The study on the complex nature of a predator-prey model with fractional-order derivatives incorporating refuge and nonlinear prey harvesting

Kottakkaran Sooppy Nisar; G Ranjith Kumar; K Ramesh; Kottakkaran Sooppy Nisar; G Ranjith Kumar; K Ramesh

doi:10.3934/math.2024657

AIMS Mathematics

2024, Volume 9, Issue 5: 13492-13507. doi: 10.3934/math.2024657

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The study on the complex nature of a predator-prey model with fractional-order derivatives incorporating refuge and nonlinear prey harvesting

1.
Department of Mathematics, College of Science and Humanities in Alkharj, Prince Sattam bin Abdulaziz University, 11942, Saudi Arabia; Email: n.sooppy@psau.edu.sa
2.
Saveetha School of Engineering, SIMATS, Chennai, India
3.
Department of Mathematics, Anurag University, Venkatapur, Hyderabad-500088, Telangana, India; Email: ranjithreddy1982@gmail.com, krameshrecw@gmail.com

Received: 08 March 2024 Revised: 24 March 2024 Accepted: 01 April 2024 Published: 11 April 2024
MSC : 34A08, 37N25, 34C23

The main objective of our research was to explore and develop a fractional-order derivative within the predator-prey framework. The framework includes prey refuge and selective nonlinear harvesting, where the harvesting progressively approaches a threshold value as the density of the harvested population advances. For memory effect, a non-integer order derivative is better than an integer-order derivative. The solutions to the fractional framework were shown to be existence, uniqueness, non-negativity, and boundedness. Matignon's condition was used for analysing local stability, and a suitable Lyapunov function provided global stability. While discussing the Hopf bifurcation's existence condition, we explored derivative order and refuge as bifurcation parameters. We aimed at redefining the predator-prey framework to incorporate fractional order, refuge, and harvesting. This kind of nonlinear harvesting is more realistic and reasonable than the model with constant yield harvesting and constant effort harvesting. The Adams-Bashforth-Moulton PECE algorithm in MATLAB software was used to simulate the proposed outcomes, investigate the impact on various factors, and analyse harvesting's effect on non-integer order predator-prey interactions.
- fractional order,
- refuge,
- harvesting,
- stability,
- Hopf bifurcation
Citation: Kottakkaran Sooppy Nisar, G Ranjith Kumar, K Ramesh. The study on the complex nature of a predator-prey model with fractional-order derivatives incorporating refuge and nonlinear prey harvesting[J]. AIMS Mathematics, 2024, 9(5): 13492-13507. doi: 10.3934/math.2024657

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Abstract

The main objective of our research was to explore and develop a fractional-order derivative within the predator-prey framework. The framework includes prey refuge and selective nonlinear harvesting, where the harvesting progressively approaches a threshold value as the density of the harvested population advances. For memory effect, a non-integer order derivative is better than an integer-order derivative. The solutions to the fractional framework were shown to be existence, uniqueness, non-negativity, and boundedness. Matignon's condition was used for analysing local stability, and a suitable Lyapunov function provided global stability. While discussing the Hopf bifurcation's existence condition, we explored derivative order and refuge as bifurcation parameters. We aimed at redefining the predator-prey framework to incorporate fractional order, refuge, and harvesting. This kind of nonlinear harvesting is more realistic and reasonable than the model with constant yield harvesting and constant effort harvesting. The Adams-Bashforth-Moulton PECE algorithm in MATLAB software was used to simulate the proposed outcomes, investigate the impact on various factors, and analyse harvesting's effect on non-integer order predator-prey interactions.

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