Extinction and wavefront propagation in a reaction-diffusion model of a structured population with distributed maturation delay

Abstract

Starting from an age-structured model, we derive a partial differential equation satisfied by the total number of mature adult members of a population, on an infinite one-dimensional domain. The formulation involves a distribution of possible ages of maturation and uses a probability density function on which ecologically realistic assumptions are made. It is found that the existence and value of a positive equilibrium solution depends on the mean maturation delay. When no positive equilibrium exists, we prove global attractivity of the zero solution. For a particular ecologically reasonable choice of the distribution function, we show that travelling fronts exist connecting the zero equilibrium with the positive one provided the mean maturation delay is sufficiently small, and the dependence of the front's propagation speed on the mean delay is discussed.