A simple population model with a stochastic carrying capacity

Abstract

Many ecosystems are subject to external perturbations such as pollution, land clearing and sudden shocks to their environment. However, most current models used do not take the changing environment into consideration. In cases where the changes in the environment are taken into account this is usually done by specifying some time-dependent function for the carrying capacity that reflects the observed behaviour of the changing environment. Here we employ an alternative approach where the carrying capacity, a proxy for the state of the environment, is used as a state variable in the governing equations. Thereby, any changes to the environment can be naturally reflected in the survival, movement and competition of the species within the ecosystem. In this paper a simple ecosystem consisting of a single species and its variable environment is studied. Specifically, a logistic population model that incorporates a stochastic carrying capacity is investigated. The carrying capacity is treated as a state variable and is described by a stochastic differential equation. The statistical properties of the carrying capacity and the population are analysed using the Monte Carlo method giving; the expected time evolution of the population and its variance, the probability distribution of the population and the mean-time to extinction.