FISHKRY BULLETIN VOL 77, NO a 



sketched in Figure 12. When 7^2, it is impossible 

 to overfish the tuna into extinction (compare Fig- 

 ure 12 with Figure 2, which corresponds to the 

 case y = 1 ). The reason for this behavior is that, as 

 the tuna level decreases, the rate of formation of 

 complexes, aKT^. decreases much more rapidly 

 since y s^ 2. When T is small, it is unlikely that a 

 complex will form. This result should be con- 

 trasted with the case of y = 1, in which is it possi- 

 ble to overfish the tuna to extinction. 

 From Fx]uation (RlOi we have 



Y/E 



hxoyaKT Sp 

 a + bE\o 



Thus, if /i •  hE\„ we obtain 

 r' cc YIE 



(B17I 



iBISi 



In the intermediate region /J ~ hEx^^ it appears 

 that no simple biomass index is available. 



The determination of the appropriate biomass 

 index depends upon the size q{ hE\„l(i. This is a 

 natural measure since it compares the rate at 

 which complexes are dissociated due to fishing 

 with the natural dissociation rate /i. 



Miiltistep (jjllision Model 



The model in the last section is somewhat un- 

 realistic in that the complex with y tuna schools is 

 formed only if the y schools collide siniultaiiei}usl\' 

 w ith an attractor. Hence, the model did not allow 

 lor complexes with y - 1, y - 2. ... ,1 tuna school 

 per complex. A more realistic model is one in 

 which the tuna-attractor complexes form by a 

 multistep mechanism: 



so that 



T cc (y/£:i'>. 



iBU)i 



Thus (V''£i' ^ is a possible biomass index, if /i > 

 hExu- 



U liEx„ fi. then 



y,T + A': 



C, 



73^ + C2^^C..j 



(B21i 



y/E * ^—^^ So. 



(B20) 



In this limit a possible biomass index is (Y')'^. 

 Thus, the catch itself is a biomass index. 



< 

 a: 



< 

 cc 



UJ X 



S o 



I- •- 

 — < 



cr 

 o 



UJ 



a kt'' 



aKT't(E,y) 



bE\o 



K -^ harve.st of I 7 schools 



i = \ 



where / = 1 , 



More detail could lie added, e.g., when a complex 



C'l is fished, / = 1, 



,/ tuna schools might be re- 



NUMBER OF CORE SCHOOLS (T; 



moved with probabilities p,,. When all y/,. = 1, 

 Equation iB21i is undoubtedly the most realistic 

 model presented here. (Since the probability that 

 two core schools are added at the same instant is 

 essentially zero, the idea of stepwise addition of 

 schools seems justified.) The kinetic equations cor- 

 responding to the reactions in Equation (B21) are 

 I for y ^ 1 for all / ) 



Ci = Q , AT - (/iiCt + hExnC\ + 02^1 T) + {i^Co 



Figure 12. — Graphical determination of the steady-state tuna 

 population (T^i for the one-step kinetic model, when fishing 

 mortal ity is greater than the natural dissociation rate i see text). 



C„ = a„C„_ir-(/3„C„ + 6£xoC„ 



+ Q„.^ir„7') +/i„.nC„„, (n>2) (B22I 



334 



