Acta Applicandae Mathematicae

, Volume 115, Issue 1, pp 91–104

Hopf Bifurcations in a Predator-Prey Diffusion System with Beddington-DeAngelis Response


DOI: 10.1007/s10440-010-9593-3

Cite this article as:
Zhang, JF., Li, WT. & Yan, XP. Acta Appl Math (2011) 115: 91. doi:10.1007/s10440-010-9593-3


This paper is concerned with a two-species predator-prey reaction-diffusion system with Beddington-DeAngelis functional response and subject to homogeneous Neumann boundary conditions. By linearizing the system at the positive constant steady-state solution and analyzing the associated characteristic equation in detail, the asymptotic stability of the positive constant steady-state solution and the existence of local Hopf bifurcations are investigated. Also, it is shown that the appearance of the diffusion and homogeneous Neumann boundary conditions can lead to the appearance of codimension two Bagdanov-Takens bifurcation. Moreover, by applying the normal form theory and the center manifold reduction for partial differential equations (PDEs), the explicit algorithm determining the direction of Hopf bifurcations and the stability of bifurcating periodic solutions is given. Finally, numerical simulations supporting the theoretical analysis are also included.


Predator-prey systemDiffusionStabilityHopf bifurcationBogdanov-Takens bifurcation

Mathematics Subject Classification (2000)


Copyright information

© Springer Science+Business Media B.V. 2010

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsLanzhou UniversityLanzhouPeople’s Republic of China
  2. 2.Department of MathematicsLanzhou Jiaotong UniversityLanzhouPeople’s Republic of China