BREIWICK ET AL.: ESTIMATED INITIAL POPULATION SIZE OF BOWHEAD WHALE 



increases proportionally, attaining its maximum 

 level when the stock is near zero. However, it is 

 also recognized that response in rate may occur 

 with some lag and thus various lag periods are 

 assumed. 



In order to construct the model , various parame- 

 ter estimates are needed. These are discussed be- 

 low. 



Net Recruitment Rate 



Since information on the maximum net re- 

 cruitment rate is lacking, a range of values (0.01- 

 0.05) was used, based on analogy with other 

 baleen whale stocks for which such data are avail- 

 able. For example, Allen (1972) showed calculated 

 rates for the fin whale, Balaenoptera physalus, (as 

 a proportion of exploited stock) to be mostly in the 

 range 0.021-0.036. The Scientific Committee of the 

 International Whaling Commission (Interna- 

 tional Whaling Commission 1978) calculated 

 maximum gross recruitment rates for the sei 

 whale, B. borealis, (as a proportion of exploited 

 female stock) as 0.26 which implies a net recruit- 

 ment rate of the exploited stock to be 0.06. If we 

 express these rates as a proportion of the total 

 stock, they are in the range of 0.01-0.05. (Fur- 

 thermore, estimates of the 1848 stock level became 

 unstable and did not converge if maximum net 

 recruitment rates >0.05 were used in our itera- 

 tion.) 



Natural Mortality 



Similarly, there is no information available on 

 natural mortality in bowheads, and a range of 

 values of 0.04-0.08 was used. These correspond to 

 mortality estimates for other baleen species (Doi 

 et al. 1967; International Whaling Commission 

 1971). 



Lag Time 



We have no data on the growth and age of this 

 species, and there were no regulations in the 

 fishery. Nor do we have any information on the lag 

 that may occur between the reduction of stock 

 density and response of the population through its 

 presumed increase in recruitment. In a similar 

 study carried out for porpoise stocks involved in 

 the yellowfin tuna purse seine fishery (National 

 Marine Fisheries Service^), lag periods of 1, 3, and 

 5 yr were used. Because the population response 



in larger animals might be delayed, we have arbi- 

 trarily tried four lag periods: 1, 3, 5, and 7 yr. As 

 will be shown, this parameter has minor effect. 



Model Development 



The assumptions outlined above can be formu- 

 lated in mathematical terms as follows. The re- 

 cruitment model is 



r, =M -Kl -P^_JP )(r - M) 



max' 



(1) 



where r^ = recruitment rate in season t 



Pi__^ = population size at the beginning 



of season t - t (t = lag time 



assumed for population response ) 



Pq = initial population size (start of 



1848 season) 



M = natural mortality rate 



(r - M)^^^ - maximum net recruitment rate. 



The extrapolation model also uses the (approxi- 

 mate) recursion formula (Allen 1966): 



Pt.i = iPt-C,)e-'' +R, 



(2) 



where Rf - r, Pi_^ is the gross recruitment be- 

 tween the beginning of season t and season ^ -^ 1. 

 and C„ Pt are catch in season t and population 

 size at the beginning of season t. A further ap- 

 proximation made is thate"'^= 1 -M . Equation (2) 

 provides a good approximation if the catching 

 season is relatively short and natural mortality is 

 low. 



The iterative procedure consists of specifying a 

 current stock level, natural mortality rate, max- 

 imum net recruitment rate, and iterating on Pq 

 in Equation (2). Thus 



Pn-S^P0'Pl'--^Pn-l^ 



(3) 



where P„ is some current stock level. Due to the 

 lag time involved in Equation (2), it is not practical 

 to invert Equation (3) and solve for Py explicitly; 

 hence the iterative solution of Equation (3) is ob- 

 tained. 



^National Marine Fisheries Service. 1976. Report of the 

 workshop on stock assessment of porpoises involved in the east- 

 em Pacific yellowfin tuna fishery. Adm. Rep. LJ- 76-29, 54 p. 

 Southwest Fish. Cent., Natl. Mar. Fish. Serv, NOAA. PO. Box 

 271, La Jolla, CA 92038. 



847 



