CLARK and MANGEL. AGGREGATION AND FISHERY DYNAMICS 



The instantaneous rate of harvest, Y, is the prod- 

 uct of (the number of complexes) X (the encounter 

 rate bE) x (the capture ratio Xo' ^ ' the number of 

 schools per complex). Thus 



Y = 



bExoyaKT^Sp 

 P + bE\o 



If y = 1. Equation iBlO) becomes 



y = 



bExoaKTSg 

 i5 + bE\o 



(BIO) 



(BID 



which is, with the exception of S^. identical to 

 Equation (llA). The additional factor S„ arises 

 here because we are considering numbers of tuna 

 schools, whereas model A of the main part works 

 directly with tuna biomass. 



Model A is thus a subcase of the model in this 

 section. Hence, we have provided a second physi- 

 cal picture for the mechanism which generates 

 model A used in the paper. 



Equation (Bll) exhibits a saturation as E in- 

 creases and is similar to results obtained in the 

 Michaelis-Menten approach to enzyme kinetics 

 (White et al. 1973). This is not unexpected, since 

 our models are based on the assumption that the 

 attractors "catalyze" the fishery. 



The tuna and attractor steady states are deter- 

 mined from the steady-state versions of Equations 

 iB5i and (B6i. Adding Equations (B6) and (B7) 

 gives 



Sk 



0, 



(B12) 



w^hich we assume has a solution A' = K,.. (Note that 

 this model does not allow for the loss of attractors 

 due to fishing.) The steady-state tuna population 

 satisfies 



0=,,-aKr^.^, [^^] 



(B13) 



gr = oAT'[ 



bExo-ii{y-l) 

 a + bExo 



= aAT7(£,7). 



tB14) 



Since the case in which 7 = 1 was analyzed in 

 the body of the paper, we shall not consider that 



case here. We shall briefly consider the case of -y a 

 2. This case may be of little interest in the actual 

 tuna fishery, but there may be other instances 

 where y 3= 2 is interesting le.g., animal popula- 

 tions). 



When E is small, so that ^^Xu < f^^y - H- the 

 coefficient /'(£',>') is negative. Equation (B14) has a 

 graphical solution sketched in Figure 11. The 

 steady-state tuna level, 7",, is greater than the 

 "natural level" T^, but this is explained as follows. 

 At any time there are a certain number of tuna 

 schools bound in the complexes. The remaining, 

 uncomplexed, tuna achieve the steady state level 

 r,|. The total number of tuna, however, is T„ plus 

 the number in the complexes. 



AsE increases, a level of effort is reached so that 

 /"(£,y) = 0. At this point. Equation iB14) becomes 



8t 



0. 



(B15) 



The tuna level is at the natural steady state, be- 

 cause tuna are removed from the complexes as 

 quickly as they enter the complexes. 



AsE increases further, /"(S.y) -►/■,, where/'j = 1 

 is the limit as E -*■ ^ of /(fi.y). Equation (B14) 

 becomes 



g^ = uKT\ 



(B16) 



The graphical solution of Equation (B16) is 



< 



< 



I 

 u 



< 



aKT' f (E,y ) 



NUMBER OF CORE SCHOOLS (T) 



Figure U. — Graphical determination of the steady-state tuna 

 population I Tj I for the one-step kinetic model, when the natural 

 dissociation rate is greater than fishing mortality isee text). 



333 



