Stability and Hopf bifurcation in a prey-predator model with memory-based diffusion

Shu Li; Zhenzhen Li; Binxiang Dai

doi:10.3934/dcdsb.2022025

Article Contents

2022, Volume 27, Issue 11: 6885-6906. Doi: 10.3934/dcdsb.2022025

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Stability and Hopf bifurcation in a prey-predator model with memory-based diffusion

School of Mathematics and Statistics, Central South University, Changsha, Hunan, 410083, China

^*Corresponding author: Binxiang Dai
^*Corresponding author: Binxiang Dai

Received: April 2021

Revised: December 2021

Early access: February 18, 2022

Published online: February 18, 2022

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Abstract

In this paper, we consider a predator-prey model with memory-based diffusion. We first analyze the stability of all steady states in detail. Then by analyzing the distribution of eigenvalues, we find that the average memory period can cause the stability change of the positive steady state, and Hopf bifurcation occurs at the positive steady state. Moreover, from the central manifold theorem and the normal form theory, we give the direction and stability of Hopf bifurcation. The results show that, under certain conditions, a family of spatially inhomogeneous periodic solutions will bifurcate from the positive steady state when the average memory period appear.

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Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.

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Figure 1. Numerical simulation of (5) for $ \tau = 0 $ with parameter condition $ {\bf(P1)} $, the unique positive equilibrium $ E_* = (0.8, 0.2) $ of (5) is locally asymptotically stable.

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Figure 2. Numerical simulation of (5) for $ \tau = 3 $ with parameter condition $ {\bf(P1)} $, the unique positive equilibrium $ E_* = (0.8, 0.2) $ of (5) is locally asymptotically stable.

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Figure 3. Numerical simulation of (5) for $\tau=0$ with parameter condition ${\bf(P2)}$, the unique positive equilibrium $E_*=(0.25,0.75)$ of (5) is locally asymptotically stable.

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