ANALYSIS OF A SIMPLE MODEL FOR 

 ESTIMATING HISTORICAL POPULATION SIZES 



T. D. Snqth' and T. Polacheck=^ 



ABSTRACT 



Estimates of historical abundance of animal populations are important in many management deci- 

 sions. Historical estimates based on a simple model of population growth have been made for several 

 populations of dolphin involved with the yellowfin tuna purse seine fishery. We used the data for the 

 bridled dolphin, Stenella attenuata. to investigate the behavior of the model by which these historical 

 estimates were calculated. For populations with low net reproductive rates, the effect of bias in the 

 estimates of the input parameters on the estimated historical abundances was approximately linear 

 and additive. When all the input parameters were independently estimated, the variances of the 

 historical abundance estimates were dominated by the variance of the initial abundance estimate and 

 the coefficient of variation of the historical estimate was less than the largest coefficient of variation of 

 any parameter. 



Many decisions about the management of animal 

 populations are based on the estimates of abun- 

 dance of the population relative to its historical or 

 preexploitation size. These estimates are basic to 

 any application of the theory of maximum sus- 

 tained yield as incorporated in several interna- 

 tional marine mammal management agreements 

 such as the North Pacific Fur Seal Treaty and the 

 International Whaling Convention. Similarly, the 

 concept of "optimum sustainable populations" as 

 specified in the recent Marine Mammal Protection 

 Act of 1972 (MMPA) has been defined in terms of 

 comparing the present size of a population with its 

 original size (Southwest Fisheries Center^). 

 Schools of dolphin of several species (primarily 

 Stenella attenuata and S. longirostris) have been 

 used by purse seine fishermen in the eastern tropi- 

 cal Pacific to locate yellowfin tuna, Thunnus alba- 

 cares, since 1959, as described by Perrin (1969). 

 Significant numbers of dolphin have been killed 

 by becoming entangled in the purse seines. In 

 order to make management decisions under the 

 MMPA about these dolphin populations, the Na- 

 tional Marine Fisheries Service (NMFS) needed 



'Department of Zoology, University of Hawaii, Honolulu, 

 Hawaii; present address: Southwest Fisheries Center, National 

 Marine Fisheries Service, NOAA, P.O. Box 271, La Jolla, CA 

 92038. 



^Department of Zoology, University of Hawaii, Honolulu, 

 Hawaii; present address: Department of Biology, University of 

 Oregon, Eugene, OR 97477. 



^Southwest Fisheries Center. 1976. Report of the Workshop on 

 Stock Assessment of Porpoises Involved in the Eastern Pacific 

 Yellowfin Tuna Fishery. Southwest Fish. Cent. La Jolla Lab., 

 Natl. Mar. Fish. Serv., NOAA, Admin. Rep. LJ-76-29, 53 p. 



estimates of the preexploitation abundance of the 

 various populations. The NMFS convened a work- 

 shop of scientists to obtain the estimates based on 

 a simple model of population change (see footnote 

 3). This paper evaluates the behavior of estimates 

 of abundance obtained from their approach. This 

 is important in order to be able to evaluate the 

 degree of confidence to be placed in such estimates, 

 and hence in management plans based on them. 



METHODS AND MATERIALS 



The model used to estimate preexploitation 

 abundance is based on a common discrete model of 

 population growth: 



AT^^j = N-K^HN-K^) (b-d) (1) 



where N^. = the abundance at time r 

 b = the birth rate 

 d = the natural death rate 

 K^ = the number of animals killed, as- 

 sumed to occur at the beginning 

 of time interval t 

 N^^ 1 = the abundance 1 time unit later. 



Reversing the procedure (i.e., solving the above 

 equation for N,. ) results in the expression 



^ l+i?_ ^ 



(2) 



Manuscript accepted May 1978. 



FISHERY BULLETIN; VOL. 76. NO. 4, 1979. 



where Nr now is the estimate of abundance 1 yr 



771 



