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

Dingyong Bai; Jianshe Yu; Yun Kang

doi:10.3934/dcdss.2020132

Article Contents

2020, Volume 13, Issue 11: 2949-2973. Doi: 10.3934/dcdss.2020132

This issue Previous Article Preface: Recent advances in bifurcation theory and application Next Article A nutrient-prey-predator model: Stability and bifurcations

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
^* Corresponding author: Jianshe Yu

Received: November 30, 2018

Revised: January 31, 2019

Early access: November 2019

Published: July 2020

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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 $ \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.

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Mathematics Subject Classification: Primary: 35B36, 45M10; Secondary: 92C15.

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Figure 1. 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) $.

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Figure 2. 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) $.

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Figure 3. 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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