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IMA Journal of Applied Mathematics Advance Access originally published online on March 19, 2009
IMA Journal of Applied Mathematics 2009 74(4):574-603; doi:10.1093/imamat/hxp013
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© The Author 2009. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.

Bifurcation and stability analysis for a delayed Leslie–Gower predator–prey system

Sanling Yuan{dagger}

College of Science, Shanghai University for Science and Technology, Shanghai 200093, People's Repulic of China

Yongli Song

Department of Mathematics, Tongji University, Shanghai 200092, People's of Republic of China

{dagger} Email: yuansanling{at}263.net

Received on June 26, 2008; Revision received January 27, 2009. Accepted on February 11, 2009

A delayed Leslie–Gower predator–prey system is considered in this paper. It is assumed that the predator and the prey species have the same feedback delay to their growth. Using the delay as a bifurcation parameter, our results show that the positive equilibrium can only be asymptotically stable or unstable depending on the delays and that Hopf bifurcations can occur as the delay crosses some critical values. The model can exhibit an interesting property, i.e. under certain conditions, the positive equilibrium may switch a finite number of times between being stable and unstable, but always becomes unstable eventually. By deriving the equation describing the flow on the centre manifold, we can determine the direction of the Hopf bifurcations and the stability of the bifurcating periodic solutions. In addition, special attention is paid to the global continuation of local Hopf bifurcations. Using a global Hopf bifurcation result of Wu (1998, Trans. Am. Math. Soc., 350, 4799–4838 for functional differential equations, we may show the global existence of periodic solutions. Computer simulations illustrate the results.

Keywords: time delay; local Hopf bifurcation; global Hopf bifurcation; periodic solutions.


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