Bifurcation analysis of a stage-structured predator-prey model with prey refuge

Qing Zhu; Huaqin Peng; Xiaoxiao Zheng; Huafeng Xiao

doi:10.3934/dcdss.2019141

Article Contents

2019, Volume 12, Issue 7: 2195-2209. Doi: 10.3934/dcdss.2019141

This issue Previous Article The mean and noise of FPT modulated by promoter architecture in gene networks Next Article

Bifurcation analysis of a stage-structured predator-prey model with prey refuge

a.
School of Mathematics and Statistics, Guangxi Normal University, Guilin 541004, China
b.
School of Mathematics and Information Science, Guangzhou University, Guangzhou 510006, China
c.
School of Mathematical Sciences, Qufu Normal University, Qufu 273165, China

^* Corresponding author: Huaqin Peng
^* Corresponding author: Huaqin Peng

Received: June 30, 2017

Revised: June 30, 2018

Published: June 2019

The research is supported by NNSF of China grant 11301102, 11771104, 11701113, Guangzhou Education Bureau 1201431215 and Key Laboratory of Mathematics and Statistical Model of Guangxi Colleges Open Foundation 2017GXKLMS007

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Abstract

A stage-structured predator-prey model with prey refuge is considered. Using the geometric stability switch criteria, we establish stability of the positive equilibrium. Stability and direction of periodic solutions arising from Hopf bifurcations are obtained by using the normal form theory and center manifold argument. Numerical simulations confirm the above theoretical results.

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Mathematics Subject Classification: Primary: 34K18, 34K20; Secondary: 92D25.

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Figure 1. For $ \tau\in[0,\bar{\tau}) $, the graph of $ S_0(\tau) $ and $ S_1(\tau) $

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Figure 2. For $ \tau = 0.6<\tau^* $ and the initial value $ " 1.0 , 1.0 " $, the positive equilibrium point of system (49) is stable

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Figure 3. For τ = 22 > τ^* and the initial value " 0.81, 0.61 ", system (49) exhibits a periodic solution

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