Volume 38, Issue 14 pp. 2994-3006
Research Article

Bifurcation analysis of a diffusive predator–prey system with a herd behavior and quadratic mortality

Zhou Xu

Zhou Xu

Department of Mathematics, Tongji University, Shanghai, 200092 China

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Yongli Song

Corresponding Author

Yongli Song

Department of Mathematics, Tongji University, Shanghai, 200092 China

Correspondence to: Yongli Song, Department of Mathematics, Tongji University, Shanghai 200092, China

E-mail: [email protected]

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First published: 25 September 2014
Citations: 25

Abstract

In this paper, a diffusive predator–prey system, in which the prey species exhibits herd behavior and the predator species with quadratic mortality, has been studied. The stability of positive constant equilibrium, Hopf bifurcations, and diffusion-driven Turing instability are investigated under the Neumann boundary condition. The explicit condition for the occurrence of the diffusion-driven Turing instability is derived, which is determined by the relationship of the diffusion rates of two species. The formulas determining the direction and the stability of Hopf bifurcations depending on the parameters of the system are derived. Finally, numerical simulations are carried out to verify and extend the theoretical results and show the existence of spatially homogeneous periodic solutions and nonconstant steady states. Copyright © 2014 John Wiley & Sons, Ltd.

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