FISHERY BULLETIN VOL 77. NO 2 



Note that if we set /3, = fi., = and y, = y., = 1 . 

 Equation (B33) becomes 



bExoSoOiKT 



bExo + oc2T 



With the exception of the multiplicative term ( 1 + 

 a.,T/hEx„). Equation (B34) is equivalent to Equa- 

 tion 111) (model B) in the body of the paper. We 

 shall show that the model presented in this section 

 contains model B as a special case and also 

 exhibits "abrupt" transitions, between multiple 

 steady states. 



As E -► ^, the harvest rate saturates and 



y-y„ = ai7iSoX,r'' 



(B35) 



Hence, when E is large, iV)' ''' is a biomass esti- 

 mate. 



When E is small. Equation iB33) becomes 



bExoSoOiiKT'' 720 2 



Y^ 7. (7i+-^T'M, IB36) 



Pi P2 



which can be written as 



Y/E ^ hiT" +h2T'' 



where h-^ 



bXoSoO^iKyi /Ji72"2 



■,/'2 



12& 



(837) 



iB38) 



2P2 



Unlike the one-step model, in the multistep model 

 YIE is not a useful biomass estimate at any level of 

 effort. 



The steady-state tuna level is determined from 

 the steady-state version of Equation (B25). After 

 Equations (B29) and (B32) are used for the values 

 of C| and C, and the resulting expression is sim- 

 plified, we obtain 



gr = aiT' K 



A{aJ.E,T) 



iB39) 



p{a,li,E,T) 

 where A = (7i + 1) [)3i/32 + |3i bExo 1 

 + (y2-W2T'' 

 + dibExo + (bExo)^ lB40) 



P = ^S,i32 



(B41) 



^bExoifii *li2 + bExo+oi2T'' ). 



Because A and p are so complicated, Equation 

 (B39) is difficult to analyze as it stands. To 

 simplify the analysis, we assume that /3, = fi^ ~ 0. 

 Physically, this means that the rate of dissociation 

 of complexes due to fishing is much greater than 

 the natural dissociation rate of complexes. Since 

 our major interest is in the qualitative behavior of 

 Equation ( B39), this assumption seems acceptable. 



If /3, = li.2 = 0. Equation (B39l becomes 



gr 



a^T" KbExo 

 bExo+a2T'' 



{B42) 



which can be analyzed. We denote by /■(£, y^,y2,T) * 

 the right-hand side of Equation (B42). The solu- 

 tions of Equation (B42) will be discussed according 

 to the values of y, and y.,- A complete analysis of 

 Equation (B42) is very involved. We shall present 

 a partial analysis, in order to illustrate the types of 

 behavior which may occur. We first consider the 

 casein which y, = yj = 1. Equation (B42) becomes 



gr = 



a^KbExoT 

 bExo -^ 02T ' 



(B43) 



which is analogous to Equation ( 1 IB) of the body of 

 the paper. Consequently, we shall not pursue the 

 analysis here. In the analysis of Equation (IIB), 

 we showed that Equation ( B43 ) may have multiple 

 steady states. As effort increases, a transition be- 

 tween the steady state where the tuna level is high 

 and the steady state where 7" = is possible if «,A' 

 '> gT' *0' 'the "catastrophe" condition). 



In the one-step model, a complex containing two 

 tuna schools was formed only if the two schools, at 

 once, came into close contact with an attractor. 

 That model did not exhibit multiple steady states, 

 or even the possibility of overfishing the tuna into 

 extinction. 



On the other hand, if the complex that contains 

 two schools is formed by a stepwise process, so that 

 schools are added to an attractor one at a time, 

 "catastrophic" behavior and extinction of the tuna 

 are possible. 



Sudden transitions in population (catastrophes) 

 are usually difficult to predict. However, the 

 model presented here leads to a natural measure 

 of overfishing. From Equations (B29) and (B31), 

 when E is large we have 



336 



