Qualitative Behavior of an Exponential Symmetric Difference Equation System

Abstract

We examine the unboundedness, persistence, boundedness, uniqueness, and existence of non-negative equilibrium of an exponential symmetric difference equations system: ${\Omega }_{n+1}={\alpha }_1+{\beta }_1{\Omega }_n+{\gamma }_1{\Omega }_{n-1}{e}^{-({\Omega }_n+{\varpi }_n)}$, ${\varpi }_{n+1}={\alpha }_2+{\beta }_2{\varpi }_n+{\gamma }_2{\varpi }_{n-1}{e}^{-({\Omega }_n+{\varpi }_n)},$ $n=0,1,\cdots$, whereby initial values ${\Omega }_{-1},{\varpi }_{-1},{\Omega }_0,{\varpi }_0$ and parameters ${\alpha }_1,{\alpha }_2$ are non-negative real numbers and ${\beta }_1,{\beta }_2\in (0,1)$ and ${\gamma }_1,{\gamma }_2\le 0$. We will discuss asymptotic global and local stability and the convergence rate of this system. Ultimately, to check our results, we set out some numerical explanations.

1. Introduction

Strong dynamics emerge exponentially from discrete dynamical structures. To deliver population models, the aforementioned formations can be used. Due to their better computational capability, discrete dynamical systems produce results that are far more useful than allied formations based on differential equations. Due to their superior computational performance, discrete dynamical systems outperform allied forms in differential equations. It is obvious that nonoverlapping procreation difference equations are more important than seeing how population models behave. Difference equations further examine naturally as discrete analogs of delay differential and differential equations and become applied in finance, biological, physical, and social sciences.For some results about difference equations, we refer the reader to [1,2,3,4,5,6,7,8,9,10,11,12,13,14]. Population forms incorporate exponential difference equations and their stability evaluation although complex, however exciting. Symmetric systems of difference equations are essential for the advancement of many fields, from decision-making to life (See Refs. [15,16] and references cited therein). The exponential form is one of the most important varieties of difference equations. These have numerous uses in daily life. For instance, El-Metwally et al. [17] elaborated the behavior of a population model,

$${\Omega }_{n+1}=\alpha +\beta {\Omega }_{n-1}{e}^{-{\Omega }_n}.$$

Papaschinopoulos et al. [18] dealt with the behavior of his solutions for the symmetric systems:

$${\Omega }_{n+1}={\alpha }_1+{\beta }_1{\varpi }_{n-1}{e}^{-{\Omega }_n},{\varpi }_{n+1}={\alpha }_2+{\beta }_2{\Omega }_{n-1}{e}^{-{\varpi }_n},$$

$${\Omega }_{n+1}={\alpha }_1+{\beta }_1{\varpi }_{n-1}{e}^{-{\varpi }_n},{\varpi }_{n+1}={\alpha }_2+{\beta }_2{\Omega }_{n-1}{e}^{-{\Omega }_n}.$$

Khan et al. [10] studied the behavior of a recursive symmetric equations model

$${\Omega }_{n+1}=\frac{{\Omega }_1{e}^{-{\varpi }_n}+\chi e^{-{\varpi }_{n-1}}}{\gamma +{\Omega }_1{\Omega }_n+\chi {\Omega }_{n-1}},{\varpi }_{n+1}=\frac{\mu e^{-{\Omega }_n}+{\chi }_1{e}^{-{\Omega }_{n-1}}}{{\gamma }_1+\mu {\varpi }_n+{\chi }_1{\varpi }_{n-1}},$$

where as the initial values ${\Omega }_{-1},{\Omega }_0,{\varpi }_{-1},{\varpi }_0$ and the parameters $\chi ,\mu ,\gamma ,{\Omega }_1,{\gamma }_1$ and ${\chi }_1$ are the non-negative real numbers. For some others related articles, we refer to [19,20,21,22,23,24,25,26,27,28,29,30]. For some outcomes of systems of exponential type difference formulas we can see Refs. [13,30,31].

The purpose of this study is to examine the behavior of non-negative solutions to the following symmetric system of rational difference equations in exponential form, including their boundedness, persistence, local and global behavior, and rate of convergence:

$${\Omega }_{n+1}={\alpha }_1+{\beta }_1{\Omega }_n+{\gamma }_1{\Omega }_{n-1}{e}^{-({\Omega }_n+{\varpi }_n)},$$

$${\varpi }_{n+1}={\alpha }_2+{\beta }_2{\varpi }_n+{\gamma }_2{\varpi }_{n-1}{e}^{-({\Omega }_n+{\varpi }_n)},$$

(1)

$n=0,1,\cdots$, whereby initial values ${\Omega }_{-1},{\varpi }_{-1},{\Omega }_0,{\varpi }_0$ and parameters ${\alpha }_1,{\alpha }_2$ are non-negative real numbers and ${\beta }_1,{\beta }_2\in (0,1)$ and ${\gamma }_1,{\gamma }_2\le 0$. We will go over the global and local asymptotic stability, and convergence rate of this model. Ultimately, to inspect our results, we laid out a variety of numerical descriptions.The importance of this type of system lies in its applications in biology, economics, and others.

