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Stability and bifurcation in a harvested one-predator–two-prey model with delays

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  • Liu, Zhihua
  • Yuan, Rong

Abstract

It is known that one-predator–two-prey system with constant rate harvesting can exhibit very rich dynamics. If such a system contains time delayed component, it can have more interesting behavior. In this paper we study the effects of the time delay on the dynamics of the harvested one-predator–two-prey model. It is shown that time delay can cause a stable equilibrium to become unstable. By choosing the delay τ as a bifurcation parameter, we show that Hopf bifurcation can occur as the delay τ crosses some critical values. The direction and stability of the Hopf bifurcation are investigated by following the procedure of deriving normal form given by Faria and Magalhães. An example is given and numerical simulations are finally performed for justifying the theoretical results.

Suggested Citation

  • Liu, Zhihua & Yuan, Rong, 2006. "Stability and bifurcation in a harvested one-predator–two-prey model with delays," Chaos, Solitons & Fractals, Elsevier, vol. 27(5), pages 1395-1407.
    • Handle: RePEc:eee:chsofr:v:27:y:2006:i:5:p:1395-1407
    DOI: 10.1016/j.chaos.2005.05.014
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    Cited by:

    1. Zhang, Long & Teng, Zhidong, 2008. "Permanence for a class of periodic time-dependent predator–prey system with dispersal in a patchy-environment," Chaos, Solitons & Fractals, Elsevier, vol. 38(5), pages 1483-1497.
    2. Zhang, Xue & Zhang, Qing-Ling & Liu, Chao & Xiang, Zhong-Yi, 2009. "Bifurcations of a singular prey–predator economic model with time delay and stage structure," Chaos, Solitons & Fractals, Elsevier, vol. 42(3), pages 1485-1494.
    3. Jiang, Zhichao & Wei, Junjie, 2008. "Stability and bifurcation analysis in a delayed SIR model," Chaos, Solitons & Fractals, Elsevier, vol. 35(3), pages 609-619.
    4. Xu, Rui & Ma, Zhien, 2008. "Stability and Hopf bifurcation in a ratio-dependent predator–prey system with stage structure," Chaos, Solitons & Fractals, Elsevier, vol. 38(3), pages 669-684.
    5. Ling, Li & Wang, Weiming, 2009. "Dynamics of a Ivlev-type predator–prey system with constant rate harvesting," Chaos, Solitons & Fractals, Elsevier, vol. 41(4), pages 2139-2153.
    6. Zhang, Long & Teng, Zhidong, 2008. "Boundedness and permanence in a class of periodic time-dependent predator–prey system with prey dispersal and predator density-independence," Chaos, Solitons & Fractals, Elsevier, vol. 36(3), pages 729-739.

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