Bifurcations and chaos control for a discrete fractional-order Leslie-Gower model incorporating fear in predators and Allee effect in prey

Abdulaziz Almaslokh; Ibrahim M. E. Abdelstar; A. A. Elsadany; Abdulaziz Almaslokh; Ibrahim M. E. Abdelstar; A. A. Elsadany

doi:10.3934/math.2026295

AIMS Mathematics

2026, Volume 11, Issue 3: 7155-7182. doi: 10.3934/math.2026295

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Bifurcations and chaos control for a discrete fractional-order Leslie-Gower model incorporating fear in predators and Allee effect in prey

1.
Department of Mathematics, Faculty of Sciences and Humanities in Al-Kharj, Prince Sattam bin Abdulaziz University, Al-Kharj 11942, Saudi Arabia
2.
Department of Mathematics, Faculty of Science, Al-Azhar University, Nasr City, P.O.Box: 11884, Cairo, Egypt

Received: 14 January 2026 Revised: 09 March 2026 Accepted: 11 March 2026 Published: 18 March 2026
MSC : 37N25, 37N30, 34D35, 34D05, 39A28, 39A30

In this research, we analyze a discrete-time fractional-order modified Leslie-Gower predator-prey model that incorporates the Allee effect on prey and the fear effect on predators. The findings demonstrate complex dynamic behaviors resulting from these elements. First, we show the existence of equilibrium points and the conditions for their stability. Moreover, the incorporation of Allee and fear effects leads to various bifurcations, such as Neimark-Sacker and flip bifurcations. We perform numerical simulations with different values of the fractional parameter $ q $ to illustrate the complex temporal dynamics of the model. The theoretical results are corroborated and confirmed through numerical simulations.
- Leslie-Gower predator-prey model,
- fractional-order,
- fear effect,
- Allee effect,
- Neimark-Sacker,
- flip bifurcations
Citation: Abdulaziz Almaslokh, Ibrahim M. E. Abdelstar, A. A. Elsadany. Bifurcations and chaos control for a discrete fractional-order Leslie-Gower model incorporating fear in predators and Allee effect in prey[J]. AIMS Mathematics, 2026, 11(3): 7155-7182. doi: 10.3934/math.2026295

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Abstract

In this research, we analyze a discrete-time fractional-order modified Leslie-Gower predator-prey model that incorporates the Allee effect on prey and the fear effect on predators. The findings demonstrate complex dynamic behaviors resulting from these elements. First, we show the existence of equilibrium points and the conditions for their stability. Moreover, the incorporation of Allee and fear effects leads to various bifurcations, such as Neimark-Sacker and flip bifurcations. We perform numerical simulations with different values of the fractional parameter $ q $ to illustrate the complex temporal dynamics of the model. The theoretical results are corroborated and confirmed through numerical simulations.

References

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