Bifurcation behaviours in a delayed three-species food-chain model with Holling type-II functional response

Abstract

This article is concerned with the local stability of a positive equilibrium and the Hopf bifurcation of a delayed three-species food-chain system with the Holling type-II functional response. Some new sufficient conditions ensuring the local stability of a positive equilibrium and the existence of Hopf bifurcation for the system are established. Some explicit formulae are derived to determine the direction of bifurcations and the stability of bifurcating periodic solutions using the normal form theory and the centre manifold theory. Numerical simulations for justifying the theoretical analysis are also provided. Finally, biological explanations and main conclusions are included.

Keywords:

• predator–prey model
• holling type-II functional response
• time delay
• stability
• Hopf bifurcation

AMS Subject Classifications::

• 34K20
• 34K25

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Erratum

Acknowledgement

This work is supported by National Natural Science Foundation of China(No.11261010, No.10961008), Soft Science and Technology Program of Guizhou Province(No.2011LKC2030), Natural Science and Technology Foundation of Guizhou Province(J[2012]2100), Governor Foundation of Guizhou Province([2012]53) and Doctoral Foundation of Guizhou University of Finance and Economics (2010).