Formation of time patterns in a diffusive plant herbivore system with toxin-determined functional response

In this paper, we consider a diffusive plant-herbivore system with the toxin-determined functional response and subject to the homogeneous Neumann boundary conditions on the bounded one-dimensional spatial domain. The impacts of diffusion on the dynamical behaviors are investigated and it is found that although the appearance of diffusion does not affect the stability of constant steady states, it can lead to the occurrence of Hopf bifurcation of spatially inhomogeneous periodic solutions at the constant positive steady state. The conclusions show that the occurrence of spatial diffusion can induce more complex dynamical behaviors.