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

We propose a reaction-diffusion extension of a two species ecotoxicologicalmodel 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 productionof toxin. Additionally, nonlocal spatial effects are present because of the combinationof delay and diffusion.The stability of the various uniform equilibria of the model are studiedby using linearised analysis, on an infinite spatial domain. It is shown thatsimple exponentially decaying delay kernels cannot destabilise the coexistenceequilibrium state.In the case of a finite spatial domain, with purely temporal delays, a nonlinearconvergence result is proved using ideas of Lyapunov functionals togetherwith invariant set theory. The result is also applicable to the purely temporalsystem studied by other investigators and, in fact, extends their results.