which shrimp become liable to selective capture 

 but, generally speaking, have not yet attained 

 commercial acceptability) . In practice, for a given 

 set of conditions, p is usually negligible and t P ' and t„ 

 may be considered equivalent. The remaining 

 notation arises from the need, during the model's 

 development, to expand the exponential term of 

 the von Bertalanffy growth equation. 

 Thus, if we let 6=3, 



w. 



--WJ\-e 



-Ka-t„)\b 



o>)" 



may be algebraically transformed to 



w t =w a '22n n e- nKl '- , i> 



71=0 



where the appropriate coefficients of the binomial 

 expansion are 



Qo= + l, fii=-3, 2 =+3, 3 =-l 



with n taking the values 0, 1,2, and 3. Although 

 earlier analysis of the weight-length relationship 

 suggested that growth in pink shrimp is not truly 

 isometric, use of the foregoing model assumes, for 

 practical purposes, that it is. Actually, the minor 

 effects of a departure from isometry should be 

 taken into account by appropriately varying W*>, 

 but failure to do so here does not lessen the 

 validity of later findings. 



From the biomass equation just postulated, it 

 follows that an expression for the mean weekly 

 yield of the fished population may be represented 



by 



w- 



--FP 



w 



And since R, the recruitment, is the only one 

 of the parameters describing Pi v that is not 

 ordinarily amenable to measurement, further 

 modification results in 



R R a fy>F+M+nK 



(l-e-< F + M +»m) (4) 



hereinafter referred to as the expected yield in 

 weight per recruit. 



DETERMINATION OF YIELD MAXIMUMS 



By varying t„,, and hence p and X, curves of 

 yield as a function of age can be generated with 



332 



equation (4) for fixed values of the remaining 

 parameters. The process obviously entads prior 

 establishment of some absolute minimum value 

 for t p , this value corresponding in subsequent 

 analyses to a size below which all shrimp are not 

 only commercially unacceptable, but also incom- 

 pletely vulnerable to the gear in common use. 



Yield Curve for Observed Parameter Values 



We have, from the Tortugas experiment and 

 other sources, the following values for the indicated 

 parameters : 



M=0.55 <i=0.68 week 



2^=0.96 <x=83 weeks 



K=0.07 t„= 15 weeks 7 



W„=42.0g. 



Recall also that \=t\ — t p . and p=t p , — t p . Sub- 

 stituting these in equation (4) and solving it for 

 each of not less than nine carefully spaced values 

 of t p ., the yield curve farthest to the left in figure 12 

 was obtained. It is immediately apparent from 

 this figure that in a population whose development 

 is governed by growth and natural mortality of the 

 magnitude observed for the Tortugas pink shrimp 

 population, peak biomass is attained somewhat 

 before, rather than after, the average shrimp 

 reaches a size equivalent to the present commercial 

 minimum. Growth, although relatively high in 

 contrast to that determined for a variety of species 

 supporting other commercial fisheries, evidently 

 cannot compensate (to man's economic advantage) 

 for losses accruing to a high natural mortality. 

 Nor is postponement of full-scale exploitation 

 indicated unless reduced environmental effects 

 can be expected to result in a markedly lower 

 natural mortality. Under observed conditions, 

 delaying exploitation until the shrimp reach a 

 greater initial size would indeed result in a negli- 

 gible gain in yields of so-called premium shrimp, 

 but only at the expense of a significant reduc- 

 tion in the total yield of all sizes. 



Yield Curves Theorizing Lower Levels of Natural 

 Mortality 



Of the parameter estimates obtained in this 

 study, by far the firmer and at the same tune the 

 least subject to wide temporal variation are those 



7 Age roughly corresponding to the minimum commercial size of 70 headless - 

 count. 



U.S. FISH AND WILDLIFE SERVICE 



