Stability of nonlinear population systems with individual scale and migration

Wei Gong; Zhanping Wang; Wei Gong; Zhanping Wang

doi:10.3934/math.2023006

AIMS Mathematics

2023, Volume 8, Issue 1: 125-147. doi: 10.3934/math.2023006

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Research article

Stability of nonlinear population systems with individual scale and migration

Wei Gong ^1,2,
Zhanping Wang ^{1
,
,}

1.
School of Mathematics and Statistics, Ningxia University, Yinchuan 750021, China
2.
School of Science, Ningxia Medical University, Yinchuan 750004, China

Received: 17 June 2022 Revised: 07 September 2022 Accepted: 18 September 2022 Published: 27 September 2022
MSC : 34D20, 34M45, 93D20

In this paper, we study the stability of a nonlinear population system with a weighted total size of scale structure and migration in a polluted environment, where fertility and mortality depend on the density in different ways. We first prove the existence and uniqueness of the equilibrium point via a contraction mapping and give the expression for the equilibrium point. Some conditions for asymptotic stability and instability are presented by means of a characteristic equation. When the effect of density restriction on mortality is not considered, the threshold value of equilibrium stability can be obtained as $ \Lambda = 0. $ When $ \Lambda < 0, $ the equilibrium is asymptotically stable, and when $ \Lambda > 0, $ the equilibrium is unstable. In addition, the upwind difference method is used to discrete the model, and two examples are given to show the evolution of species.
- population model,
- polluted environment,
- individual scale,
- equilibrium solution,
- stability
Citation: Wei Gong, Zhanping Wang. Stability of nonlinear population systems with individual scale and migration[J]. AIMS Mathematics, 2023, 8(1): 125-147. doi: 10.3934/math.2023006

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Abstract

In this paper, we study the stability of a nonlinear population system with a weighted total size of scale structure and migration in a polluted environment, where fertility and mortality depend on the density in different ways. We first prove the existence and uniqueness of the equilibrium point via a contraction mapping and give the expression for the equilibrium point. Some conditions for asymptotic stability and instability are presented by means of a characteristic equation. When the effect of density restriction on mortality is not considered, the threshold value of equilibrium stability can be obtained as $ \Lambda = 0. $ When $ \Lambda < 0, $ the equilibrium is asymptotically stable, and when $ \Lambda > 0, $ the equilibrium is unstable. In addition, the upwind difference method is used to discrete the model, and two examples are given to show the evolution of species.

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