Elimination, permanence, and exclusion in a competition model under Allee effects

Yaw Chang; Wei Feng; Michael Freeze; Xin Lu; Charles Smith; Yaw Chang; Wei Feng; Michael Freeze; Xin Lu; Charles Smith

doi:10.3934/math.2023391

AIMS Mathematics

2023, Volume 8, Issue 4: 7787-7805. doi: 10.3934/math.2023391

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Research article

Elimination, permanence, and exclusion in a competition model under Allee effects

Department of Mathematics and Statistics, University of North Carolina Wilmington, 601 S. College Road, Wilmington, NC 28403-5970, USA

Received: 27 November 2022 Revised: 02 January 2023 Accepted: 08 January 2023 Published: 31 January 2023
MSC : 35B35, 35B40, 35B41, 35K57, 92D25

In this paper, we study a system of nonlinear partial differential equations that models the population dynamics of two competitive species both under Allee effects. The consideration of the model includes Logistic growth with Allee effects, Lotka-Volterra competition, diffusion, initial density and boundary conditions on the habitat. In the reaction-diffusion system, we employ the method of upper and lower solutions to address questions on self-elimination or persistence, as well as permanence or competitive exclusion. Specific conditions on biological parameters are explicitly given for extinction, coexistence and competitive exclusion of the species under various boundary conditions. Numerical simulations for the model are demonstrated to illustrate our results from mathematical analysis.
- reaction-diffusion systems,
- competition with Allee effects,
- Permanence and competitive exclusion
Citation: Yaw Chang, Wei Feng, Michael Freeze, Xin Lu, Charles Smith. Elimination, permanence, and exclusion in a competition model under Allee effects[J]. AIMS Mathematics, 2023, 8(4): 7787-7805. doi: 10.3934/math.2023391

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Abstract

In this paper, we study a system of nonlinear partial differential equations that models the population dynamics of two competitive species both under Allee effects. The consideration of the model includes Logistic growth with Allee effects, Lotka-Volterra competition, diffusion, initial density and boundary conditions on the habitat. In the reaction-diffusion system, we employ the method of upper and lower solutions to address questions on self-elimination or persistence, as well as permanence or competitive exclusion. Specific conditions on biological parameters are explicitly given for extinction, coexistence and competitive exclusion of the species under various boundary conditions. Numerical simulations for the model are demonstrated to illustrate our results from mathematical analysis.

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