Interpreting how nonlinear diffusion affects the fate of bistable populations using a discrete modelling framework

Abstract

Understanding whether a population will survive and flourish or become extinct is a central question in population biology. One way of exploring this question is to study population dynamics using reaction-diffusion equations, where migration is usually represented as a linear diffusion term, and birth-death is represented with a bistable source term. While linear diffusion is most commonly employed to study migration, there are several limitations of this approach, such as the inability of linear diffusion-based models to predict a well-defined population front. One way to overcome this is to generalise the constant diffusivity, $D$, to a nonlinear diffusivity function $D(C)$, where $C>0$ is the density. While it has been formally established that the choice of $D(C)$ affects long-term survival or extinction of a bistable population, working solely in a classical continuum framework makes it difficult to understand precisely how the choice of $D(C)$ affects survival or extinction. Here, we address this question by working with a simple discrete simulation model that is easy to interpret. The continuum limit of the discrete model is a nonlinear reaction-diffusion equation, where the flux involves a nonlinear diffusion term and the source term is given by the strong Allee effect bistable model. We study population extinction/survival using this very intuitive discrete framework together with numerical solutions of the reaction-diffusion continuum limit equation. This approach provides clear insight into how the choice of $D(C)$ either encourages or suppresses population extinction relative to the classical linear diffusion model.