Discrete and Continuous Dynamical Systems - Series B (DCDS-B)

Dynamical complexity of a prey-predator model with nonlinear predator harvesting

Pages: 423 - 443, Volume 20, Issue 2, March 2015      doi:10.3934/dcdsb.2015.20.423

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R. P. Gupta - Department of Mathematics and Statistics, Indian Institute of Technology Kanpur, Kanpur-208016, India (email)
Peeyush Chandra - Department of Mathematics and Statistics, Indian Institute of Technology Kanpur, Kanpur-208016, India (email)
Malay Banerjee - Department of Mathematics and Statistics, Indian Institute of Technology Kanpur, Kanpur, Uttar Pradesh, India (email)

Abstract: The objective of this paper is to study systematically the dynamical properties of a predator-prey model with nonlinear predator harvesting. We show the different types of system behaviors for various parameter values. The results developed in this article reveal far richer dynamics compared to the model without harvesting. The occurrence of change of structure or bifurcation in a system with parameters is a way to predict global dynamics of the system. It has been observed that the model has at most two interior equilibria and can exhibit numerous kinds of bifurcations (e.g. saddle-node, transcritical, Hopf-Andronov and Bogdanov-Takens bifurcation). The stability (direction) of the Hopf-bifurcating periodic solutions has been obtained by computing the first Lyapunov number. The emergence of homoclinic loop has been shown through numerical simulation when the limit cycle arising though Hopf-bifurcation collides with a saddle point. Numerical simulations using MATLAB are carried out as supporting evidences of our analytical findings. The main purpose of the present work is to offer a complete mathematical analysis for the model.

Keywords:  Prey-predator model, harvesting, stability, bifurcation, phase portraits.
Mathematics Subject Classification:  Primary: 70K05, 34C23; Secondary: 34D20.

Received: May 2013;      Revised: July 2014;      Available Online: January 2015.

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