This paper is formulated as seven basic sections. Section 3 is dedicated to the study of the boundedness and persistence of solutions. In Section 4, we study the uniqueness and existence of an equilibrium for (1). We present the necessary conditions for achieving stability in a qualitative way in Section 5. In Section 6, we deal with the rate of convergence of solutions, which converges to our fixed point of (1). Our goal in Section 7 is scrutinizing the unbounded solutions for (1).

2. Preliminaries

We now present a set of important concepts and theorems that we need while studying this paper.

Let h and l are intervals in $\mathbb{R}$ and let $U:h^2\times l^2\to h,V:h^2\times l^2\to l$ are differentiable functions. Then, for the initials $({\Omega }_i,{\varpi }_i)\in h\times l$ with $i\in \{-1,0\}$, the system

$$\begin{array}{ccc}\hfill {\Omega }_{n+1}& =& U({\Omega }_n,{\Omega }_{n-1},{\varpi }_n,{\varpi }_{n-1}),\hfill \\ \hfill {\varpi }_{n+1}& =& V({\Omega }_n,{\Omega }_{n-1},{\varpi }_n,{\varpi }_{n-1}),n=0,1,\cdots ,\hfill \end{array}$$

(2)

has a solution ${\left\{({\Omega }_n,{\varpi }_n)\right\}}_{n=-1}^{\infty }$. The point $(\overline{\Omega },\overline{\varpi })\in h\times l$ is designated as an equilibrium of (2) if $\overline{\Omega }=U(\overline{\Omega },\overline{\Omega },\overline{\varpi },\overline{\varpi })$ and $\overline{\varpi }=V(\overline{\Omega },\overline{\Omega },\overline{\varpi },\overline{\varpi })$.

The following definition gives us the necessary conditions for stability and instability of various kinds.

Definition 1

([12]). If $(\overline{\Omega },\overline{\varpi })$ be the equilibrium for (2), then

(i) 

$(\overline{\Omega },\overline{\varpi })$ is revealed as stable if ∀$\epsilon >0$, ∃$\delta >0$ provided that initials $({\Omega }_i,{\varpi }_i)$, $i\in \{-1,0\}$, $\parallel {\sum }_{i=-1}^0({\Omega }_i,{\varpi }_i)-(\overline{\Omega },\overline{\varpi })\parallel <\delta$ implies $\parallel ({\Omega }_n,{\varpi }_n)-(\overline{\Omega },\overline{\varpi })\parallel <\epsilon$∀$n>0$.

(ii) 

$(\overline{\Omega },\overline{\varpi })$, if not stable, is designated unstable.

(iii) 

If ∃$\eta >0$ provided that $({\Omega }_n,{\varpi }_n)\to (\overline{\Omega },\overline{\varpi })$ as $n\to \infty$ and $\parallel {\sum }_{i=-2}^0({\Omega }_i,{\varpi }_i)-(\overline{\Omega },\overline{\varpi })\parallel <\eta$, then $(\overline{\Omega },\overline{\varpi })$ is designated an asymptotically stable.

(iv) 

If $({\Omega }_n,{\varpi }_n)\to (\overline{\Omega },\overline{\varpi })$ as $n\to \infty$, then $(\overline{\Omega },\overline{\varpi })$ is designated as a global attractor.

(v) 

If $(\overline{\Omega },\overline{\varpi })$ is a global attractor and stable, then it is said to be a global asymptotic attractor.

Definition 2

([32]). Let $(\overline{\Omega },\overline{\varpi })$ be a fixed point of the next map

$$F=\left(U,{\Omega }_n,V,{\varpi }_n\right),$$

wherever V and also U are differentiable maps. System(2) can be written ${\Omega }_{n+1}=U\left({\Omega }_n\right)$ where ${\Omega }_n={({\Omega }_n,{\varpi }_n,{\Omega }_{n-1},{\varpi }_{n-1})}^t$. The linearized system of (2) about $(\overline{\Omega },\overline{\varpi })$ is ${\varpi }_{n+1}=J_F{\varpi }_n,$ where ${\varpi }_n={\Omega }_n-\overline{\Omega }$ and $\overline{\Omega }=(\overline{\Omega },\overline{\varpi },\overline{\Omega },\overline{\varpi })$ and $J_F$ is a Jacobian matrix of (2) evaluated at $(\overline{\Omega },\overline{\varpi })$.

Lemma 1

([33]). Recognize the system $H\left(v_n\right)=v_{n+1}$, $n=0,1,\cdots$, $\overline{v}$ is an equilibrium of H. $\overline{v}$ is asymptotically locally stable if all eigenvalues of $J_H$ about $\overline{v}$ lie in $\left|\chi \right|<1$. Besides that, $\overline{v}$ is unstable if $\left|\chi \right|>1$.

