A Nonlocal Reaction-Diffusion Model for a Single Species with Stage Structure and Distributed Maturation Delay

We propose a delay differential equation model for a single species with stage-structure in which the maturation delay is modelled as a distribution, to allow for the possibility that individuals may take different amounts of time to mature. General birth and death rate functions are used. We find that the dynamics of the model depends largely on the qualitative form of the birth function, which depends on the total number of adults. If it is monotonic increasing and a non-zero equilibrium exists, then the equilibrium is globally stable for all maturation delay distributions with compact support. For the case of a finite spatial domain with impermeable boundaries, a reaction-diffusion extension of the model is rigorously derived using an approach based on the von Foerster diffusion equation. The resulting reaction-diffusion system is nonlocal. The dynamics of the reaction-diffusion system again depends largely on the qualitative form of the birth function. If the latter is nonmonotone with a single hump, then the dynamics depends largely on whether the equilibrium is to the left or right of the hump, with oscillatory dynamics a possibility if it is sufficiently far to the right.