FISHERY BULLETIN: VOL. 81. NO. 1 



rate (birth rate less natural death rate) multi- 

 plied by the number of animals actually repro- 

 ducing in a given year. The number of repro- 

 ducing animals is approximated by assuming 

 that one-half the animals killed in a year repro- 

 duce before dying. Solving this relationship for 

 N t , one obtains 





(i) 



Repeatedly applying this equation to estimate 

 the population size for any number of years (s) 

 prior to the year (c) for which an independent 

 estimate of population size {N ) is available yields 

 in general 



N s = 



N c 



-z 1 - . (2) 



n(i + r\ j=1 n/i + r) 



The 1979 workshop (footnote 3) extended this 

 procedure by calculating the recruitment rate 

 Ri, i years prior to the present, using the density- 

 dependent relationship (Allen 1981) 



Ri — -R TO <1 



(3) 



N p is the estimated population size at the begin- 

 ning of the first year of exploitation, p years 

 earlier; R,„, the maximum net recruitment rate; 

 Z, the density-dependent exponent; and N,- and 

 N p , estimated from Equation (2). Because N p in 

 Equation (3) is not known until the series in 

 Equation (2) has been calculated, an iterative 

 procedure is required to solve the equations for 

 historical population size. Equations (1) and (3) 

 together form a special case of the generalized 

 production model of Pella and Tomlinson (1969). 

 In Equation (3) the net recruitment rate is 

 maximum (R m ) when the population size ap- 

 proaches zero, decreasing to zero as the popula- 

 tion size approaches N p . Z determines the popu- 

 lation size at which the rate of change of the 

 population is maximum, the maximum net pro- 

 ductivity level (MNPL). The values of Z corre- 

 spond to the MNPL approximately as (Polacheck 

 1982) 



MNPL 



N P 



(1 + Zf z 



(4) 



If Z=l, then the MNPL is one-half the equilibri- 

 um population size; if Z is >1, then MNPL is 

 greater than one-half the equilibrium size. The 

 fraction of the maximum reproductive rate, R m , 

 realized at a given population size, increases as 

 the value of Z increases. 



Statistical properties of the estimate of N p and 

 the ratio N c /Np are examined in detail in Smith 

 and Polacheck (1979), wherein methods are de- 

 veloped for calculating the variances. Tests of 

 sensitivity of the estimates of N p to the values JV C , 

 K t , and R m show that the estimates are most sen- 

 sitive to the value of present abundance and least 

 sensitive to the net recruitment rate. Examina- 

 tion via simulation of the shape of the sampling 

 distribution shows that if N has a symmetrical 

 sampling distribution, then so does the estimate 

 Np. 



Several estimates of each parameter required 

 by the model are available in working documents 

 and technical memoranda prepared by the staff 

 of the Southwest Fisheries Center. I rely on the 

 most current estimates, primarily minor revi- 

 sions of those used by the 1979 workshop, with 

 reference to papers describing earlier estimates 

 as needed to document methods. 



POPULATIONS 



Populations affected most by the yellowfin 

 tuna purse seine fishery are of the genus Stenella, 

 and are found in the area from just south of the 

 Equator to an approximate lat. 20°N and west 

 from the Mexican and Central American coasts 

 to an approximate long. 150°W. Two populations 

 of spotted dolphins, S. attenuata, and three popu- 

 lations of spinner dolphins, S. longirostris, are 

 found in this region. 



The two spotted dolphin populations are re- 

 ferred to as "offshore" and "coastal" forms. The 

 coastal spotted dolphin population occurs near- 

 shore and around islands, while the offshore 

 spotted dolphin ranges from nearshore to an ap- 

 proximate long. 150°W. The two forms overlap 

 in range near the coast. 



Perrin (1975) distinguished these two forms of 

 S. attenuata morphologically. He noted that 1) 

 the larger coastal form occurs seaward to 50 km 

 while the offshore form occurs as nearshore as 20 

 km, and 2) the coastal form was involved in only 7 

 of 1,373 purse seine sets on dolphins observed be- 

 tween 1971 and 1974. Additional data collected 

 since then, including reexamination of speci- 

 mens collected during sets in the years 1971-74, 