Consider the system

$$\left(A\left(n\right)+B\right)v_n=v_{n+1},$$

(3)

$v_n$ is an l-dimensional vector, $B\in C^{l\times l}$ is a constant matrix, and $A:{\mathbb{Z}}^{+}\to C^{l\times l}$ satisfies

$$\parallel A\left(n\right)\parallel \to 0,$$

(4)

as $n\to \infty$, $\parallel ·\parallel$ is associated with

$$\parallel (v,u)\parallel =\sqrt{v^2+u^2}.$$

Proposition 1

([14] (Perron’s Theorem)). Suspect that (4) assumption occurs. If ${\xi }_n$ be a solution of (3), thus either ${\xi }_n=0$ ∀ n or

$$\underset{n\to \infty }{lim}(\parallel {\xi }_n{\parallel )}^{1/n}=\varrho ,$$

(5)

or

$$\underset{n\to \infty }{lim}\frac{\parallel {\xi }_{n+1}\parallel }{\parallel {\xi }_n\parallel }=\varrho ,$$

(6)

exists, and it is identical to the modulus of one of all eigenvalues of B.

Proposition 2

([12] (Comparison theorem)). Let $\alpha \in (0,\infty )$ and $\beta \in \mathbb{R}$. Let ${\left\{{\Omega }_n\right\}}_{n=-1}^{\infty }$ and ${\left\{{\varpi }_n\right\}}_{n=-1}^{\infty }$ are sequences in $\mathbb{R}$ provided that ${\Omega }_0\le {\varpi }_0$ and ${\Omega }_{-1}\le {\varpi }_{-1}$ and

$$\{\begin{array}{cc}{\Omega }_{n+1}\le \alpha {\Omega }_{n-1}+\beta \hfill & \\ \hfill {\varpi }_{n+1}=\alpha {\varpi }_{n-1}+\beta ,n=0,1,\cdots \end{array}$$

Then ${\Omega }_n\le {\varpi }_n$ for $n\ge -1$

Definition 3

([12]). The recursive equation ${\Omega }_{n+1}=F({\Omega }_n,{\Omega }_{n-1},...,{\Omega }_{n-k})$, $n=0,1,\cdots$ is called persistent if we have m and M with $0\le m\le M<\infty$ provided that for initials ${\Omega }_0,{\Omega }_{-1},\cdots and {\Omega }_{-k}\in (0,\infty )$, there is a non-negative integer N which depends on initials provided that $m\le {\Omega }_n\le M$ ∀$n\ge N$.

3. Boundedness and Persistence

This section is dedicated to the study of the boundedness and persistence of solutions.

The next theorem clearly demonstrates that any non-negative solution $\left\{({\Omega }_n,{\varpi }_n)\right\}$ of (1) persists and is bounded.

Theorem 1.

If

$${\beta }_1+{\gamma }_1\rho <1,{\beta }_2+{\gamma }_2\rho <1,{\beta }_1+\sqrt{{\left({\beta }_1\right)}^2+4{\gamma }_1\rho }<2,{\beta }_2+\sqrt{{\left({\beta }_2\right)}^2+4{\gamma }_2\rho }<2,$$

(7)

wherein $\rho =e^{-({\alpha }_1+{\alpha }_2)}$ then every solution $\left\{({\Omega }_n,{\varpi }_n)\right\}$ of (1) is bounded and persists.

Proof. 

Let $\left\{({\Omega }_n,{\varpi }_n)\right\}$ be a solution of (1). From (1), we have

$${\Omega }_n\ge {\alpha }_1{\varpi }_n\ge {\alpha }_2,n=0,1,\cdots .$$

(8)

From (1) and (8), we have

$${\Omega }_{n+1}\le {\alpha }_1+{\beta }_1{\Omega }_n+{\gamma }_1{\Omega }_{n-1}\rho ,{\varpi }_{n+1}\le {\alpha }_2+{\beta }_2{\varpi }_n+{\gamma }_2{\varpi }_{n-1}\rho ,n=0,1,\cdots .$$

(9)

Consider

$$Z_{n+1}={\alpha }_1+{\beta }_1{Z}_n+{\gamma }_1{Z}_{n-1}\rho ,T_{n+1}={\alpha }_2+{\beta }_2{T}_n+{\gamma }_2{T}_{n-1}\rho ,n=0,1,\cdots .$$

(10)

Therefore,

$$\begin{array}{c}\hfill Z_n=\frac{{\alpha }_1}{1-{\beta }_1-{\gamma }_1\rho }+r_1{(\frac{{\beta }_1-\sqrt{{({\beta }_1)}^2+4{\gamma }_1\rho }}{2})}^n+r_2{(\frac{{\beta }_1+\sqrt{{({\beta }_1)}^2+4{\gamma }_1\rho }}{2})}^n,\\ \hfill T_n=\frac{{\alpha }_2}{1-{\beta }_2-{\gamma }_2\rho }+r_3{(\frac{{\beta }_2-\sqrt{{({\beta }_2)}^2+4{\gamma }_2\rho }}{2})}^n+r_4{(\frac{{\beta }_2+\sqrt{{({\beta }_2)}^2+4{\gamma }_2\rho }}{2})}^n,\end{array}$$

(11)

$n=0,1,\cdots$, and $r_1,r_2,r_3,r_4$ depend on the initials $Z_0,Z_{-1},T_{-1}$, and $T_0$. By using (7) and (11) we have

$$\begin{array}{c}\hfill Z_n\le \frac{{\alpha }_1}{1-{\beta }_1-{\gamma }_1\rho }+\left|r_1\right|+\left|r_2\right|,T_n\le \frac{{\alpha }_2}{1-{\beta }_2-{\gamma }_2\rho }+\left|r_3\right|+\left|r_4\right|.\end{array}$$

These inequalities imply that $Z_n$ and $T_n$ are bounded sequences.

