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The Gestation Delay: A Factor Causing Complex Dynamics in Gause-Type Competition Models

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  • Zizhen Zhang
  • Ranjit Kumar Upadhyay
  • Rashmi Agrawal
  • Jyotiska Datta

Abstract

In this paper, we consider a Gause-type model system consisting of two prey and one predator. Gestation period is considered as the time delay for the conversion of both the prey and predator. Bobcats and their primary prey rabbits and squirrels, found in North America and southern Canada, are taken as an example of an ecological system. It has been observed that there are stability switches and the system becomes unstable due to the effect of time delay. Positive invariance, boundedness, and local stability analysis are studied for the model system. Conditions under which both delayed and nondelayed model systems remain globally stable are found. Criteria which guarantee the persistence of the delayed model system are derived. Conditions for the existence of Hopf bifurcation at the nonzero equilibrium point of the delayed model system are also obtained. Formulae for the direction, stability, and period of the bifurcating solution are conducted using the normal form theory and center manifold theorem. Numerical simulations have been shown to analyze the effect of each of the parameters considered in the formation of the model system on the dynamic behavior of the system. The findings are interesting from the application point of view.

Suggested Citation

  • Zizhen Zhang & Ranjit Kumar Upadhyay & Rashmi Agrawal & Jyotiska Datta, 2018. "The Gestation Delay: A Factor Causing Complex Dynamics in Gause-Type Competition Models," Complexity, Hindawi, vol. 2018, pages 1-21, November.
    • Handle: RePEc:hin:complx:1589310
    DOI: 10.1155/2018/1589310
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    Cited by:

    1. Pati, N.C. & Ghosh, Bapan, 2022. "Delayed carrying capacity induced subcritical and supercritical Hopf bifurcations in a predator–prey system," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 195(C), pages 171-196.
    2. Zhang, Zizhen & Kundu, Soumen & Tripathi, Jai Prakash & Bugalia, Sarita, 2020. "Stability and Hopf bifurcation analysis of an SVEIR epidemic model with vaccination and multiple time delays," Chaos, Solitons & Fractals, Elsevier, vol. 131(C).
    3. Mishra, P. & Raw, S.N. & Tiwari, B., 2019. "Study of a Leslie–Gower predator-prey model with prey defense and mutual interference of predators," Chaos, Solitons & Fractals, Elsevier, vol. 120(C), pages 1-16.
    4. Bhunia, Bidhan & Ghorai, Santu & Kar, Tapan Kumar & Biswas, Samir & Bhutia, Lakpa Thendup & Debnath, Papiya, 2023. "A study of a spatiotemporal delayed predator–prey model with prey harvesting: Constant and periodic diffusion," Chaos, Solitons & Fractals, Elsevier, vol. 175(P1).

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