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Global stability and bifurcation analysis of a delay induced prey-predator system with stage structure

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Abstract

This paper describes a delay induced prey–predator system with stage structure for prey. The dynamical characteristics of the system are rigorously studied using mathematical tools. The coexistence equilibria of the system is determined and the dynamic behavior of the system is investigated around coexistence equilibria. Sufficient conditions are derived for the global stability of the system. The optimal harvesting problem is formulated and solved in order to achieve the sustainability of the system, keeping the ecological balance, and maximize the monetary social benefit. Maturation time delay of prey is incorporated and the existence of Hopf bifurcation phenomenon is examined at the coexistence equilibria. It is shown that the time delay can cause a stable equilibrium to become unstable and even a switching of stabilities. Moreover, we use normal form method and center manifold theorem to examine the nature of the Hopf bifurcation. Finally, some numerical simulations are given to verify the analytical results, and the system is analyzed through graphical illustrations.

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Acknowledgements

The first author gratefully acknowledges Director, INCOIS for his encouragement and unconditional help. This is INCOIS contribution number 138.

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Authors and Affiliations

  1. Information Services and Ocean Sciences Group, Indian National Centre for Ocean Information Services, Hyderabad, Ocean Valley, Pragathinagar (BO), Nizampet (SO), Hyderabad, 500090, Andhra Pradesh, India

    Kunal Chakraborty

  2. Department of Mathematics, Hooghly Womens’ College, Pipulpati, Hooghly, 712103, West Bengal, India

    Samadyuti Haldar

  3. Department of Mathematics, Bengal Engineering and Science University, Shibpur, Howrah, 711103, West Bengal, India

    T. K. Kar

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  1. Kunal Chakraborty
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  2. Samadyuti Haldar
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  3. T. K. Kar
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Correspondence to Kunal Chakraborty.

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Chakraborty, K., Haldar, S. & Kar, T.K. Global stability and bifurcation analysis of a delay induced prey-predator system with stage structure. Nonlinear Dyn 73, 1307–1325 (2013). https://doi.org/10.1007/s11071-013-0864-1

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  • DOI: https://doi.org/10.1007/s11071-013-0864-1

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