Now let $\left\{(Z_n,T_n)\right\}$ such that

$$Z_{-1}={\Omega }_{-1},Z_0={\Omega }_0,T_{-1}={\varpi }_{-1},T_0={\varpi }_0.$$

(12)

Therefore, from (9) and (12), we have by comparison proposition 2,

$${\Omega }_n\le Z_n\le \frac{{\alpha }_1}{1-{\beta }_1-{\gamma }_1\rho }+\left|r_1\right|+\left|r_2\right|,$$

$$\begin{array}{c}\hfill {\varpi }_n\le T_n\le \frac{{\alpha }_2}{1-{\beta }_2-{\gamma }_2\rho }+\left|r_3\right|+\left|r_4\right|.\end{array}$$

(13)

From (8) and (13), we get

$$\begin{array}{c}\hfill {\alpha }_1\le {\Omega }_n\le Z_n,{\alpha }_2\le {\varpi }_n\le T_n,n=0,1,\cdots .\end{array}$$

Since $Z_n$ and $T_n$ are bounded sequences, then $({\Omega }_n,{\varpi }_n)$ are bounded and persists. □

Theorem 2.

Let $\left\{({\Omega }_n,{\varpi }_n)\right\}$ be a non-negative solution of (1) and

$${\beta }_1+{\gamma }_1\rho <1,{\beta }_2+{\gamma }_2\rho <1,{\beta }_1+\sqrt{{\left({\beta }_1\right)}^2+4{\gamma }_1\rho }<2,$$

$${\beta }_2+\sqrt{{\left({\beta }_2\right)}^2+4{\gamma }_2\rho }<2,$$

where $\rho =e^{-({\alpha }_1+{\alpha }_2)}$. Then the interval $\left[{\alpha }_1,\frac{{\alpha }_1}{1-{\beta }_1-{\gamma }_1\rho }\right]\times \left[{\alpha }_2,\frac{{\alpha }_2}{1-{\beta }_2-{\gamma }_2\rho }\right]$ is an invariant set for (1).

Proof. 

Let $\left\{({\Omega }_n,{\varpi }_n)\right\}$ be a solution of (1) with initials ${\Omega }_0,{\Omega }_{-1}\in s=[{\alpha }_1,\frac{{\alpha }_1}{1-{\beta }_1-{\gamma }_1\rho }]$ and ${\varpi }_0,{\varpi }_{-1}\in t=[{\alpha }_2,\frac{{\alpha }_2}{1-{\beta }_2-{\gamma }_2\rho }]$.

Then it follows, from (1), that

$${\Omega }_1={\alpha }_1+{\beta }_1{\Omega }_0+{\gamma }_1{\Omega }_{-1}{e}^{-({\Omega }_0+{\varpi }_0)}$$

$$\le {\alpha }_1+{\beta }_1\left[\frac{{\alpha }_1}{1-{\beta }_1-{\gamma }_1\rho }\right]+{\gamma }_1\left[\frac{{\alpha }_1}{1-{\beta }_1-{\gamma }_1\rho }\right]e^{-({\alpha }_1+{\alpha }_2)}=\frac{{\alpha }_1}{1-{\beta }_1-{\gamma }_1\rho }.$$

Similarly ${\varpi }_1\le \frac{{\alpha }_2}{1-{\beta }_2-{\gamma }_2\rho }$. Hence, ${\Omega }_1\in s$ and ${\varpi }_1\in t$.

Currently, suppose the outcome holds true for $n=k$, $k>1$. It means that ${\Omega }_k\in s$ and ${\varpi }_k\in t$.

$${\Omega }_{k+1}={\alpha }_1+{\beta }_1{\Omega }_k+{\gamma }_1{\Omega }_{k-1}{e}^{-({\Omega }_k+{\varpi }_k)}$$

$$\le {\alpha }_1+{\beta }_1\left[\frac{{\alpha }_1}{1-{\beta }_1-{\gamma }_1\rho }\right]+{\gamma }_1\left[\frac{{\alpha }_1}{1-{\beta }_1-{\gamma }_1\rho }\right]\rho =\frac{{\alpha }_1}{1-{\beta }_1-{\gamma }_1\rho }.$$

Similarly, ${\varpi }_{k+1}\le \frac{{\alpha }_2}{1-{\beta }_2-{\gamma }_2\rho }$. Thus, the proof follows by induction. □

4. Uniqueness and Existence

The following theorem determines the uniqueness and existence of an equilibrium for (1).

Theorem 3.

