Spatiotemporal dynamics of a diffusive predator-prey model with generalist predator

Abstract

In this paper, we study the spatiotemporal dynamics of a diffusive predator-prey model with generalist predator subject to homogeneous Neumann boundary condition. Some basic dynamics including the dissipation, persistence and non-persistence(i.e., one species goes extinct), the local and global stability of non-negative constant steady states of the model are investigated. The conditions of Turing instability due to diffusion at positive constant steady states are presented. A critical value \begin{document}$ \rho $\end{document} of the ratio \begin{document}$ \frac{d_2}{d_1} $\end{document} of diffusions of predator to prey is obtained, such that if \begin{document}$ \frac{d_2}{d_1}>\rho $\end{document} , then along with other suitable conditions Turing bifurcation will emerge at a positive steady state, in particular so it is with the large diffusion rate of predator or the small diffusion rate of prey; while if \begin{document}$ \frac{d_2}{d_1} , both the reaction-diffusion system and its corresponding ODE system are stable at the positive steady state. In addition, we provide some results on the existence and non-existence of positive non-constant steady states. These existence results indicate that the occurrence of Turing bifurcation, along with other suitable conditions, implies the existence of non-constant positive steady states bifurcating from the constant solution. At last, by numerical simulations, we demonstrate Turing pattern formation on the effect of the varied diffusive ratio \begin{document}$ \frac{d_2}{d_1} $\end{document} . As \begin{document}$ \frac{d_2}{d_1} $\end{document} increases, Turing patterns change from spots pattern, stripes pattern into spots-stripes pattern. It indicates that the pattern formation of the model is rich and complex.