Stability and bifurcation analysis of predator-prey model with Allee effect using conformable derivatives

Volume 36, Issue 3, pp 299--316 https://dx.doi.org/10.22436/jmcs.036.03.05

Publication Date: August 07, 2024 Submission Date: September 18, 2023 Revision Date: May 13, 2024 Accteptance Date: July 08, 2024

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Authors

M‎. ‎B‎. ‎ Almatrafi - Department of Mathematics‎, ‎College of Science, ‎Taibah University, ‎Al-Madinah Al-Munawarah, ‎Saudi Arabia. M. Berkal - Department of Applied Mathematics, ‎University of Alicante, ‎03690 Alicante, ‎Spain.

Abstract

‎Some organisms coexist on the expense of others‎. ‎This coexistence is called predation which has been successfully investigated using differential equations‎. ‎In this work‎, ‎we aim to analyse a fractional order predator-prey dynamical system with Allee effect using bifurcation theory‎. ‎The Allee effect is a density-dependent phenomenon where the population growth and individual fitness increase as population density increases‎. ‎Several mechanisms‎, ‎such as cooperative feeding‎, ‎mate limitation‎, ‎and predator satiation‎, ‎can cause Allee effects‎. ‎{The piecewise-constant approximation method and the conformable derivatives are utilized to discretise the propose model.} We explore equilibrium points‎, ‎the local stability‎, ‎the Neimark-Sacker bifurcation‎, ‎periodic-doubling bifurcation‎, ‎chaos control‎, ‎and numerical simulations of the proposed model‎. ‎The linear theory of stability is used to examine the local attractivity of the fixed points‎. ‎Our findings include that the coexistence equilibrium point is locally stable‎, ‎source‎, ‎unstable under certain constraints‎. ‎We also prove that the considered discrete model goes through Neimark-Sacker and periodic-doubling bifurcations according to specific conditions‎. ‎The used techniques can be applied for other nonlinear discrete systems‎.

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Keywords

• Stability
• Allee effect
• prey-predator model
• Neimark-Sacker bifurcation
• period-doubling bifurcation
• fractional derivatives
• chaos control‎

MSC

•  37N25
•  37N30
•  34D35
•  34D05
•  34D08
•  39A28
•  39A30
•  39A05

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