If the following condition holds,

$$-\frac{\left({\alpha }_1{\beta }_2-{\alpha }_1\right){\left(G1\left({\Omega }_1\right)\right)}^2+\left(-2{\Omega }_1{\alpha }_1+2{\beta }_2{\Omega }_1{\alpha }_1-{\alpha }_2{\alpha }_1\right)G1\left({\Omega }_1\right)}{\left({\alpha }_1+G1\left({\alpha }_1\right)\right){\alpha }_1\left(-{\alpha }_1+{\alpha }_1+{\beta }_1{\alpha }_1\right)\left(\left(-1+{\beta }_2\right)G1\left({\alpha }_1\right)-{\alpha }_1-{\alpha }_2+{\beta }_2{\alpha }_1\right)}$$

$$-\frac{-{{\Omega }_1}^2{\alpha }_1-{\alpha }_2{\alpha }_1+{\beta }_2{{\Omega }_1}^2{\alpha }_1-{\alpha }_2{{\Omega }_1}^2+{\alpha }_2{\beta }_1{{\Omega }_1}^2}{\left({\alpha }_1+G1\left({\alpha }_1\right)\right){\alpha }_1\left(-{\alpha }_1+{\alpha }_1+{\beta }_1{\alpha }_1\right)\left(\left(-1+{\beta }_2\right)G1\left({\alpha }_1\right)-{\alpha }_1-{\alpha }_2+{\beta }_2{\alpha }_1\right)}<0,$$

(14)

where $1-{\beta }_1-{\gamma }_1\rho \ne 0$, $({\gamma }_1{\alpha }_1+{\alpha }_2{\gamma }_1)e^{-{\alpha }_1-{\alpha }_2}-{\alpha }_2+{\beta }_1{\alpha }_1+{\alpha }_2{\beta }_1\ne 0$, and ${\beta }_1<0$ or ${\gamma }_1<0$, ${\Omega }_1=\frac{{\alpha }_1}{1-{\beta }_1-{\gamma }_1\rho }$, ${\Omega }_2=\frac{{\alpha }_2}{1-{\beta }_2-{\gamma }_2\rho }$ and $G\left(x\right)=ln((x-{\alpha }_1-{\beta }_{1}x)/\left({\gamma }_{1}x\right)),$ then (1) has only one non-negative equilibrium $(\overline{\Omega },\overline{\varpi })$ in $\left[{\alpha }_1,{\Omega }_1\right]\times \left[{\alpha }_2,{\Omega }_2\right].$

Proof. 

Presume

$$x={\beta }_{1}x+{\alpha }_1+{\gamma }_{1}x{e}^{-(x+y)},y={\alpha }_2+{\beta }_{2}y+{\gamma }_{2}y{e}^{-(x+y)}.$$

(15)

So,

$$y=-ln\left[\frac{x-{\alpha }_1-{\beta }_{1}x}{{\gamma }_{1}x}\right]-x,x=-ln\left[\frac{y-{\alpha }_2-{\beta }_{2}y}{{\gamma }_{2}y}\right]-y.$$

Defining

$${\rm Y}\left(x\right)=-ln\left[\frac{\varrho \left(x\right)-{\alpha }_2-{\beta }_2\varrho \left(x\right)}{{\gamma }_2\varrho \left(x\right)}\right]-x-\varrho \left(x\right),$$

(16)

where $\varrho \left(x\right)=-ln\left[\frac{x-{\alpha }_1-{\beta }_{1}x}{{\gamma }_{1}x}\right]-x$. We get, ${\rm Y}\left({\alpha }_1\right)>0$ if and only if

$${\gamma }_2>\frac{e^{{\xi }_1+{\alpha }_1}\left({\xi }_1-{\alpha }_2-{\beta }_2{\xi }_1\right)}{{\xi }_1},$$

where ${\zeta }_1=-({\alpha }_1+ln\left(\frac{-{\beta }_1}{{\gamma }_1}\right))$.

Additionally, ${\rm Y}\left({\Omega }_1\right)<0$ if and only if

$${\gamma }_2<$$

$$-\frac{\left(-{\gamma }_1{\alpha }_1+{\beta }_2{\gamma }_1{\alpha }_1+{\beta }_2{\gamma }_1{\alpha }_2\right)e^{-{\alpha }_1-{\alpha }_2}+\left(-{\alpha }_2+{\beta }_1{\alpha }_1+{\alpha }_2{\beta }_1\right){\beta }_2-{\beta }_1{\alpha }_1}{e^{-{\alpha }_1-{\alpha }_2}\left(\left({\gamma }_1{\alpha }_1+{\alpha }_2{\gamma }_1\right)e^{-{\alpha }_1-{\alpha }_2}-{\alpha }_2+{\beta }_1{\alpha }_1+{\alpha }_2{\beta }_1\right)}.$$

