Spatiotemporal patterns of a diffusive plant–herbivore model with toxin-determined functional responses: Multiple bifurcations

Abstract

In this paper, a homogeneous diffusive plant–herbivore model with toxin-determined functional response subject to the homogeneous Neumann boundary condition in the one dimensional spatial open bounded domain is considered. By using Hopf bifurcation theorem and steady state bifurcation theorem due to Yi et al. (2009), we are able to show the existence of Hopf bifurcating periodic solutions (spatially homogeneous and non-homogeneous) and bifurcating non-constant steady state solutions. In particular, under certain conditions, the globally asymptotic stability of the positive constant steady state solutions and the non-existence of non-constant positive steady state solutions are investigated. These results allow the clear understanding of the mechanisms of the spatiotemporal pattern formations of this ecology model. In particular, our numerical results authenticate that toxicant parameter will play important roles in the stability and instability of the periodic solutions.