# American Institute of Mathematical Sciences

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

 1 School of Mathematics and Information Science, Guangzhou University, Guangzhou 510006, China 2 Center for Applied Mathematics, Guangzhou University, Guangzhou 510006, China 3 Science and Mathematics Faculty, College of Integrative Sciences and Arts, Arizona State University, Mesa, AZ 85212, USA 4 Simon A. Levin Mathematical, Computational, and Modeling Sciences Center, Arizona State University, Tempe, AZ 85281, USA

* Corresponding author: Jianshe Yu

Received  December 2018 Revised  February 2019 Published  November 2019

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 $\rho$ of the ratio $\frac{d_2}{d_1}$ of diffusions of predator to prey is obtained, such that if $\frac{d_2}{d_1}>\rho$, 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 $\frac{d_2}{d_1}<\rho$, 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 $\frac{d_2}{d_1}$. As $\frac{d_2}{d_1}$ 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.

Citation: Dingyong Bai, Jianshe Yu, Yun Kang. Spatiotemporal dynamics of a diffusive predator-prey model with generalist predator. Discrete & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2020132
##### References:

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##### References:
Stationary hot spots pattern in model (2). The parameter values are taken as (32) and $(d_1, d_2) = (0.028, 0.226)$. The zero-flux boundary condition is used and initial condition is small perturbation around the homogeneous steady-state $E^* = (1.5934, 2.0768)$.
Stationary stripes pattern in model (2). The parameter values are taken as (32) and $(d_1, d_2) = (0.028, 0.27)$. The zero-flux boundary condition is used and initial condition is small perturbation around the homogeneous steady-state $E^* = (1.5934, 2.0768)$.
Stationary spots-stripes pattern in model (2). The parameter values are taken as (32) and $(d_1, d_2) = (0.028, 0.45)$. The zero-flux boundary condition is used and initial condition is small perturbation around the homogeneous steady-state $E^* = (1.5934, 2.0768)$
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