Thus, ${\rm Y}\left(x\right)$ contains one solution in the interval $\left[{\alpha }_1,{\Omega }_1\right]$. Furthermore, one has, by using both of (14) and (16),

$${{\rm Y}}^{\prime }\left(x\right)=-\frac{\left({\alpha }_1{\beta }_2-{\alpha }_1\right){\left(G1\left(x\right)\right)}^2+\left(-2x{\alpha }_1+2{\beta }_{2}x{\alpha }_1-{\alpha }_2{\alpha }_1\right)G1\left(x\right)}{\left(x+G1\left(x\right)\right)x\left(-x+{\alpha }_1+{\beta }_{1}x\right)\left(\left(-1+{\beta }_2\right)G1\left(x\right)-x-{\alpha }_2+{\beta }_{2}x\right)}$$

$$-\frac{-x^2{\alpha }_1-{\alpha }_2{\alpha }_1+{\beta }_2{x}^2{\alpha }_1-{\alpha }_2{x}^2+{\alpha }_2{\beta }_1{x}^2}{\left(x+G1\left(x\right)\right)x\left(-x+{\alpha }_1+{\beta }_{1}x\right)\left(\left(-1+{\beta }_2\right)G1\left(x\right)-x-{\alpha }_2+{\beta }_{2}x\right)}$$

$$<-\frac{\left({\alpha }_1{\beta }_2-{\alpha }_1\right){\left(G1\left({\Omega }_1\right)\right)}^2+\left(-2{\Omega }_1{\alpha }_1+2{\beta }_2{\Omega }_1{\alpha }_1-{\alpha }_2{\alpha }_1\right)G1\left({\Omega }_1\right)}{\left({\alpha }_1+G1\left({\alpha }_1\right)\right){\alpha }_1\left(-{\alpha }_1+{\alpha }_1+{\beta }_1{\alpha }_1\right)\left(\left(-1+{\beta }_2\right)G1\left({\alpha }_1\right)-{\alpha }_1-{\alpha }_2+{\beta }_2{\alpha }_1\right)}$$

$$-\frac{-{{\Omega }_1}^2{\alpha }_1-{\alpha }_2{\alpha }_1+{\beta }_2{{\Omega }_1}^2{\alpha }_1-{\alpha }_2{{\Omega }_1}^2+{\alpha }_2{\beta }_1{{\Omega }_1}^2}{\left({\alpha }_1+G1\left({\alpha }_1\right)\right){\alpha }_1\left(-{\alpha }_1+{\alpha }_1+{\beta }_1{\alpha }_1\right)\left(\left(-1+{\beta }_2\right)G1\left({\alpha }_1\right)-{\alpha }_1-{\alpha }_2+{\beta }_2{\alpha }_1\right)}$$

$$<0.$$

□

5. Global and Local Stability

In this section of the study we present the necessary conditions for achieving stability in a qualitative way.

Theorem 4.

The equilibrium $(\overline{\Omega },\overline{\varpi })\in$$\left[{\alpha }_1,\frac{{\alpha }_1}{1-{\beta }_1-{\gamma }_1\rho }\right]\times \left[{\alpha }_2,\frac{{\alpha }_2}{1-{\beta }_2-{\gamma }_2\rho }\right]$ of (1) is asymptotically locally stable if

$${\gamma }_1{\gamma }_2({\Omega }_1+{\Omega }_2+1){\rho }^2+\rho (1+{\beta }_2)\{({\Omega }_1+1){\gamma }_1+{\gamma }_2({\Omega }_2+1)\}+(1+{\beta }_2){\beta }_1+{\beta }_2<1.$$

(17)

where ${\Omega }_1=\frac{{\alpha }_1}{1-{\beta }_1-{\gamma }_1\rho }$, ${\Omega }_2=\frac{{\alpha }_2}{1-{\beta }_2-{\gamma }_2\rho }$, and $\rho =e^{-({\alpha }_1+{\alpha }_2)}$.

Proof. 

The linearized system of (1) is generated by

$${\Phi }_{n+1}=E{\Phi }_n,$$

where ${\Phi }_n=\left(\begin{array}{c}{\Omega }_n\\ {\varpi }_n\\ {\Omega }_{n-1}\\ {\varpi }_{n-1}\end{array}\right)$, $E=\left(\begin{array}{cccc}{\beta }_1-{\gamma }_1\overline{\Omega }{e}^{-(\overline{\Omega }+\overline{\varpi })}& -{\gamma }_1\overline{\Omega }{e}^{-(\overline{\Omega }+\overline{\varpi })}& {\gamma }_1{e}^{-(\overline{\Omega }+\overline{\varpi })}& 0\\ -{\gamma }_2\overline{\varpi }{e}^{-(\overline{\Omega }+\overline{\varpi })}& {\beta }_2-{\gamma }_2\overline{\varpi }{e}^{-(\overline{\Omega }+\overline{\varpi })}& 0& {\gamma }_2{e}^{-(\overline{\Omega }+\overline{\varpi })}\\ 1& 0& 0& 0\\ 0& 1& 0& 0\end{array}\right).$

