Discrete and Continuous Dynamical Systems - Series S (DCDS-S)

Turing instability in a coupled predator-prey model with different Holling type functional responses

Pages: 1621 - 1628, Volume 4, Issue 6, December 2011      doi:10.3934/dcdss.2011.4.1621

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Zhifu Xie - Department of Mathematics and Computer Science, Virginia State University, Petersburg, Virginia 23806, United States (email)

Abstract: In a reaction-diffusion system, diffusion can induce the instability of a positive equilibrium which is stable with respect to a constant perturbation, therefore, the diffusion may create new patterns when the corresponding system without diffusion fails, as shown by Turing in 1950s. In this paper we study a coupled predator-prey model with different Holling type functional responses, where cross-diffusions are included in such a way that the prey runs away from predator and the predator chase preys. We conduct the Turing instability analysis for each Holling functional response. We prove that if a positive equilibrium solution is linearly stable with respect to the ODE system of the predator-prey model, then it is also linearly stable with respect to the model. So diffusion and cross-diffusion in the predator-prey model with Holling type functional responses given in this paper can not drive Turing instability. However, diffusion and cross-diffusion can still create non-constant positive solutions for the model.

Keywords:  Predator-prey model; Reaction-diffusion system; Cross-diffusion; Turing instability; Stationary pattern; Holling type functional response.
Mathematics Subject Classification:  Primary: 92C15, 35K57; Secondary: 37L15, 92D40.

Received: April 2009;      Revised: November 2009;      Available Online: December 2010.