Nonlinear Dynamics

, Volume 78, Issue 1, pp 49–70 | Cite as

Bifurcation analysis of a diffusive ratio-dependent predator–prey model

  • Yongli Song
  • Xingfu Zou
Original Paper


In this paper, a ratio-dependent predator–prey model with diffusion is considered. The stability of the positive constant equilibrium, Turing instability, and the existence of Hopf and steady state bifurcations are studied. Necessary and sufficient conditions for the stability of the positive constant equilibrium are explicitly obtained. Spatially heterogeneous steady states with different spatial patterns are determined. By calculating the normal form on the center manifold, the formulas determining the direction and the stability of Hopf bifurcations are explicitly derived. For the steady state bifurcation, the normal form shows the possibility of pitchfork bifurcation and can be used to determine the stability of spatially inhomogeneous steady states. Some numerical simulations are carried out to illustrate and expand our theoretical results, in which, both spatially homogeneous and heterogeneous periodic solutions are observed. The numerical simulations also show the coexistence of two spatially inhomogeneous steady states, confirming the theoretical prediction.


Predator–prey Diffusion Ratio dependent Turing instability Hopf bifurcation Pitchfork bifurcation  Normal form 



The first author is supported by the State Key Program of National Natural Science of China (No. 11032009), the Scientific Research Foundation for the Returned Overseas Chinese Scholars, the Fundamental Research Funds for the Central Universities, and the Program for New Century Excellent Talents in University (NCET-11-0385); and the second author is partially supported by Natural Science and Engineering Council of Canada.


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Copyright information

© Springer Science+Business Media Dordrecht 2014

Authors and Affiliations

  1. 1.Department of MathematicsTongji UniversityShanghai People’s Republic of China
  2. 2.Department of Applied MathematicsUniversity of Western OntarioLondonCanada

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