Prager and MacCall: Contaminant and climate effects on spawning of three pelagic fishes 



mechanistic population model that somehow extrapo- 

 lates known effects on individual organisms to effects 

 at the population level. The second approach is en- 

 tirely at the population level: construction of statisti- 

 cal models that correlate historical changes in popula- 

 tion properties (especially properties thought to be 

 sensitive to environmental fluctuation) to changes in 

 contaminant loadings. 



Neither of these approaches is completely satisfac- 

 tory. Weighing against the first approach is the fact 

 that the existing theory of population biology does 

 not allow extrapolation from physiology and behavior 

 of individuals to the net productivity of the popula- 

 tion. Indeed, biological populations seem to have 

 emergent properties unknowable from the observable 

 properties of individuals (Mayr 1982). A similar prob- 

 lem is that combinations of contaminants may work in 

 unforeseen ways; one might say that such combina- 

 tions have their own emergent properties. Thus the 

 biological activity and availability of mixtures of con- 

 taminants in the coastal environment are difficult, if 

 not impossible, to extrapolate from laboratory studies 

 of exposure to individual contaminants. Given the 

 seemingly insurmountable barriers to constructing 

 mechanistic bottom-up models of populations chroni- 

 cally exposed to low levels of contaminants, we used 

 the alternative approach: we constructed empirical 

 statistical models of contaminant effects on popula- 

 tions. The empirical approach, of course, also has many 

 limitations; these are discussed at length in later 

 sections. 



A common interpretation of Eq. 1 is that a represents 

 the net fecundity of a unit of spawning biomass and P 

 reflects the degree of density-dependence in the stock's 

 recruitment. 



The Ricker model is often expressed in terms of the 

 natural logarithm of recruits per spawner: 



log(f?/P) = a + 6P + e, 



(2) 



where a = log(a), b = -p\ and £ = log(£). (The sign 

 change in b is made merely to simplify the notation; b 

 is negative in a compensatory stock.) Assuming that 

 the total number of eggs spawned is proportional to 

 the spawning biomass P, the quantity RIP is an index 

 of an egg's probability of survival to recruitment. This 

 seems to be a more appropriate quantity to use for 

 detecting exogenous effects than recruitment itself, 

 which is usually held to depend in the first order upon 

 stock size. We refer to the quantity RIP as "spawning 

 success" and to its natural logarithm as "log spawning 

 success." The model of log spawning success (Eq. 2) is 

 linear in the parameters and contains an additive er- 

 ror structure, and thus can be fit by ordinary least- 

 squares (OLS) regression. 



Eq. 2 can easily be modified to incorporate external 

 variables, such as climate or contaminant effects, that 

 might affect spawning success. Suppose we have m 

 such variables, \x u x 2 , ..., xj, for which we wish to 

 estimate parameters 19,, 9 2 , ..., 9,„|. Then a model in- 

 cluding these variables is 



Model structure 



\og(R/P) = a + bP + I e,x, + e. 



(3) 



i=l 



Evidence suggests that the youngest stages of fishes 

 should be most sensitive to the effects of contaminants 

 (Weis & Weis 1989). For that reason, our model of 

 contaminant effects addresses survival from the egg 

 stage to age of recruitment (roughly the first year of 

 life). The model is based on the widely-used Ricker 

 ( 1954) model offish recruitment: 



In this expanded model, m explanatory variables and 

 population size affect log spawning success in an addi- 

 tive manner. The application of this linearized model 

 is described in a later section. Other adaptations of 

 the Ricker model to represent contaminant effects were 

 made by Goodyear (1983) and Vaughan et al. (1984). 



R = aPe^ p C,, 



(1) 



Data sources and processing 



where R = recruitment in number of fish or 



biomass, 

 P = parent spawning-stock size (usually 



spawning biomass), 

 a, (3 = estimated parameters of the Ricker 



model, 

 e = base of natural logarithms, and 

 C, = a lognormally-distributed stochastic 



component with zero mean. 



Two main categories of data were used in this 

 study: data on fish abundances, and explanatory data 

 on the environment. Data on fish populations com- 

 prised time-series of age-structured abundance esti- 

 mates from virtual population analyses (VPA). We ex- 

 amined three coastal pelagic stocks off southern 

 California: northern anchovy Engraulis mordax, Pa- 

 cific sardine Sardinops sagax, and chub mackerel 

 Scomber japonicus (known locally as Pacific mackerel). 



