Bifurcation analysis of a delayed predator–prey system with strong Allee effect and diffusion

Abstract

This article is concerned with a predator–prey system with two delays and a unique positive equilibrium E _*. The dynamics are studied in terms of the local stability of E _* and of the description of the Hopf bifurcation that is proven to exist as the delays cross some critical values. The direction of Hopf bifurcation and stability of bifurcated periodic solution are discussed by employing the normal form theory and the centre manifold. We also consider a reaction-diffusion system with Neumann conditions and the results obtained for the case without diffusion are applied.

Keywords:

• diffusion
• strong Allee effect
• discrete delay
• Hopf bifurcation

AMS Subject Classifications::

• 34C23
• 34C25
• 92D25

Acknowledgement

This research was supported by the National Natural Science Foundation of China (No.11031002).