We consider a partially coupled diffusive population model in which the state variables represent the densities of the immature and mature population of a single species. The equation for the mature population can be considered on its own, and is a delay differential equation with a delay-dependent coefficient. For the case when the immatures are immobile, we prove that travelling wavefront solutions exist connecting the zero solution of the equation for the matures with the delay-dependent positive equilibrium state. As a perturbation of this case we then consider the case of low immature diffusivity showing that the travelling front solutions continue to persist. Our findings are contrasted with recent studies of the delayed Fisher equation. Travelling fronts of the latter are known to lose monotonicity for sufficiently large delays. In contrast, travelling fronts of our equation appear to remain monotone for all values of the delay.
This paper is concerned with a delay differential equation model for the interaction between two species, the adult members of which are in competition. The competitive effects are of the Lotka-Volterra kind, and in the absence of competition it is assumed that each species evolves according to the predictions of a simple age-structured model which reduces to a single equation for the total adult population. For each of the two species the model incorporates a time delay which represents the time from birth to maturity of that species. Thus, the time delays appear in the adult recruitment terms.The dynamics of the model are determined, and global stability results are established for each equilibrium. The equilibria of the model involve the maturation delays. The criteria for global convergence to each equilibrium are sharp and involve these delays.A reaction-diffusion extension of the model is also studied for the case when only the adult members of each species can diffuse. We prove the existence of a traveling front solution connecting the two boundary equilibria for the case when there is no coexistence equilibrium. This represents invasion by the stronger species of territory previously inhabited only by the weaker. The proof of the existence of such a front uses Wu and Zou's theory for traveling front solutions of delayed reaction-diffusion systems.
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.
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