the harvest rate is a nonUnear function of effort 

 and saturates asE -> ^-. Consequently Y/E is not a 

 valid biomass estimate. We discuss other possible 

 biomass indices, the behavior of ^(^ £i as a func- 

 tion of effort and the sensitivity of the results to 

 the parameters which appear in the kinetic equa- 

 tions. 



Next, a multistep complex formation process is 

 considered. A two-step model is analyzed in full 

 detail. Submodel B is contained as a special case. 

 In addition to exhibiting all of the features of the 

 one-step model a multistep mechanism may lead 

 to "catastrophic" behavior. The catastrophic be- 

 havior was not built into the model but arises 

 naturally from the dynamics. 



The models presented in this appendix ( particu- 

 larly the multistep model) are based on what ap- 

 pear to be reasonable assumptions about the 

 schooling behavior of tuna and formation of the 

 complexes. 



The ultimate behavior of the system (fishery + 

 tuna + porpoises) does not appear to be an artifact 

 of the models, but a result of the basic assumptions 

 that the tuna form into schools and that the fishery 

 seeks tuna schools associated with attractors. In 

 fact, Thom's ( 1975) theorem on the structural sta- 

 bility (robustness! of unfoldings asserts that small 

 modifications of our models will not alter the qual- 

 itative behavior. 



The analysis of discrete-time versions of our 

 models is relatively intractable. Numerical 

 studies are underway. We do not expect the results 

 will be qualitatively different from the 

 continuous-time results. The analysis presented 

 in Appendix A supports this expectation. 



We have not included spatial effects (e.g., diffu- 

 sion) in our kinetic equations. The addition of dif- 

 fusion greatly complicates the analysis of the 

 kinetic equati(ms. However, preliminary work 

 based on the recent theory of Aronson and Wein- 

 berger (1975) has been carried out. treating the 

 kinetic equations with spatial dependence. We ex- 

 pect that if diffusion is added to the models in this 

 appendix, the transitions between high and low- 

 tuna steady states may occur at effort levels lower 

 than those predicted by the models without diffu- 

 sion. 



Single-.Stt'p ( ollision Model 



In this model we assume that y tuna schools 

 collide, at once, with one attractor to form a com- 

 plex: 



K+yT-' 



FISHERY BULLETIN: VOL, 77, NO, 2 



a 



(B3! 



The rate constants a, fj measure the association 

 and dissociation rates of the complex. The com- 

 plexes are fished at a rate/)/? with capture ratio x„: 



\obE 

 C K + harvest of 7 schools. (84) 



Elquation iB.3) indicates that y schools must be 

 present for a complex to form. In particular, if -y > 

 1 this model does not allow for the formation of 

 "partial" complexes, with fewer than y tuna 

 schools in the complex. It is clear that this assump- 

 tion is restrictive; later we relax it and allow for 



complexes with 1,2 y tuna schools. 



The kinetic equations corresponding to Equa- 

 tions (B.3) and (B4) are 



f = gr -aKT' + ^yC 



(B5) 



k = gn -aKT' +liC + hE\„C (B6) 



C = aKT'' -iiC-bE\oC: (B7) 



in p]quations (B5) and [BGigj. and^'^. are the tuna 

 and attractor growth functions, respectively (^^. = 

 for logs). 



The term proportional to T^arises in the follow- 

 ing way. Consider a small area of ocean, a. The 

 probability,/), that a tuna school is in a should be 

 proportional to alii and to T: 



T = h'T. 



(B8) 



If a complex containing y tuna schools is to form, y 

 schools must be in o. Since the tuna schools move | 

 independently and randomly, the probability of 

 finding y schools in a is proportional top'' = /^T'. 

 (A more precise analysis would lead iokTiT - 1) 

 (T-2)...(7'-y-i-l) instead of^7'\ since once a 

 school is in a specified area of ocean, there remain I 

 r - 1 schools to be distributed over the ocean. ' 

 Once the location of two schools has been specified, 

 there remain T - 2 schools, etc. When T is large, 

 as we are assuming, ^T'' is a good approximation 

 to the exact expression.) 



The steady-state number of complexes is deter- 

 mined by setting C = 0. We obtain 



C 



aKT' 

 (3 + bExo 



(89) 



332 



