Linear and nonlinear stability in a diffusional ecotoxicological model with time delays

We propose a reaction-diffusion extension of a two species ecotoxicological model with time-delays proposed by Chattopadhyay et al. (1997). Each species has the capacity to produce a substance toxic to its competitor, and a distributed time-delay is incorporated to model lags in the production of toxin. Additionally, nonlocal spatial effects are present because of the combination of delay and diffusion. The stability of the various uniform equilibria of the model are studied by using linearised analysis, on an infinite spatial domain. It is shown that simple exponentially decaying delay kernels cannot destabilise the coexistence equilibrium state. In the case of a finite spatial domain, with purely temporal delays, a nonlinear convergence result is proved using ideas of Lyapunov functionals together with invariant set theory. The result is also applicable to the purely temporal system studied by other investigators and, in fact, extends their results.