been widely spectulated, are not unreasonable 

 when one considers the species' high reproductive 

 potential, short life expectancy, gregarious habits, 

 and, presumably, its sensitivity to environmental 

 fluctuations. Such characteristics, as they con- 

 trol population development, are not atypical of 

 Arthropoda in general. 



Returning to the matter of discontinuity in 

 temporal distribution of recaptures during the 

 Tortugas experiment (table 7, fig. 10), observe now 

 that its further reconciliation is possible. The 

 value obtained for X, 0.55, which is henceforth 

 assumed for the reasons noted earlier to have been 

 practically all due to natural causes, is readily 

 contained in the total-loss coefficients estimated 

 for both partially and fully exploited phases, viz, 

 0.76 and 1.51, respectively. If natural mortality 

 is presumed to have been effectively constant 

 through the transition period, then the difference 

 in residuals (fishing mortalities, F=0.2l versus 

 7^=0.96) may be immediately attributed to cessa- 

 tion of selective fishing accompanied by an abrupt 

 rise in fishing rate upon completion of recruitment. 

 It appears that had the low rate of recapture 

 established during the partially exploited phase 

 continued, subsequent computations would also 

 have led to an estimate for X (hence M) compa- 

 rable to that determined from existing data for the 

 fully exploited phase. 



That the instantaneous rates of mortality do not 

 remain constant over extended periods of time but 

 diminish with age is suggested by a small number 

 of marked shrimp having been recaptured late in 

 the experiment. Note in figure 10 that their cor- 

 responding logarithm fell far to the right of the 

 line fitted to those of adjusted numbers of recap- 

 tures made earlier in the experiment. 



In summary, the foregoing results are inter- 

 preted to reflect exclusively conditions in that age 

 group of shrimp represented by the experimental 

 population, and hence should only be applied with 

 great caution in other fisheries, or to other age 

 groups at different seasons. Every age group (as 

 herein defined) is but one of a succession of groups 

 that may overlap several seasons. Each is there- 

 fore likely to be subjected at every stage of de- 

 velopment to different levels of exploitation as well 

 as to changing ecological conditions, with the high 

 expectation that mortality parameters will vary 

 accordingly. 



YIELD IN WEIGHT AS A FUNCTION OF 

 AGE (SIZE) AT RECRUITMENT 



To answer the fundamental question posed 

 earlier, it remains now to express the interaction 

 of population growth and mortality in terms of 

 expected yield when age (or size) at recruitment is 

 varied over a wide range of values. Thus, so that 

 the commercial return from a particular age group 

 (or, in general, all age groups combined) will be 

 maximal, at what average age of shrimp, under the 

 conditions of observed growth and mortality, 

 should harvesting begin? 



THEORETICAL POPULATION MODEL 



A satisfactory answer may be provided through 

 application of any one of several mathematical 

 population models, or analogs, which have been 

 developed to facilitate study of the dynamics of 

 open, self-maintaining biological systems (e.g., 

 Watt, 1956; Beverton and Holt, 1957; Ricker, 

 1958) . Notwithstanding its possible shortcomings, 

 all of which are thoroughly discussed at appro- 

 priate stages in its derivation, the simple model 

 developed and applied by Beverton and Holt 

 (1957, pp. 35-38 and 309-327) was chosen for the 

 purposes of this study because it offered the most 

 straightforward solution to a practical problem. 



Deterministic in nature, i.e., growth and mor- 

 tality are presumed effectively constant from 

 recruitment onward to the end of the species 

 fishable life span, the expression for the shrimp 

 population's mean weekly biomass over this period 

 is given by 



0„e 



-nK(l'-l ) 



P '»= R *- M ' W S F+M+nK 



(1- 



■(F+M+nK)\ 



where F and M are, respectively, the coefficients 

 of instantaneous fishing and natural mortality; 

 W a , K, and t are the growth parameters defined 

 earlier; R represents the number of shrimp re- 

 cruited weekly to the fished population; t p > is the 

 age, corresponding to the minimum commercial 

 size referred to above, at which recruitment is no 

 longer influenced by selective fishing; \=t\ — t p > 

 indicates the population's fishable life span (t\ 

 being the species mean life expectancy) ; and 

 p=tf,' — t p designates an interval during which 

 some recruitment occurs because of selective 

 fishing but population decline is mainly attrib- 

 utable to natural causes (t„ indicating the age at 



DYNAMICS OF A PENAEID SHRIMP POPULATION 



331 



