Feedback stabilization for prey predator general model with diffusion via multiplicative controls

Ilyasse Lamrani; Imad El Harraki; M. A. Aziz-Alaoui; Fatima-Zahrae El Alaoui; Ilyasse Lamrani; Imad El Harraki; M. A. Aziz-Alaoui; Fatima-Zahrae El Alaoui

doi:10.3934/math.2023122

AIMS Mathematics

2023, Volume 8, Issue 1: 2360-2385. doi: 10.3934/math.2023122

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Feedback stabilization for prey predator general model with diffusion via multiplicative controls

1.
TSI Team, Department of Mathematics, Moulay Ismail University, Faculty of Sciences, Meknes, Morocco
2.
OMACSIA Team, Laboratory of Applied Mathematics and Business Intelligence, Department of Earth Sciences, Rabat Superior National School of Mines, Rabat, Morocco
3.
Normandie Univ, UNIHAVRE, LMAH, FR-CNRS-3335, ISCN, Le Havre 76600, France

Received: 30 July 2022 Revised: 01 October 2022 Accepted: 25 October 2022 Published: 02 November 2022
MSC : 92D25, 35K57, 93D15, 93D23

In this paper, we consider a predator–prey model given by a reaction–diffusion system. This model encompasses the classic Holling Ⅰ, Holling Ⅱ, Holling Ⅲ, and Holling Ⅳ functional responses. We investigate the stabilization problem of the considered system using multiplicative controls. By linearizing the system and using the maximum principle, we construct a multiplicative control that exponentially stabilizes the system towards its steady-state solutions. The proposed feedback control allows us to reach a large class of steady-state solutions. The global well-posedness is obtained via Banach fixed point. Applications and numerical simulations to Holling responses Ⅰ, Ⅱ, Ⅲ, and Ⅳ are presented.
- prey predator with diffusion,
- Holling responses,
- feedback stabilization,
- multiplicative controls,
- numerical simulations
Citation: Ilyasse Lamrani, Imad El Harraki, M. A. Aziz-Alaoui, Fatima-Zahrae El Alaoui. Feedback stabilization for prey predator general model with diffusion via multiplicative controls[J]. AIMS Mathematics, 2023, 8(1): 2360-2385. doi: 10.3934/math.2023122

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Abstract

In this paper, we consider a predator–prey model given by a reaction–diffusion system. This model encompasses the classic Holling Ⅰ, Holling Ⅱ, Holling Ⅲ, and Holling Ⅳ functional responses. We investigate the stabilization problem of the considered system using multiplicative controls. By linearizing the system and using the maximum principle, we construct a multiplicative control that exponentially stabilizes the system towards its steady-state solutions. The proposed feedback control allows us to reach a large class of steady-state solutions. The global well-posedness is obtained via Banach fixed point. Applications and numerical simulations to Holling responses Ⅰ, Ⅱ, Ⅲ, and Ⅳ are presented.

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