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Analysis of Bogdanov–Takens bifurcations in a spatiotemporal harvested-predator and prey system with Beddington–DeAngelis-type response function

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Abstract

In this article, we consider a predator–prey system with constant rate of harvesting, which exhibits Hopf and Bogdanov–Takens bifurcations under certain parametric conditions. The parametric space under which the system enters into Hopf bifurcation is investigated. By constructing suitable Lyapunov function, global stability results are obtained. Here, death rate and harvesting rate are taken as the Bogdanov–Takens bifurcation parameters. The canonical form of Bogdanov–Takens bifurcation is derived with the use of repeated nonlinear analytic transformation of coordinates. Later, we include the spatiotemporal effect on the same system and observed some relevant outcomes like Turing pattern, Turing–Bogdanov–Takens bifurcation, Turing–Hopf bifurcation and asynchrony of predator and prey in the space. The present study renders important tools for investigations of the dynamics of biotic organisms (predator and prey) for the management and control of overharvesting. Extensive numerical examples are given in support of the physical existence of the model system under consideration.

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Acknowledgements

The authors are grateful to the Department of Mathematics and Statistics, Aliah University, for furnishing opportunities to perform the present work. The authors do appreciate Dr. Mehboob Hoque, Department of Biological Sciences, Aliah University, for his ample help to improve the quality of the manuscript. The corresponding author Dr. Sarwardi is thankful to Dr. Mohd. Hafiz Mohd., School of Mathematical Sciences, Universiti Sains Malaysia, for verifying numerical results of the present refined manuscript.

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  1. Department of Mathematics and Statistics, Aliah University, IIA/27, New Town, Kolkata, 700 160, India

    Sahabuddin Sarwardi, Md. Manarul Haque & Sajjad Hossain

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  1. Sahabuddin Sarwardi
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Sarwardi, S., Haque, M.M. & Hossain, S. Analysis of Bogdanov–Takens bifurcations in a spatiotemporal harvested-predator and prey system with Beddington–DeAngelis-type response function. Nonlinear Dyn 100, 1755–1778 (2020). https://doi.org/10.1007/s11071-020-05549-y

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