Diffusion-driven stability and bifurcation in a predator–prey system with Ivlev-type functional response

Abstract

A diffusive predator–prey system with Ivlev-type functional response subject to Neumann boundary conditions is considered. Hopf and steady-state bifurcation analysis are carried out in detail. First, the stability of the positive equilibrium and the existence of spatially homogeneous and inhomogeneous periodic solutions are investigated by analysing the distribution of the eigenvalues. The direction and stability of Hopf bifurcation are determined by the normal form theory and the centre manifold reduction for partial functional differential equations and then steady-state bifurcation is studied. Finally, some numerical simulations are carried out for illustrating the theoretical results.

Keywords:

• prey–predator system
• Ivlev-type functional response
• diffusion
• Hopf bifurcation
• periodic solutions

AMS Subject Classifications::

• 35B32
• 35K57
• 92D25

Acknowledgements

The authors are grateful to the anonymous referees for their helpful comments and valuable suggestions. This research is supported by National Natural Science Foundation of China (No. 11031002).