Appendix A. Calculation of population prediction interval (PPI).
Let Nt be the population size in year t in a population far below the carrying capacity K and Xt = ln Nt. We assume that change in the logarithm of population size among years is normally distributed. If r is the deterministic growth rate, i.e., , and is the environmental stochasticity (Lande et al. 2003), the stochastic growth rate is:
assuming that population size is sufficiently large to ignore demographic stochasticity.
The likelihood function (omitting the constant factors) then becomes
and is maximized numerically with respect to the two unknown parameters r and .
To construct the PPI we generated parametric bootstrap-replicates by simulating a new data set from the estimated model and estimating new values of the parameters for each new data set. Predictions of future population sizes were done by simulating the process beginning with the last recorded population size and choosing new values of s and from the distribution of the bootstrap-replicates for each simulated process. Quantiles of the distribution of population sizes are used as the PPI at a given time t. Simulations have shown that this approximation is usually good when there is no density regulation, although there may be a small deviation between the range of the interval and the theoretical value. This was corrected by additional simulations following procedures in Engen et al. (2001).