Hopf bifurcation analysis in a diffusive predator-prey system with delay and surplus killing effect

Zuolin Shen; Junjie Wei

doi:10.3934/mbe.2018031

Article Contents

2018, Volume 15, Issue 3: 693-715. Doi: 10.3934/mbe.2018031

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Hopf bifurcation analysis in a diffusive predator-prey system with delay and surplus killing effect

Zuolin Shen and
Junjie Wei^,

Department of Mathematics, Harbin Institute of Technology, Harbin, Heilongjiang, 150001, China

* Corresponding author. Email: weijj@hit.edu.cn
* Corresponding author. Email: weijj@hit.edu.cn

Received: March 13, 2017

Accepted: September 29, 2017

Published: May 2018

This research is supported by National Natural Science Foundation of China (Nos.11371111 and 11771109).

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Abstract

A diffusive predator-prey system with a delay and surplus killing effect subject to Neumann boundary conditions is considered. When the delay is zero, the prior estimate of positive solutions and global stability of the constant positive steady state are obtained in details. When the delay is not zero, the stability of the positive equilibrium and existence of Hopf bifurcation are established by analyzing the distribution of eigenvalues. Furthermore, an algorithm for determining the direction of Hopf bifurcation and stability of bifurcating periodic solutions is derived by using the theory of normal form and center manifold. Finally, some numerical simulations are presented to illustrate the analytical results obtained.

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Mathematics Subject Classification: Primary: 35K57, 35R10; Secondary: 35P15.

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Figure 1. The critical curve on $(m_1,\gamma)$ plane. Ⅰ: $E_*(u_*,v_*)$ is global asymptotically stable; Ⅱ: $E_*(u_*,v_*)$ is local asymptotically stable; Ⅲ: $E_*(u_*,v_*)$ disappears while $E_1(0,1)$ is global asymptotically stable. The parameters are chosen as follows: $\alpha=0.3$, $K_2=0.2$, $\theta=0.5$ with $m_2=\alpha m_1/K_2$.

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Figure 2. The positive equilibrium is asymptotically stable when $\tau\in[0, \tau^*)$, where $\tau=2＜\tau^*\approx4.6242$.

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Figure 3. The bifurcating periodic solution is stable, where $\tau=5>\tau^*\approx4.6242$.

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Figure 4. The axial equilibrium $E_1(0,1)$ is global asymptotically stable.

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