The characteristic polynomial of the Jacobain matrix E about $(\overline{\Omega },\overline{\varpi })$ is given by

$$P\left(\chi \right)={\chi }^4+A{\chi }^3+B{\chi }^2+C\chi +D,$$

(18)

where

$$A=-{\beta }_2+{\gamma }_2\overline{\varpi }{e}^{-(\overline{\Omega }+\overline{\varpi })}-{\beta }_1+{\gamma }_1\overline{\Omega }{e}^{-(\overline{\Omega }+\overline{\varpi })}$$

$$B=-{\gamma }_2{e}^{-(\overline{\Omega }+\overline{\varpi })}-{\gamma }_1{e}^{-(\overline{\Omega }+\overline{\varpi })}+{\beta }_2{\beta }_2-{\beta }_2{\gamma }_1\overline{\Omega }{e}^{-(\overline{\Omega }+\overline{\varpi })}+{\beta }_1{\gamma }_2\overline{\varpi }{e}^{-(\overline{\Omega }+\overline{\varpi })}$$

$$C={\beta }_1{\gamma }_2{e}^{-(\overline{\Omega }+\overline{\varpi })}-{\gamma }_1{\gamma }_2\overline{\Omega }{e}^{-2(\overline{\Omega }+\overline{\varpi })}+{\gamma }_1{\beta }_2{e}^{-(\overline{\Omega }+\overline{\varpi })}+{\gamma }_1{\gamma }_2\overline{\varpi }{e}^{-2(\overline{\Omega }+\overline{\varpi })},$$

$D={\gamma }_1{\gamma }_2{e}^{-2(\overline{\Omega }+\overline{\varpi })}.$ Let $\varphi \left(\chi \right)={\chi }^4$ and $\psi \left(\chi \right)=A{\chi }^3+B{\chi }^2+C\chi +D$.

We can claim that condition (17) happens and $\left|\chi \right|=1$. Thus,

$$\left|\psi \left(\chi \right)\right|<{\gamma }_1{\gamma }_2(\overline{\Omega }+\overline{\varpi }+1){\overline{\rho }}^2+\overline{\rho }(1+{\beta }_2)\{(\overline{\Omega }+1){\gamma }_1+{\gamma }_2(\overline{\varpi }+1)\}+(1+{\beta }_2){\beta }_1+{\beta }_2$$

$$<{\gamma }_1{\gamma }_2({\Omega }_1+{\Omega }_2+1){\rho }^2+\rho (1+{\beta }_2)\{({\Omega }_1+1){\gamma }_1+{\gamma }_2({\Omega }_2+1)\}+(1+{\beta }_2){\beta }_1+{\beta }_2<1,$$

where $\overline{\rho }=e^{-(\overline{\Omega }+\overline{\varpi })}.$ Then, $\varphi \left(\chi \right)$ and $\varphi \left(\chi \right)+\psi \left(\chi \right)$ have the exact same number of zeros in $1>\left|\chi \right|$. □

Theorem 5.

$(\overline{\Omega },\overline{\varpi })\in$$\left[{\alpha }_1,\frac{{\alpha }_1}{1-{\beta }_1-{\gamma }_1\rho }\right]\times \left[{\alpha }_2,\frac{{\alpha }_2}{1-{\beta }_2-{\gamma }_2\rho }\right]$ is a global attractor.

Proof. 

Let $\left\{({\Omega }_n,{\varpi }_n)\right\}$ be a non-negative and bounded solution of (1) and let ${lim}_{n\to \infty }sup{\Omega }_n=S_1<\infty ,{lim}_{n\to \infty }inf{\Omega }_n=I_1>0,$ ${lim}_{n\to \infty }sup{\varpi }_n=S_2<\infty$ and ${lim}_{n\to \infty }inf{\varpi }_n=I_2>0$ where $I_i,S_i\in (0,\infty ),i=1,2$. Then, from (1), we have

$$\begin{array}{c}\hfill S_1\le {\alpha }_1+{\beta }_1{S}_1+{\gamma }_1{S}_1{e}^{-(I_1+I_2)},I_1\ge {\alpha }_1+{\beta }_1{I}_1+{\gamma }_1{I}_1{e}^{-(S_1+S_2)}.\end{array}$$

(19)

And

$$\begin{array}{c}\hfill S_2\le {\alpha }_2+{\beta }_2{S}_2+{\gamma }_2{S}_2{e}^{-(I_1+I_2)},I_2\ge {\alpha }_2+{\beta }_2{I}_2+{\gamma }_2{I}_2{e}^{-(S_1+S_2)}.\end{array}$$

(20)

From (19), we have

$$\begin{array}{c}\hfill I_1{S}_1\le {\alpha }_1{I}_1+{\beta }_1{I}_1{S}_1+{\gamma }_1{I}_1{S}_1{e}^{-(I_1+I_2)},\end{array}$$

(21)

$$\begin{array}{c}\hfill I_1{S}_1\ge {\alpha }_1{S}_1+{\beta }_1{I}_1{S}_1+{\gamma }_1{I}_1{S}_1{e}^{-(S_1+S_2)}.\end{array}$$

(22)

From (20), we get

$$\begin{array}{c}\hfill I_2{S}_2\le {\alpha }_2{I}_2+{\beta }_2{I}_2{S}_2+{\gamma }_2{I}_2{S}_2{e}^{-(I_1+I_2)},\end{array}$$

(23)

$$\begin{array}{c}\hfill I_2{S}_2\ge {\alpha }_2{S}_2+{\beta }_2{I}_2{S}_2+{\gamma }_2{I}_2{S}_2{e}^{-(S_1+S_2)}.\end{array}$$

(24)

From (21)–(24), we get

$${\alpha }_1{S}_1+{\gamma }_1{I}_1{S}_1{e}^{-(S_1+S_2)}\le {\alpha }_1{I}_1+{\gamma }_1{I}_1{S}_1{e}^{-(I_1+I_2)},$$

(25)

$${\alpha }_2{S}_2+{\gamma }_2{I}_2{S}_2{e}^{-(S_1+S_2)}\le {\alpha }_2{I}_2+{\gamma }_2{I}_2{S}_2{e}^{-(I_1+I_2)}$$

(26)

From (25) and (26) we have $S_1\le I_1$ and $S_2\le I_2$. Hence, $S_1=I_1$ and $S_2=I_2$. □

Theorem 6.

