FISHERY BULLETIN: VOL. 82, NO. 3 



1 - 



qf f _ qf -t' h (C/f') 



CIN 



F = 



qft' 

 1+q t' h N 



Inserting this definition of F as a substitute in 

 Graham's equilibrium yield model, then 



Y =B 



e e 



= B„ 



Fe=* 



fi (B-BVB, 



f -f'/a+q -t' h -B e ) 



(5) 



where fi x is the maximum stock biomass, k is the 

 instantaneous rate of increase of the stock as den- 

 sity approaches zero, B e is the stock mass at an 

 equilibrium position, Y e is the equilibrium yield, 

 and F e is the fishing mortality rate which main- 

 tains the stock at B e (Ricker 1975). 



As expected a plot of Y e against B e or F e will 

 yield a symmetrical hyperbola, Figure 2 A; a plot of 

 q against B will denote that q is a constant re- 

 gardless of the relative magnitude of t' h to t'. This 

 is not the case, however, if we ignore the effect that 

 stock density has on searching time. For example, 

 a plot of Y against f  t ' results in an asymmetri- 

 cal hyperbola skewed to the right, Figure 2B. The 

 distortion increases as the ratio t'Jt' approaches 1 

 and is a result of the increased time the fleet must 

 spend searching for fish as population biomass 

 approaches zero, Figure 2C. Additionally, if 

 searching time is assumed to be independent of 

 stock abundance, q will be incorrectly measured 

 asg/(l + q  t' h  B e ) and will appear to be inversely 

 related to population density, Figure 2D. 



Note that when t ' h >0, f • t' will not peak at B g 

 = (for the situation described in Equation (5)). 

 Instead, it will peak at some intermediate level of 

 B and will then decline, Figure 2E. Thus, even 

 for the hypothetical equilibrium fishery, f  t' 

 would have to "voluntarily" decline from a maxi- 

 mum level as B e approaches zero. 



DISCUSSION 



The assumptions inherent in Equation (5) limit 

 its direct application as a qualitative model of 

 existing fisheries. However, the general behavior 

 described in Figure 2B, D has been noted in sev- 

 eral recent papers (Fox 1974; Pope and Garrod 

 1975; Schaaf 1975b; MacCall 1976; Ultang 1976; 

 Garrod 1977; Peterman 1980; Peterman and Steer 

 1981; Bannerot and Austin 1983). Two important 



examples occur with the Pacific sardine and At- 

 lantic menhaden fisheries. 



In their analysis of the available catch and ef- 

 fort data on the California based fishery on Pacific 

 sardine, Fox (1974) and MacCall (1976) were forced 

 to relax the usual restriction of a constant catch- 

 ability coefficient which is independent of popula- 

 tion size. Rather, they applied a density-dependent 

 catchability coefficient of the form 



aN? 



(6) 



where a and /3 are constants, assuming a constant 

 catch per unit effort. The general patterns pre- 

 dicted by these analyses are similar to those in 

 Figure 2B, D, with MacCall noting an inverse re- 

 lationship between the apparent q and population 

 abundance, and Fox noting a collapse of the 

 fishery in plots of catch versus vessel-months, Fig- 

 ure 3. Both of these patterns may be the result of 

 an inability to describe mathematically how the 

 time available for searching increased as the 

 population of sardines declined. 



It is generally assumed that Atlantic menhaden 

 have been overfished since the early sixties. Sup- 

 port for this conclusion was derived from the 

 surplus-production work of Schaaf and Huntsman 

 (1972), later updated by Schaaf (1975a). In both 

 studies the available effort index (vessel-weeks) 

 was modified in an attempt to correct for changes 

 in fishing efficiency with time. Under the assump- 

 tion of a constant q and lacking detailed informa- 

 tion on vessel characteristics and catch, the au- 

 thors adjusted effort by "multiplying the effort 

 observed in each year by the relative change in q", 

 using either 1965 (Schaaf and Huntsman) or 1971 

 (Schaaf 1975a) as a base year. The resulting pat- 

 tern (Fig. 4A) strongly suggests that the fishery 

 was operating on the descending arm of the 

 catch-effort curve. 



In another paper, Schaaf (1975b) observed an 

 inverse relationship between his estimates of 

 catchability coefficient and the population density, 

 generating a pattern, like MacCall's (1976), simi- 

 lar to Figure 2D and at least partially explained 

 by the lack of information on density-dependent 

 searching time. However, Schaaf's apparent 

 density-dependent estimate of q violates his early 

 assumption of a constant q for use in standardiz- 

 ing the available effort data. The point is not triv- 

 ial. Without this adjustment, the available effort 

 data suggest that the Atlantic menhaden fishery 

 is operating on the ascending arm of the catch- 

 effort curve, Figure 4B. 



450 



