Stability and Neimark–Sacker bifurcation in Runge–Kutta methods for a predator–prey system

Abstract

We investigate the discretization of a predator–prey system with two delays under the general Runge–Kutta methods. It is shown that if the exact solution undergoes a Hopf bifurcation at τ=τ*, then the numerical solution undergoes a Neimark–Sacker bifurcation at τ(h)=τ*+O(h ^p ) for sufficiently small step size h, where p≥1 is the order of the Runge–Kutta method applied. The direction of Neimark–Sacker bifurcation and stability of bifurcating invariant curve are the same as that of delay differential equation.

Keywords:

• predator–prey system
• the Runge–Kutta method
• Neimark–Sacker bifurcation

2000 AMS Subject Classifications :

• 39A11
• 65P30
• 92D25

Acknowledgements

This work is supported by the NSF of P.R. China (No. 10671047). We would like to thank the referees and the associate editors for their valuable suggestions that enabled us to improve the presentation of this paper.