



If searching effort is properly standardized, we 

 will have 



a/A = bE, 



where E = effort 



6 = a constant. 



Hence 



\ =bE K. 



(8) 



'Broadhead and Orange 1 19601 imply that Q' is nearly con- 

 stant, although It may in some cases be slightly density depen- 

 dent. However, for skipjack tuna, in the eastern Pacific, school 

 size and population size as indexed by CPUE are highly corre- 

 lated (but the two estimates are not independent), J, Joseph. 

 Director of Investigations. Inter-American Tropical Tuna Com- 

 mission, La Jolla, CA 94720, pers. commun. July 1978. 



The average number of attractors located by the 

 fleet in time t is 



k = \t =bEKt. 



Thus the total catch rate of tuna, Y. is given by 



Y = bEKxoQ <9) 



where \q = capture ratio (average fraction cap- 

 tured w-hen a school is encounter- 

 ed). 



LetS(t) denote the total number of tuna present 

 at time t in surface schools: S = KQ. Our model 

 then implies that 



ds \aKN- /« - b\oES (Model A) 



^^ ]aKN(l-SIS*)-bxoES (Model B) 



where S* = KQ* represents the total "carrying 

 capacity" of the surface school attractors. (Note 

 that, replacing oKN by pN = flow rate from sub- 

 surface to surface populations, we could simply 

 adopt Equation (10) as the basic hypothesis of our 

 model, eliminating any particular assumption re- 

 garding the attractive mechanism for surface 

 schools.) 



Let us assume for the moment that an equilib- 

 rium is achieved rapidly in the surface fishery, 

 relative to adjustments in the underlying popula- 

 tion N. (The dynamics of the underlying popula- 



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