Stability property of the boundary equilibria of a symbiotic model of commensalism and parasitism with harvesting in commensal populations

Xiaowan Liu; Qin Yue; Xiaowan Liu; Qin Yue

doi:10.3934/math.20221034

AIMS Mathematics

2022, Volume 7, Issue 10: 18793-18808. doi: 10.3934/math.20221034

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Research article

Stability property of the boundary equilibria of a symbiotic model of commensalism and parasitism with harvesting in commensal populations

Xiaowan Liu ,
Qin Yue ^,

College of Finance and Mathematics, West Anhui University, Liuan 237000, China

Received: 02 June 2022 Revised: 21 July 2022 Accepted: 02 August 2022 Published: 23 August 2022
MSC : 34C25, 92D25, 34D20, 34D40

This article demonstrates the stability property of two boundary equilibria of a symbiotic model of commensalism and parasitism with harvesting in the commensal population. The model was proposed by Nurmaini Puspitasari, Wuryansari Muharini Kusumawinahyu, Trisilowati (2021). We first give two numeric examples to show that the corresponding results of the mentioned paper may be incorrect. Then, by analysis of the characteristic roots of the characteristic equations, we obtain sufficient conditions that ensure the locally asymptotic stability of the equilibria. After that, by applying the standard comparison theorem, some novel results on the global attractivity of these two equilibria are obtained respectively. Our results complement and supplement some known results.
- commensalism,
- parasitism,
- comparison theorem,
- global attractivity
Citation: Xiaowan Liu, Qin Yue. Stability property of the boundary equilibria of a symbiotic model of commensalism and parasitism with harvesting in commensal populations[J]. AIMS Mathematics, 2022, 7(10): 18793-18808. doi: 10.3934/math.20221034

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Abstract

This article demonstrates the stability property of two boundary equilibria of a symbiotic model of commensalism and parasitism with harvesting in the commensal population. The model was proposed by Nurmaini Puspitasari, Wuryansari Muharini Kusumawinahyu, Trisilowati (2021). We first give two numeric examples to show that the corresponding results of the mentioned paper may be incorrect. Then, by analysis of the characteristic roots of the characteristic equations, we obtain sufficient conditions that ensure the locally asymptotic stability of the equilibria. After that, by applying the standard comparison theorem, some novel results on the global attractivity of these two equilibria are obtained respectively. Our results complement and supplement some known results.

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