Spatial, temporal and spatiotemporal patterns of diffusive predator-prey models with mutual interference

Hong Bo Shi, Shigui Ruan

Research output: Contribution to journalArticlepeer-review

48 Scopus citations


In this paper, the spatial, temporal and spatiotemporal dynamics of a reaction-diffusion predator-prey system with mutual interference described by the Crowley-Martin-type functional response, under homogeneous Neumann boundary conditions, are studied. Preliminary analysis on the local asymptotic stability and Hopf bifurcation of the spatially homogeneous model based on ordinary differential equations is presented. For the reaction-diffusion model, firstly the invariance, uniform persistence and global asymptotic stability of the coexistence equilibrium are discussed. Then it is shown that Turing (diffusion-driven) instability occurs, which induces spatial inhomogeneous patterns. Next it is proved that the model exhibits Hopf bifurcation which produces temporal inhomogeneous patterns. Furthermore, at the points where the Turing instability curve and Hopf bifurcation curve intersect, it is demonstrated that the model undergoes Turing-Hopf bifurcation and exhibits spatiotemporal patterns. Finally, the existence and non-existence of positive non-constant steady states of the reaction-diffusion model are established. Numerical simulations are presented to verify and illustrate the theoretical results.

Original languageEnglish (US)
Pages (from-to)1534-1568
Number of pages35
JournalIMA Journal of Applied Mathematics (Institute of Mathematics and Its Applications)
Issue number5
StatePublished - Oct 28 2014


  • Hopf bifurcation
  • Turing instability
  • Turing-Hopf bifurcation
  • diffusive predator-prey model
  • mutual interference
  • positive non-constant steady states

ASJC Scopus subject areas

  • Applied Mathematics


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