Let us consider two-dimensional discrete dynamical system of the form

$\begin{array}{r}{x}_{n+1}=f({x}_{n},{y}_{n}),\\ {y}_{n+1}=g({x}_{n},{y}_{n}),\phantom{\rule{1em}{0ex}}n=0,1,\dots ,\end{array}$

(2)

where

$f:I\times J\to I$ and

$g:I\times J\to J$ are continuously differentiable functions and

*I*,

*J* are some intervals of real numbers. Furthermore, a solution

${\{({x}_{n},{y}_{n})\}}_{n=0}^{\mathrm{\infty}}$ of system (2) is uniquely determined by initial conditions

$({x}_{0},{y}_{0})\in I\times J$. An equilibrium point of (2) is a point

$(\overline{x},\overline{y})$ that satisfies

$\begin{array}{c}\overline{x}=f(\overline{x},\overline{y}),\hfill \\ \overline{y}=g(\overline{x},\overline{y}).\hfill \end{array}$

Let

$(\overline{x},\overline{y})$ be an equilibrium point of a map

$F(x,y)=(f(x,y),g(x,y))$, where

*f* and

*g* are continuously differentiable functions at

$(\overline{x},\overline{y})$. The linearized system of (2) about the equilibrium point

$(\overline{x},\overline{y})$ is

${X}_{n+1}=F({X}_{n})={F}_{J}{X}_{n},$

where ${X}_{n}=\left(\begin{array}{c}{x}_{n}\\ {y}_{n}\end{array}\right)$ and ${F}_{J}$ is Jacobian matrix of system (2) about the equilibrium point $(\overline{x},\overline{y})$.

Let

$(\overline{I},\overline{S})$ be the equilibrium point of system (1), then one has

$\overline{I}=\overline{S}(1-{e}^{-\alpha \overline{I}}),\phantom{\rule{2em}{0ex}}\overline{S}=\overline{S}{e}^{-\alpha \overline{I}}+\beta .$

Then, it follows that

$(\overline{I},\overline{S})=(\beta ,\frac{\beta}{1-{e}^{-\alpha \beta}})$ is the unique positive equilibrium point of system (1). Moreover, the Jacobian matrix

${F}_{J}(\overline{I},\overline{S})$ of system (1) about the equilibrium point

$(\overline{I},\overline{S})$ is given by

${F}_{J}(\overline{I},\overline{S})=\left(\begin{array}{cc}\alpha \overline{S}{e}^{-\alpha \overline{I}}& 1-{e}^{-\alpha \overline{I}}\\ -\alpha \overline{S}{e}^{-\alpha \overline{I}}& {e}^{-\alpha \overline{I}}\end{array}\right).$

The characteristic polynomial of

${F}_{J}(\overline{I},\overline{S})$ is given by

$P(\lambda )={\lambda}^{2}-{e}^{-\alpha \overline{I}}(1+\alpha \overline{S})\lambda +\alpha \overline{S}{e}^{-\alpha \overline{I}}.$

(3)

**Lemma 1** [15]

*Consider the second*-

*degree polynomial equation* ${\lambda}^{2}+p\lambda +q=0,$

(4)

*where* *p* *and* *q* *are real numbers*.

*Then*,

*the necessary and sufficient condition for both roots of Equation* (4)

*to lie inside the open disk* $|\lambda |<1$ *is* **Lemma 2** [16]

*Assume that* ${X}_{n+1}=F({X}_{n})$, $n=0,1,2,\dots $ , *is a system of difference equations and* $\overline{X}$ *is the fixed point of* *F*. *If all eigenvalues of the Jacobian matrix* ${J}_{F}$ *about* $\overline{X}$ *lie inside the open unit disk* $|\lambda |<1$, *then* $\overline{X}$ *is locally asymptotically stable*. *If one of them has a modulus greater than one*, *then* $\overline{X}$ *is unstable*.

**Theorem 3** *Assume that* ${e}^{\alpha \beta}(1+\alpha \beta )<\frac{1+{e}^{2\alpha \beta}}{2}$. *Then*, *the unique positive equilibrium point* $(\overline{I},\overline{S})=(\beta ,\frac{\beta}{1-{e}^{-\alpha \beta}})$ *is locally asymptotically stable*.

