FISHERY BULLETIN: VOL, 77, NO, 2 



stochastic considerations are taken up in a forth- 

 coming paper by Mangel. ) Two important omis- 

 sions from our models are: a) age structure and b) 

 spatial distribution of the tuna population; the 

 multispecies aspect is also not covered. These 

 omissions were dictated by our desire to concen- 

 trate on the novel features of our work, viz the 

 schooling strategy and its implications. Further 

 research will be required (probably based primar- 

 ily on simulation techniques) if more sophisti- 

 cated, disaggregated models are to be studied. 



SCHOOL FORMATION SUBMODELS 



We imagine a given number, K, of school "at- 

 tractors," such as porpoise schools, or collections of 

 floating debris. (Our models also apply to nonpor- 

 poise and nonlog schools provided that the ex- 

 change process between subsurface and surface 

 schools satisfies the appropriate hypotheses, see 

 Equations (10).) Tuna from an underlying, or 

 "background," population associate with these at- 

 tractors according to one of the submodels A or B 

 below; the attractors are independent of one 

 another and do not interchange associated tuna. 

 Let N denote the number of tuna present in the 

 background (subsurface) population. The number 

 of tuna in an individual generic school is denoted 

 by Q = QW). (A full list of variables and parame- 

 ters is given in Table 1.) 



Table l — Basic parameters and variables of the models. Sym- 

 bols endemic to the appendices are pven below. 



Q(0 = Q* d-c-oe" '), 



(4) 



Model A 



where Cq = 1 - QqIQ* . 



Tuna associate with a given attractor at a rate 

 aN proportional to the background population, 

 and dissociate at a rate /3Q proportional to the 

 current school size: 



— =OrN 



m. 



(2) 



(The dissociated tuna return to the background 

 population, see Equation (15).) For fixed N the 

 resulting equilibrium school size Q* is given by 



aN 



(3) 



If Q(0) = Q„, Equation (2) has the solution (for 

 fixed AT): 



Thus in model A, the equilibrium size of schools is 

 directly proportional to the background tuna 

 population. (Since we treat the number of attrac- 

 tors, K, as fixed, we do not discuss the possibility 

 that school size could also depend on K .) 



Model B 



In this alternative submodel, we assume that 

 the maximum school size is a constant, Q *, which 

 is independent of the background tuna population. 

 Equation (2) is replaced by 





(5) 



•Mangel, M. 1978. Aggregation, bifurcation, and extinc- 

 tion in exploited animal populations. Cent. Nav Prof Pap 

 224. Center for Naval Analyses, 1401 Wilson Boulevard, Ar- 

 lington, VA 22209. 



where Q' = fixed maximum school size. 

 Thus we now have (for fixed N) 



320 



