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

Abstract

Understanding whether a population will survive 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 nonlinear 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 generalize the constant diffusivity, D , to a nonlinear diffusivity function D ( C ) , where C > 0 is the population density. While the choice of D ( C ) affects long-term survival or extinction of a bistable population, working solely in a continuum framework makes it difficult to understand how the choice of D ( C ) affects survival or extinction. We address this question by working with a discrete simulation model that is easy to interpret. 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.