CLARK and MANGEL: AGGREGATION AND FISHERY DYNAMICS 



exp \aKT) > g, 



i.e., if and only if'the intrinsic schooling rate (over 

 the duration of the fishing season) exceeds the 

 intrinsic growth rate. 



It is also clear that the yield-effort curves for 

 this model have the same appearance as in Figure 

 3. Hence the behavior of the two models is closely 

 analogous; bifurcations do not arise. 



The discrete-time version of model B is obtained 

 by replacing the expression ( aKN ~ fiS) in Equa- 

 tion ( Al I by «A'A'( 1 - SIS- ). This gives rise to a 

 nonlinear escapement-recuitment relationship 



^'^AR) 



It can be shown (we omit details i that 



limR.o^K («) = exp(-aKr) 

 lim£.,.vl'e {/?) - exp(-aKT)-R 

 limfi,„ (/?-*£-(/?)) = bxnS'TE. 



The resulting dynamics can be described in 



APPENDIX B 



terms of Figure 10. If aAT < a = \ng the model is 

 noncatastrophic (Figure 10(a)), having a single 

 equilibrium P* (escapement) which approaches 

 P > as £ -► +-^-. (If g > 2 the equilibrium at P* 

 may be unstable, even "chaotic," for small E ( May 

 1974), but this possibility will not concern us 

 here.) But if aKT > era second, unstable, equilib- 

 riumP ' emerges, and a bifurcation occurs at some 

 critical effort level E = E. 



To summarize, this appendix has demonstrated 

 that the qualitative predictions of our schooling 

 strategy models are independent of the basic popu- 

 lation dynamics of the tuna population. Although 

 we have explicitly established this fact only for 

 two specific models, it should be clear that the 

 theory will remain valid for a large variety of 

 other models, including alternative forms of the 

 growth and stock-recruitment functions and in- 

 cluding delayed-recruitment models as well as 

 cohort models. In all cases, the nature of yield- 

 effort curves will depend critically upon a) the 

 relationship between intrinsic schooling rate and 

 biotic potential and b) the schooling strategy of 

 tuna to the extent that school size is sensitive to 

 the total tuna population. 



In this appendix, we present two detailed, kinetic 

 models of the schooling behavior of tuna and 

 tuna-porpoise complex formation. The models are 

 more general that either model A or model B. 

 which are in fact special cases of the models de- 

 veloped in this appendix. Since our basic assump- 

 tions are quite different from those used in the 

 body of the paper, it is interesting that equivalent 

 results can be obtained, at least in special cases. 



The models are based on the following assump- 

 tion: in some large area of ocean, 11, there are Ttt^ 

 core tuna schools and Kit) "attractors" (porpoise 

 schools or logs) at time I . We assume that the core 

 schools move independently of each other and that 

 the motion is random. 



We first assume that when an attractor and y ( y 

 s 1 ) tuna schools "collide" (i.e., come within some 

 critical distance), a tuna-attractor complex is 

 formed. Let C(n denote the number of tuna- 

 attractor complexes at time I. The fishery is as- 

 sumed to fish only on these complexes. We shall 

 postulate different mechanisms of complex forma- 

 tion and analyze the resulting kinetic equations. 

 The kinetic equations are derived assuming a law 

 of "mass action'" similar to the one used in chemi- 

 cal kinetics (Moore 1972). 



We shall not consider the mechanism by which 

 the core tuna schools are formed. Whenever it is 

 necessary for the analysis, we shall assume that 

 the number of core schools has a logistic grow-th 

 function. This assumption is derived by firstly as- 

 suming that the biomass of tuna, Af(/), has a logis- 

 tic growth function. Namely, if no fishing occurred 

 and no complexes formed: 



f-*" 



N/No) 



(Bl) 



where .V„ is the carrying capacity of H, in terms of 

 biomass of tuna. LetS,, denote the weight of a core 

 school. Then we have 



dT 

 dt 



= rT(l -T/To) 



(B2) 



where 7"(n = .V(nS|, is the number of core schools 

 at time I. 



A model in which the tuna-attractor complex is 

 formed by one collision between y tuna schools and 

 one attractor is first analyzed. Submodel A of the 

 paper is a special case of this model. We show that 



331 



