of even the most carefully executed mark-recapture 

 experiment will satisfy all theoretical requirements, 

 he is forced when attempting such measurement to 

 rely heavily upon certain conditional assumptions 

 that may or may not be warranted. 



Potential sources of bias affecting the accuracy 

 of mortality estimates anticipated from data 

 yielded by the Tortugas experiment have already 

 been discussed. To reiterate, assumptions for 

 which reasonable substantiation was given are: 



(1) No, or only negligible, losses of experimental 

 shrimp due to rough treatment at release, or to 

 predation during and immediately after release; 



(2) no losses attributable at any time to after- 

 effects of the mark (dye); (3) little if any move- 

 ment of marked shrimp from the range of effective 

 fishing during the experiment; and (4) negligible 

 loss of recaptured shrimp because of failure to 

 report them. Information indicating the percent- 

 age loss due to nondetection was not obtained, 

 but the manner in which commercial catches were 

 processed leaves little doubt that the likelihood of 

 detecting marked shrimp was high (as was the 

 incentive to do so). Nevertheless, a necessary 

 assumption is that not only was the number of 

 recaptured but undetected shrimp low, but that 

 the ratio of undetected to detected recaptures did 

 not change during the experiment. 



THEORY AND EXPERIMENTAL RESULTS 



All present-day theory constituting the frame- 

 work of what is commonly termed "population 

 dynamics" has as its point of departure the con- 

 cept that the average rate of decline in any popula- 

 tion (fish, shellfish, etc.) is at every instant pro- 

 portional to population size. This relationship 

 may be simply expressed by the differential 

 equation 



dN 



dt 



= -ZN 



which, upon integration, gives the geometric 

 progression 



N,=N e- z ' (1) 



with common ratio e~ z , N the initial population 

 size, and N, the number in the population during 

 any of a series of equal-width time intervals t. 

 Two parameters, N and Z, characterize the ex- 

 pression, with the coefficient Z referred to as the 

 instantaneous rate of total mortality. 



The foregoing theorem proves particularly use- 

 ful in mark-recapture work since the initial size of 

 a marked population, N , is almost always known. 

 In some situations this feature readily permits the 

 separation of Z into its components, viz., (1) 

 mortality in the experimental population due to 

 recapture (fishing) , and (2) losses of marked mem- 

 bers due to all other causes. These quantities are 

 symbolized in the following analysis by the nota- 

 tion F and A', respectively (Beverton and Holt, 

 1957). Of major interest is the coefficient X, 

 part of which represents true natural mortality, 

 hereinafter denoted by the symbol M. Depend- 

 ing on the acceptability of assumptions concerning 

 the degree to which marked members are not 

 prone to loss other than through fishing and natu- 

 ral mortality, X itself can provide a reasonable 

 approximation of M. 



As revealed earlier (fig. 5), the probability of a 

 marked shrimp being recaptured varied widely 

 during the Tortugas experiment. It follows that 

 the corresponding fishing mortality fluctuated 

 accordingly, and that the effects of nonuniform 

 recapture effort would therefore have to be 

 eliminated before attempting to measure total 

 mortality, Z, and, ultimately, natural mortality, 

 M. Two approaches to the satisfactory measure- 

 ment of Z with recapture data generated under 

 such circumstances are employed herein, whereas 

 only a single alternative offered itself as a solution 

 to the more difficult problem of estimating X 

 (i.e., M). 



The first of the two methods used to determine 

 Z entailed application of an analytical method 

 developed for the simple situation where fishing 

 effort (or intensity) does not change appreciably 

 during an experiment. Its use here initially re- 

 quired that, rather than assume within each equal- 

 width time interval a fixed but, between intervals, 

 a successively different (i.e., a discontinuous) fish- 

 ing mortality, the number of recaptures accumu- 

 lating in every time interval be adjusted to a con- 

 tinuously uniform fishing effort throughout the 

 experiment (table 7). Such an approach clearly 

 infers that had a static fishing effort prevailed, 

 the pattern of population decline expressed by 

 theorem (1) — with all bias constant or negligible— 

 would have been reflected. In other words, 

 removing the confounding effects of a varying 

 fishing effort served to eliminate all but that part 

 of the overall recapture probability that would 



DYNAMICS OF A PENAEID SHRIMP POPULATION 



327 



