Basins of attraction in a modified ratio-dependent predator-prey model with prey refugee

Khairul Saleh; Khairul Saleh

doi:10.3934/math.2022816

AIMS Mathematics

2022, Volume 7, Issue 8: 14875-14894. doi: 10.3934/math.2022816

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Basins of attraction in a modified ratio-dependent predator-prey model with prey refugee

Khairul Saleh ^,

Department of Mathematics, King Fahd University of Petroleum and Minerals, Dhahran 31261, Saudi Arabia

Received: 18 February 2022 Revised: 19 May 2022 Accepted: 26 May 2022 Published: 10 June 2022
MSC : 34C60, 37C75, 37G10, 37G35

In this paper, we analyze a modified ratio-dependent predator-prey model with a strong Allee effect and linear prey refugee. The model exhibits rich dynamics with the existence of separatrices in the phase plane in-between basins of attraction associated with oscillation, coexistence, and extinction of the interacting populations. We prove that if the initial values are positive, all solutions are bounded and stay in the interior of the first quadrant. We show that the system undergoes several bifurcations such as transcritical, saddle-node, Hopf, and Bogdanov-Takens bifurcations. Consequently, a homoclinic bifurcation curve exists generating an unstable periodic orbit. Moreover, we find that the Bogdanov-Takens bifurcation acts as an organizing center for the scenario of surviving or extinction of both interacting species. Topologically different phase portraits with all possible trajectories and equilibria are depicted illustrating the behavior of the system.
- predator-prey,
- bifurcation,
- stability,
- refugee,
- basin of attraction
Citation: Khairul Saleh. Basins of attraction in a modified ratio-dependent predator-prey model with prey refugee[J]. AIMS Mathematics, 2022, 7(8): 14875-14894. doi: 10.3934/math.2022816

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Abstract

In this paper, we analyze a modified ratio-dependent predator-prey model with a strong Allee effect and linear prey refugee. The model exhibits rich dynamics with the existence of separatrices in the phase plane in-between basins of attraction associated with oscillation, coexistence, and extinction of the interacting populations. We prove that if the initial values are positive, all solutions are bounded and stay in the interior of the first quadrant. We show that the system undergoes several bifurcations such as transcritical, saddle-node, Hopf, and Bogdanov-Takens bifurcations. Consequently, a homoclinic bifurcation curve exists generating an unstable periodic orbit. Moreover, we find that the Bogdanov-Takens bifurcation acts as an organizing center for the scenario of surviving or extinction of both interacting species. Topologically different phase portraits with all possible trajectories and equilibria are depicted illustrating the behavior of the system.

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