Stability analysis of a discrete biological model (Q2788701)

This paper discusses the local and global behavior of the discrete biological model NEWLINE\[NEWLINE\begin{aligned} X_{n+1} & =X_ne^{\ln (\alpha)(1-Y_n)},\\ Y_{n+1} & =(\gamma\delta-1)Y_n\left(1+\frac{1}{\gamma\delta-1}-\frac{Y_n}{X_n}\right),\end{aligned}NEWLINE\]NEWLINE where \(\alpha,\gamma,\delta\) and the initial conditions \(X_0,\;Y_0\) are positive real values.NEWLINENEWLINEThe 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.NEWLINENEWLINEThe 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.