FISHERY BULLETIN: VOL. 84, NO. 3 



tion size, calving interval and sex ratio. Assuming 

 that calving interval was density dependent, Kasuya 

 (1976) estimated a maximum annual rate of increase 

 of 0.044 for this population of 5. coeruleoalba. 



METHODS 



The Model 



Population growth rates are estimated here using 

 the familiar Leslie matrix model (Leslie 1945). A 

 simplified parameterization is used for which sur- 

 vival rates and fecundities remain constant over 

 many age classes. Four parameters are required: 1) 

 calving interval for reproductively mature females, 

 2) average age at first birth for females, 3) annual 

 adult (noncalf) survival rate, and 4) annual calf sur- 

 vival rate. This degree of detail corresponds to the 

 practical limitations in collecting data on wild 

 dolphin stocks. 



The model is constructed with the assumption that 

 age class 1 corresponds to newly born calves (i.e., 

 censuses occur immediately after the calving sea- 

 son). In fact, the model is not dependent on discrete 

 calving seasons, but this assumption helps in con- 

 ceptualizing some elements of the model. The fecun- 

 dities (elements of the first row of the Leslie matrix) 

 represent the number of female calves born in one 

 year per female of a given age class in the previous 

 year. Fecundities for mature age classes are esti- 

 mated as the annual pregnancy rate (the inverse of 

 calving interval) multiplied by the adult survival rate 

 (the probability that a [pregnant] female will sur- 

 vive to the calving season) multiplied by 0.5 (the frac- 

 tion of female offspring). The annual pregnancy rate 

 is estimated as the percent of sexually mature 

 females which are pregnant, divided by the gesta- 

 tion period (in years). 



The choice of only two different survival rates for 

 all life stages was made because of data limitations 

 for dolphins. Perhaps a more biologically reasonable 

 assumption would be that dolphins have a U-shaped 

 mortality curve which is characteristic of mammals 

 in general (Spinage 1972; Caughley 1977; Siler 1979; 

 Smith and Polacheck 1981). Barlow 2 incorporated 

 this typical mammalian survivorship curve in models 

 of growth for spotted dolphins, Stenella attenuata. 

 Our choice of a separate survival rate for calves was 

 based on the common observation of higher mortal- 



ity in juvenile mammals (Caughley 1977; Siler 1979). 

 For convenience, juvenile mortality factors are com- 

 pressed into the first year's survival rates. This 

 simplification is justified because population growth 

 rates do not depend on the age at which juvenile 

 mortality actually occurs. We recognize that juvenile 

 mortality factors probably extend past the first year 

 of life, but insufficient data exist to justify including 

 this in our model. Higher mortality in old age was 

 not incorporated in our model, but maximum age 

 was limited to 50 yr. The survival rate at age 50 was 

 thus zero. 



We calculate population growth rates for a range 

 of the four vital rate parameters mentioned above. 

 Finite population growth rates, A, that are associ- 

 ated with these parameter values were calculated 

 by solving Lotka's characteristic equation, using 

 Newton's method. The explicit form of Lotka's equa- 

 tion used is 



50 



1 = Z A~*l 



x=l 



x ™>x 



2 Barlow, Jay. 1986. Biological limits on current growth rate 

 of a spotted dolphin population (Stenella attenuata). Unpubl. 

 manuscr. Southwest Fisheries Center La Jolla Laboratory, Na- 

 tional Marine Fisheries Service, NOAA, 8604 La Jolla Shores 

 Drive, La Jolla, CA 92038. 



where l x is the survivorship from birth to age class 

 x and m x is the fecundity of age class x. 



Below, we define the ranges used for the four 

 population parameters and describe how they were 

 selected. 



Survival Rates 



Ranges in Noncalf Survival Rates 



Few estimates of adult survival rates for dolphins 

 are available in the literature, primarily because ade- 

 quate data are difficult to collect. Kasuya (1976) pre- 

 sented annual survival rate estimates of 0.925 and 

 0.882 for exploited populations of Stenella attenuata 

 and S. coeruleoalba, respectively; however, his 

 method (log-linear regression) is biased (Barlow 

 1982), and he did not adjust for the effect of popula- 

 tion growth on age structure. A range of 0.85 to 0.97 

 was chosen for survival rates in this study. Values 

 <0.85 do not allow population growth for the ranges 

 of other parameters appropriate here, hence these 

 values were not considered. Values higher than 0.97 

 result in more than 22% of the population being over 

 50 yr old. This is inconsistent with estimates of 

 longevity for delphinids based on tooth layer counts 

 [58 yr in S. coeruleoalba (Sacher 1980), 38 yr in 

 S. attenuata (Hohn and Myrick 3 )], hence values 



3 Hohn, A. A., and A. C. Myrick, Jr. 1986. Age distribution 

 of the kill of spotted dolphins in the eastern tropical Pacific. 



528 



