A Simple Model for Population Dynamics in Stochastic Environments

1. A linear model for population dynamics in stochastic environments is introduced by linearizing the logistic equation in discrete time for small stochastic variations around a large average carrying capacity. The analysis of this model involves the relatively simple mathematics of time-series analysis and is not restricted to white noise (i.e., totally unpredictable) environments. 2. In this linear model the population size at any time is a reflection of many historical effects. The intrinsic rate of increase, r, is a measure of the responsiveness of the population to changes in k. A responsive population (r ≈ 1) does not retain historical effects for long, whereas a sluggish population (r ≈ 0) does. 3. The variance of the population size through time always increases with r. But the presence of much predictability in the environment, as measured by the serial correlation between consecutive values of K, reduces the influence of r. High serial correlation allows time for even a sluggish population to achieve an N near the current K thereby causing the population variance to nearly equal the carrying-capacity variance. 4. The predictability of the population size is in general a reflection of both its responsiveness to changes in K and the predictability of K itself. However if the population is perfectly responsive (r = 1), then its predictability becomes a direct expression of the predictability of K. Conversely, if K is unpredictable then the predictability of the population depends solely on its own responsiveness. Except in these special cases, the population's predictability depends on both factors. 5. The temporal pattern of the population size depends on the population's responsiveness and on the temporal pattern of the carrying capacity. If r < 1 then the population pattern preferentially represents the long-term components present within the carrying-capacity pattern, if r = 1 no component is preferentially represented, and if r > 1 the short-term components are preferentially represented. The population's temporal pattern is "connected" to the carrying-capacity's temporal pattern through a "filter" whose frequency characteristics are set by r. 6. The linear model and logistic model were compared through computer simulation. In the logistic, but not the linear, model deviations of N above K are restored more strongly than deviations of N below K. As a result the population size from the logistic model was invariably slightly lower than in the linear model, and the logistic population variance was slightly higher. 7. The predictions about the serial correlation between consecutive population sizes which were derived analytically from the linear model were checked with computer simulations on the logistic model. When the coefficient of variation (CV) of K was 5%, the differences between the models were not detectable with even 1,000 census points. When the CV was 15%, provided population extinction did not occur, the differences again were not detectable with 1,000 points. 8. Population extinction due to a fluctuating K is shown to be related to two issues: the change in K may be translated into a perturbation to N and the perturbation to N may be restored. The responsiveness of the population is the parameter relevant to the first issue and the rate of return to equilibrium following a perturbation is the parameter relating to the second.