Under nonlinear prey-harvesting, effect of strong Allee effect on the dynamics of a modified Leslie-Gower predator-prey model

Manoj K. Singh; Brajesh K. Singh; Poonam; Carlo Cattani; Manoj K. Singh; Brajesh K. Singh; Poonam; Carlo Cattani

doi:10.3934/mbe.2023422

Mathematical Biosciences and Engineering

2023, Volume 20, Issue 6: 9625-9644. doi: 10.3934/mbe.2023422

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Under nonlinear prey-harvesting, effect of strong Allee effect on the dynamics of a modified Leslie-Gower predator-prey model

1.
Department of Mathematics and Statistics, Banasthali Vidyapith, Village-Banasthali, District-Tonk, Rajasthan 304022, India
2.
Department of Mathematics, Babasaheb Bhimrao Ambedkar University, Lucknow 226025, India
3.
Department of Mathematics and Informatics, Azerbaijan University, J. Hajibeyli str., AZ1007, Baku, Azerbaijan
4.
Engineering School, DEIM, University of Tuscia, P.le dellUniversità, Viterbo 01100, Italy

Academic Editor: Francesco Villecco

Received: 29 December 2022 Revised: 20 February 2023 Accepted: 24 February 2023 Published: 22 March 2023

In the present study, the effects of the strong Allee effect on the dynamics of the modified Leslie-Gower predator-prey model, in the presence of nonlinear prey-harvesting, have been investigated. In our findings, it is seen that the behaviors of the described mathematical model are positive and bounded for all future times. The conditions for the local stability and existence for various distinct equilibrium points have been determined. The present research concludes that system dynamics are vulnerable to initial conditions. In addition, the presence of several types of bifurcations (e.g., saddle-node bifurcation, Hopf bifurcation, Bogdanov-Takens bifurcation, homoclinic bifurcation) has been investigated. The first Lyapunov coefficient has been evaluated to study the stability of the limit cycle that results from Hopf bifurcation. The presence of a homoclinic loop has been demonstrated by numerical simulation. Finally, possible phase drawings and parametric figures have been depicted to validate the outcomes.
- predator-prey model,
- stability,
- bifurcation,
- harvesting,
- Allee effect
Citation: Manoj K. Singh, Brajesh K. Singh, Poonam, Carlo Cattani. Under nonlinear prey-harvesting, effect of strong Allee effect on the dynamics of a modified Leslie-Gower predator-prey model[J]. Mathematical Biosciences and Engineering, 2023, 20(6): 9625-9644. doi: 10.3934/mbe.2023422

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Abstract

In the present study, the effects of the strong Allee effect on the dynamics of the modified Leslie-Gower predator-prey model, in the presence of nonlinear prey-harvesting, have been investigated. In our findings, it is seen that the behaviors of the described mathematical model are positive and bounded for all future times. The conditions for the local stability and existence for various distinct equilibrium points have been determined. The present research concludes that system dynamics are vulnerable to initial conditions. In addition, the presence of several types of bifurcations (e.g., saddle-node bifurcation, Hopf bifurcation, Bogdanov-Takens bifurcation, homoclinic bifurcation) has been investigated. The first Lyapunov coefficient has been evaluated to study the stability of the limit cycle that results from Hopf bifurcation. The presence of a homoclinic loop has been demonstrated by numerical simulation. Finally, possible phase drawings and parametric figures have been depicted to validate the outcomes.

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