Logistic growth in the presence of non-white environmental noise

Abstract

A method is given for studying realistic random fluctuations in the carrying capacity of the logistic population growth model. This method is then applied using an environmental noise based on a Poisson process, and the time-dependent moments of the population probability density calculated. These moments are expressed in terms of a parameter obtained by dividing the correlation time of the environmental fluctuations by the characteristic response time of the population. When this quotient is large (very slow fluctuations tracked by the population) or small (very rapid fluctuations which are averaged), exact solutions are obtained for the probability density itself. It is also shown that at equilibrium, the average population sizes given by these two exact solutions bound all other cases. Numerical simulations confirm these developments and point to a trade-off between population stability and average population size. Additional simulations show that the probability of becoming extinct in a given time is greatest for populations intermediate between tracking and averaging the carrying capacity fluctuations. In addition to specifying when environmental noise can be ignored, these results indicate the direction in which growth parameters evolve in a fluctuating environment.