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Appendix A. Derivation of time series model. Click here for a PDF file of the appendix.

The recruitment functions in the demographic model we used consisted of a linearized fecundity function, a linearized neonatal survival function and a constant describing mean litter size produced by breeding females. The curvature of the product of these two functions was sufficiently weak that it could be well approximated by a linear function on the logit scale. The survival and recruitment functions are of the same form (equation 1 in the main text) but have different parameter values (Table 1 main text). Because the shape of the recruitment and survival functions was similar the sum of the survival and recruitment functions could be well approximately with an equation of the form of Eq. 1, except the numerator becomes because the recruitment function has L, the average litter size of breeding females, as a constant.

Algebraically then,

(A.1)

where the are model parameters, N represents population density and NAO represents the North Atlantic Oscillation.

The time series equation derived above does not need to be simplified to the extent it is. The degree of simplification will depend on how many parameters can be confidently estimated given the length of the time series of counts available.