Stability and bifurcation in a stage-structured predator–prey system with Holling-II functional response and multiple delays

Abstract

In this paper, we analyse a delayed Holling-II predator–prey system with stage-structure for the prey. At first, we study the stability and the existence of periodic solutions via Hopf bifurcation with respect to both delays at the positive equilibrium by analysing the distribution of the roots of the associated characteristic equation. Then, the explicit formula that determine the direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions from the Hopf bifurcation are established by using the normal form method and centre manifold argument. Finally, some numerical simulations are carried out to support the main theoretical results.

Keywords:

• predator–prey system
• delays
• stability
• Hopf-bifurcation
• stage-structure
• periodic solutions

2000 AMS Subject Classifications:

• 34C23
• 34K18

Acknowledgements

This work was supported by the National Natural Science Foundation of China (61273070), a project Funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions and Natural Science Foundation of the Higher Education Institutions of Anhui Province (KJ2013A003, KJ2013B137).