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Bifurcation Patterns in a Discrete Predator–Prey Model Incorporating Ratio-Dependent Functional Response and Prey Harvesting

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Abstract

This work examines a discrete Leslie-Gower model of prey-predator dynamics with Holling type-IV functional response and harvesting effects. The study includes the existence and local stability analysis of all fixed points. Using center manifold theory, the codimension-1 bifurcations, viz. transcritical, Neimark–Sacker, fold, and period-doubling bifurcations, are determined for varying parameters. Moreover, the existence of codimension-2 Bogdanov–Takens bifurcation and Chenciner bifurcation is demonstrated, requiring two parameters to vary for the bifurcation to occur, and the non-degeneracy conditions for Bogdanov–Takens bifurcation are determined. An extensive numerical study is conducted to confirm the analytical findings. A wide range of dense, chaotic windows can be seen in the system, including period-2, 4, 8, and 16, period-doubling bifurcations, Neimark–Sacker bifurcations, and Chenciner and BT curves following two-parameters bifurcations. Further, it is also shown that the effect of harvesting parameter of the model for which the population dies out.

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Funding

Anuraj Singh’s endeavors receive support from a Core Research Grant provided by the Science Engineering Research Board, an entity of the Indian government (CRG/2021/006380). The contribution of Vijay Shankar Sharma is funded by the University Grant Commission (UGC), also under the Indian government.

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Authors and Affiliations

  1. Department of Mathematics, Maulana Azad National Institute of Technology (MANIT), Bhopal, MP, 462003, India

    Vijay Shankar Sharma

  2. Department of Engineering Sciences, ABV-Indian Institute of Information Technology and Management Gwalior, Gwalior, MP, 474015, India

    Anuraj Singh

  3. Department of Mathematics, Faculty of Science, SGT University, Gurugram, Haryana, 122505, India

    Pradeep Malik

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  1. Vijay Shankar Sharma
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  2. Anuraj Singh
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Contributions

VSS: Conceptualization, Methodology, Software, Validation, Formal analysis, Investigation, Data curation, Writing-original draft preparation. AS: Conceptualization, Methodology, Formal analysis, Resources, Supervision, Visualization, Investigation, Project administration. PM: Software, Validation, Formal analysis.

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Correspondence to Anuraj Singh.

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Sharma, V.S., Singh, A. & Malik, P. Bifurcation Patterns in a Discrete Predator–Prey Model Incorporating Ratio-Dependent Functional Response and Prey Harvesting. Qual. Theory Dyn. Syst. 23, 74 (2024). https://doi.org/10.1007/s12346-023-00929-2

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