*Proof* The characteristic polynomial of

${F}_{J}(\overline{I},\overline{S})$ about positive equilibrium point

$(\beta ,\frac{\beta}{1-{e}^{\alpha \beta}})$ is given by

$P(\lambda )={\lambda}^{2}-({e}^{-\alpha \beta}+\frac{{e}^{-\alpha \beta}\alpha \beta}{1-{e}^{-\alpha \beta}})\lambda +\frac{{e}^{-\alpha \beta}\alpha \beta}{1-{e}^{-\alpha \beta}}.$

(5)

Let

$f(\lambda )={\lambda}^{2},\phantom{\rule{2em}{0ex}}g(\lambda )=({e}^{-\alpha \beta}+\frac{{e}^{-\alpha \beta}\alpha \beta}{1-{e}^{-\alpha \beta}})\lambda -\frac{{e}^{-\alpha \beta}\alpha \beta}{1-{e}^{-\alpha \beta}}.$

Assume that

${e}^{\alpha \beta}(1+\alpha \beta )<\frac{1+{e}^{2\alpha \beta}}{2}$, and

$|\lambda |=1$. Then, one has

$\begin{array}{rcl}|g(\lambda )|& \le & ({e}^{-\alpha \beta}+\frac{{e}^{-\alpha \beta}\alpha \beta}{1-{e}^{-\alpha \beta}})+\frac{{e}^{-\alpha \beta}\alpha \beta}{1-{e}^{-\alpha \beta}}\\ =& {e}^{-\alpha \beta}+\frac{2{e}^{-\alpha \beta}\alpha \beta}{1-{e}^{-\alpha \beta}}\\ =& \frac{{e}^{\alpha \beta}+2\alpha \beta {e}^{\alpha \beta}-1}{{e}^{2\alpha \beta}-{e}^{\alpha \beta}}<1.\end{array}$

Then, by Rouche’s theorem

$f(\lambda )$ and

$f(\lambda )-g(\lambda )$ have the same number of zeroes in an open unit disk

$|\lambda |<1$. Hence, both roots

${\lambda}_{1}=\frac{1-{e}^{\alpha \beta}-{e}^{\alpha \beta}\alpha \beta +\sqrt{-4{e}^{\alpha \beta}(-{e}^{\alpha \beta}+{e}^{2\alpha \beta})\alpha \beta +{(1-{e}^{\alpha \beta}-{e}^{\alpha \beta}\alpha \beta )}^{2}}}{2({e}^{\alpha \beta}-{e}^{2\alpha \beta})}$

and

${\lambda}_{2}=\frac{-1+{e}^{\alpha \beta}+{e}^{\alpha \beta}\alpha \beta +\sqrt{-4{e}^{\alpha \beta}(-{e}^{\alpha \beta}+{e}^{2\alpha \beta})\alpha \beta +{(1-{e}^{\alpha \beta}-{e}^{\alpha \beta}\alpha \beta )}^{2}}}{2(-{e}^{\alpha \beta}+{e}^{2\alpha \beta})}$

of (5) lie in an open disk $|\lambda |<1$, and it follows from Lemma 2 that the equilibrium point $(\beta ,\frac{\beta}{1-{e}^{-\alpha \beta}})$ is locally asymptotically stable. □

The following theorem shows the necessary and sufficient condition for the local asymptotic stability of a unique positive equilibrium point of system (1).

**Theorem 4** *The unique positive equilibrium point* $(\overline{I},\overline{S})=(\beta ,\frac{\beta}{1-{e}^{-\alpha \beta}})$ *of system* (1) *is locally asymptotically stable if and only if* $\frac{1+\alpha \beta}{{e}^{\alpha \beta}}<1$.

*Proof* Let

$p=-({e}^{-\alpha \beta}+\frac{{e}^{-\alpha \beta}\alpha \beta}{1-{e}^{-\alpha \beta}})$ and

$q=\frac{{e}^{-\alpha \beta}\alpha \beta}{1-{e}^{-\alpha \beta}}$, then (5) can be written as

$P(\lambda )={\lambda}^{2}+p\lambda +q.$

Then, $|p|={e}^{-\alpha \beta}+\frac{{e}^{-\alpha \beta}\alpha \beta}{1-{e}^{-\alpha \beta}}<1+\frac{{e}^{-\alpha \beta}\alpha \beta}{1-{e}^{-\alpha \beta}}=1+q$ and $1+q=1+\frac{{e}^{-\alpha \beta}\alpha \beta}{1-{e}^{-\alpha \beta}}<2$ if and only if $\frac{1+\alpha \beta}{{e}^{\alpha \beta}}<1$. Hence, from Lemma 1, the unique positive equilibrium point $(\overline{I},\overline{S})=(\beta ,\frac{\beta}{1-{e}^{-\alpha \beta}})$ of system (1) is locally asymptotically stable if and only if $\frac{1+\alpha \beta}{{e}^{\alpha \beta}}<1$. □