Global behavior of a discrete population model

Linxia Hu; Yonghong Shen; Xiumei Jia; Linxia Hu; Yonghong Shen; Xiumei Jia

doi:10.3934/math.2024592

AIMS Mathematics

2024, Volume 9, Issue 5: 12128-12143. doi: 10.3934/math.2024592

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Research article

Global behavior of a discrete population model

1.
School of Mathematics and Statistics, Tianshui Normal University, Tianshui, Gansu 741001, China
2.
School of Mathematics and Statistics, Hexi University, Zhangye, Gansu 734000, China

Received: 30 December 2023 Revised: 13 March 2024 Accepted: 18 March 2024 Published: 27 March 2024
MSC : 39A10

In this work, the global behavior of a discrete population model

$ \begin{equation*} \begin{cases} x_{n+1}& = \alpha x_n e^{-y_n}+\beta,\\ y_{n+1}& = \alpha x_n(1-e^{-y_n}), \end{cases}\quad n = 0,1,2,\dots, \end{equation*} $

is considered, where $ \alpha\in (0, 1) $, $ \beta\in (0, +\infty) $, and the initial value $ (x_{0}, y_0)\in [0, \infty)\times [0, \infty) $. To illustrate the dynamics behavior of this model, the boundedness, periodic character, local stability, bifurcation, and the global asymptotic stability of the solutions are investigated.
- discrete population model,
- boundedness,
- prime period-two solution,
- global asymptotic stablility,
- transcritical bifurcation,
- center manifold theorem
Citation: Linxia Hu, Yonghong Shen, Xiumei Jia. Global behavior of a discrete population model[J]. AIMS Mathematics, 2024, 9(5): 12128-12143. doi: 10.3934/math.2024592

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Abstract

In this work, the global behavior of a discrete population model

$ \begin{equation*} \begin{cases} x_{n+1}& = \alpha x_n e^{-y_n}+\beta,\\ y_{n+1}& = \alpha x_n(1-e^{-y_n}), \end{cases}\quad n = 0,1,2,\dots, \end{equation*} $

is considered, where $ \alpha\in (0, 1) $, $ \beta\in (0, +\infty) $, and the initial value $ (x_{0}, y_0)\in [0, \infty)\times [0, \infty) $. To illustrate the dynamics behavior of this model, the boundedness, periodic character, local stability, bifurcation, and the global asymptotic stability of the solutions are investigated.

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