On the other hand, if p > r (Figures 2(b), 3(b)) 

 then exhaustion is possible at sufficiently high 

 levels of effort. This case is similar to the Schaefer 

 model. 



For model A, CPUE is a seriously biased index of 

 total stock abundance. The instantaneous CPUE 

 is, of course, simply an index of abundance for the 

 surface population. Sustained CPUE progres- 

 sively overestimates the decline in abundance at 

 high levels of effort. Conversely, particularly if the 

 aggregation rate is large, CPUE may underesti- 

 mate the decline in abundance at intermediate 

 levels of effort. It is clear in general that no simple 

 transformation of the CPUE index can provide an 

 unbiased estimator of abundance, for this model. 

 Any fishery exploiting a substock of a biological 

 population necessarily provides only partial in- 

 formation concerning total abundance; in the 

 event that the fishery itself affects the relation- 

 ship between the substocks, the interpretation of a 

 time series of catch-effort data becomes extremely 

 difficult. 



To summarize, if the present model realistically 

 represents the process of aggregation (via surface 

 schooling) of tuna, then CPUE data may ulti- 

 mately overestimate the decline in abundance of 

 tuna. Management policy based on such data may 

 then be unduly restrictive. The situation may be 

 very different, however, if model B is the more 

 realistic representation. We now turn to this case. 



Model B 



The solution trajectories of Equations (18) are 

 illustrated in Figure 4(a) and (b), again corres- 

 ponding to the cases aK < r and aK > r respec- 

 tively. The corresponding yield-effort curves are 

 shown in Figure 5. 



In case (a), aK < r, the system has a unique 

 stable equilibrium (iV^, Sj. As in model A, we 

 haveA'^^-»'iV -OasE -> + ==. The yield-effort curve 

 for this case has the same shape as for model A. 



A new phenomenon arises, however, in the case 

 that aK > r. For small E (see Figure 4(b)) there 

 now exist two stable equilibria, at iN.,.S,) and at 

 ( 0,0), separated by a point of unstable equilibrium. 

 As E increases, the stable and unstable equilibria 

 coalesce and then disappear, leaving only the sta- 

 ble equilibrium at (0,0). In mathematical ter- 

 minology, the Equation system (18) undergoes a 

 "bifurcation" at the critical effort level E = E^ 

 where the two equilibria coalesce. The graph of 

 systainable yield vs. effort (Figure 5(b)) becomes 



324 



FISHERY BULLETIN: VOL. 77. NO, 2 

 



3 = 



(b) aK > r 

 SUBSURFACE POPULATION (N) 



Figure 4.— Tr^ectory diagrams for model B: a stable equilib- 

 rium exists at point ( N», S^); in diagram (b) an unstable equilib- 

 rium also exists for small E, but both equilibna disappear for 

 large E. Case la): intrinsic schooling rate less than intrinsic 

 growth rate: population cannot be depleted below N by surface 

 fishery. Case (bi: intrinsic schooling rate greater than intrinsic 

 growth rate population can theoretecally be shed to arbitrarily 

 low levels; the transition from N = N^ toN = is "catastrophic"; 

 see also text and Figure 5(b), 



multivalued for this case. Model B exhibits an 

 explicit mathematical "catastrophe." 



The significance of multivalued yield-effort 

 curves for fishery management has been discussed 

 by Clark (1974, 1976); see also Anderson (1977). 

 As effort E expands from a low level, the catch 

 follows the upper stable branch (Figure 5(b)), pos- 

 sibly with some lag. But onceE exceeds the critical 

 level £, , sustainable yield drops discontinuously 

 to zero and the fish population goes into a steady 

 decline. Subsequent decreases in effort do not 

 necessarily result in recovery of the fishery, which 

 may become "trapped" at a position of low abun- 

 dance. This behavior is characteristic of the 

 "catastrophe" situation (here the so-called "fold" 

 catastrophe (Zeeman 1975)). In general, once a 

 catastrophic jump has occurred, a large-scale 

 change in the control variable (effort) is required 



