Bifurcation analysis of a generalized toxic-phytoplankton–zooplankton system with nonlinear cross-diffusion and non-local interaction effect

In this paper, a generalized toxic-phytoplankton–zooplankton system with nonlinear cross-diffusion and non-local competition effect subject to the Neumann boundary conditions is investigated. We mainly show the results on the bifurcation analysis and spatiotemporal patterns by bifurcation theory and numerical simulations. First, we discuss the stability of non-negative constant equilibria and Hopf bifurcation of the corresponding non-spatial system. Then the linear stability analysis presents that the nonlinear cross-diffusion is the crucial mechanism for the occurrence of Turing instability without non-local effect and non-local interaction effect has a significant impact on the stability of positive constant steady state solution. By choosing proper bifurcation parameter, the existence of the steady state bifurcation and non-homogeneous Hopf bifurcation at the positive constant steady state is derived by Lyapunov–Schmidt reduction. Finally, some numerical simulations are executed to demonstrate our theoretical results. Spatial patterns and spatiotemporal patterns appear near steady state-Hopf bifurcation point.