STEVENSON and CARRANZA: MAXIMUM YIELD ESTIMATES FOB. OPISTHONEMA SPP. 



the fleet expanded to new fishing grounds in re- 

 sponse to declining CPUE and a later period 

 characterized by diminished total production and 

 severely reduced CPUE throughout the entire 

 geographic range of the fishery. Estimates of 

 maximum equilibrium yield and the amount of 

 fishing effort required to produce that yield could 

 serve as "starting points" for management 

 strategies intended to conserve the Costa Rican 

 thread herring resource and achieve maximum 

 social and economic benefits from the fishery. 



The linear surplus production model was first 

 outlined by Graham (1935) and further developed 

 by Schaefer (1954). These authors postulated that 

 under equilibrium conditions the instantaneous 

 rate of surplus production from a given population 

 at any time it) is directly proportional to the 

 biomass (Bt) of the population at that time and to 

 the difference between the theoretical maximum 

 biomass (By-) and Bt and inversely proportional to 

 B^, i.e. 



ditions 

 q = the coefficient of "catchability," i.e., 

 the vulnerability of the population 

 to fishing. 



From Equations (1) and (2), at equilibrium 



dB 



dt 



Ye = Fe Be 



kB^jB^-B^) 

 B 



(4) 



Substituting YeIFe for Be in Equation (4) 



YeBe = F^B^-^F,2 

 k 



(5) 



and from Equation (3) 



Yf. = qbj. 



q'b 



/■p-2 



(6) 



dB^ kB^{B^-B^) 

 dt B 



(1) 



where k = the instantaneous rate of increase in 

 stock size at densities approaching 

 zero. 



Moreover, when the biomass which exceeds an 

 equilibrium level (Be) is being removed at the 

 same rate as it is produced, the surplus production 

 (dB/dt) is converted into an annual equilibrium 

 yield (Ye) according to the following expression: 



which defines a parabola in which Ye is a function 

 of/k. 



Dividing both sides of Equation (6) by /k yields a 

 linear relationship YE^fE = a + bfE where the 

 intercept a = qB^, and the slope b = q^BJk. The 

 effort fs which corresponds to the maximum 

 equilibrium catch Ys is determined from the rela- 

 tionship 



dYE 

 dU 



= a-2bf^ 



e 



(7) 



dB 



FeBe 



(2) 



Therefore 



where Fe = rate of fishing which maintains the 

 population at Be- 



If we assume that the rate of fishing (or instan- 

 taneous rate of fishing mortality) at equilibrium is 

 proportional to the fishing intensity (or effort) 

 then 



f. = 



26 



(8) 



Substituting Equation (8) in Equation (6), the 

 maximum equilibrium yield Ys determined from 

 the regression function is 



Fe = QfF 



(3) 



where /e = fishing effort under equilibrium con- 



Ye = Ys = — 

 ^ 46 



(9) 



Thus, when Equation (3) is valid, estimates of 

 maximum equilibrium yield and associated effort 



695 



