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Stability analysis of a discrete biological model. (English) Zbl 1342.39017
This paper discusses the local and global behavior of the discrete biological model \[ \begin{aligned} X_{n+1} & =X_ne^{\ln (\alpha)(1-Y_n)},\\ Y_{n+1} & =(\gamma\delta-1)Y_n\left(1+\frac{1}{\gamma\delta-1}-\frac{Y_n}{X_n}\right),\end{aligned} \] where \(\alpha,\gamma,\delta\) and the initial conditions \(X_0,\;Y_0\) are positive real values.
The first section is a brief literature review which includes where this discrete model comes from. In the second section “Preliminaries”, the authors give some basic definitions and some known results.
The main results of the paper are in the third section. The authors first obtain the necessary and sufficient condition about the parameters \(\alpha,\gamma,\delta\) for local asymptotic stability of the unique positive equilibrium point \((1, 1)\). Then the authors get two theorems about the global asymptotic stability of the equilibrium point \((1,1)\). Moreover, they find the rate of convergence of a solution that converges to the equilibrium point. At last, quite a few examples are given to illustrate their results.
Reviewer: Fei Xue (Hartford)

MSC:
39A30 Stability theory for difference equations
39A10 Additive difference equations
92D25 Population dynamics (general)
39A12 Discrete version of topics in analysis
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