If

$${\gamma }_1{\gamma }_2({\Omega }_1+{\Omega }_2+1){\rho }^2+\rho (1+{\beta }_2)\{({\Omega }_1+1){\gamma }_1+{\gamma }_2({\Omega }_2+1)\}+(1+{\beta }_2){\beta }_1+{\beta }_2<1,$$

where ${\Omega }_1=\frac{{\alpha }_1}{1-{\beta }_1-{\gamma }_1\rho }$, ${\Omega }_2=\frac{{\alpha }_2}{1-{\beta }_2-{\gamma }_2\rho }$, and $\rho =e^{-({\alpha }_1+{\alpha }_2)}$ then, $(\overline{\Omega },\overline{\varpi })\in$$\left[{\alpha }_1,\frac{{\alpha }_1}{1-{\beta }_1-{\gamma }_1\rho }\right]\times \left[{\alpha }_2,\frac{{\alpha }_2}{1-{\beta }_2-{\gamma }_2\rho }\right]$ of (1) is asymptotically globally stable.

Proof. 

Our proof can be proved by using Theorems (4) and (5). □

6. Convergence Rate

In this section, we deal with the rate of convergence of solutions that converges to our fixed point of (1).

We use the notion ≈ to mean that the relation is approximately equal.

Speculate that $\left\{({\Omega }_n,{\varpi }_n)\right\}$ be a solution of (1) provided that ${lim}_{n\to \infty }{\Omega }_n=\overline{\Omega }$, and ${lim}_{n\to \infty }{\varpi }_n=\overline{\varpi }$, where $\overline{\Omega }$ ∈ $\left[{\alpha }_1,\frac{{\alpha }_1}{1-{\beta }_1-{\gamma }_1\rho }\right]$ and $\overline{\varpi }\in \left[{\alpha }_2,\frac{{\alpha }_2}{1-{\beta }_2-{\gamma }_2\rho }\right]$.

We have,

$$\begin{array}{ccc}\hfill {\Omega }_{n+1}-\overline{\Omega }& =& {\alpha }_1+{\beta }_1{\Omega }_n+{\gamma }_1{\Omega }_{n-1}{e}^{-({\Omega }_n+{\varpi }_n)}-\{{\alpha }_1+{\beta }_1\overline{\Omega }+{\gamma }_1\overline{\Omega }{e}^{-(\overline{\Omega }+\overline{\varpi })}\}\hfill \end{array}$$

(27)

$$={\beta }_1({\Omega }_n-\overline{\Omega })+{\gamma }_1\{{\Omega }_{n-1}{e}^{-({\Omega }_n+{\varpi }_n)}-\overline{\Omega }{e}^{-(\overline{\Omega }+\overline{\varpi })}\}$$

$$={\beta }_1({\Omega }_n-\overline{\Omega })+{\gamma }_1{e}^{-({\Omega }_n+{\varpi }_n)}({\Omega }_{n-1}-\overline{\Omega })+{\gamma }_1\overline{\Omega }(e^{-({\Omega }_n+{\varpi }_n)}-e^{-(\overline{\Omega }+\overline{\varpi })})$$

$$={\beta }_1({\Omega }_n-\overline{\Omega })+{\gamma }_1{e}^{-({\Omega }_n+{\varpi }_n)}({\Omega }_{n-1}-\overline{\Omega })+{\gamma }_1\overline{\Omega }{e}^{-(\overline{\Omega }+\overline{\varpi })}(e^{-({\Omega }_n+{\varpi }_n-\overline{\Omega }-\overline{\varpi })}-1)$$

$$={\beta }_1({\Omega }_n-\overline{\Omega })+{\gamma }_1{e}^{-({\Omega }_n+{\varpi }_n)}({\Omega }_{n-1}-\overline{\Omega })+{\gamma }_1\overline{\Omega }{e}^{-(\overline{\Omega }+\overline{\varpi })}(-({\Omega }_n-\overline{\Omega })-({\varpi }_n-\overline{\varpi }))$$

$$+{\gamma }_1\overline{\Omega }{e}^{-(\overline{\Omega }+\overline{\varpi })}(O\left({({\Omega }_n-\overline{\Omega })}^2\right)+O\left({({\varpi }_n-\overline{\varpi })}^2\right)-1)$$

$$=({\beta }_1-{\gamma }_1\overline{\Omega }{e}^{-(\overline{\Omega }+\overline{\varpi })})({\Omega }_n-\overline{\Omega })+{\gamma }_1{e}^{-({\Omega }_n+{\varpi }_n)}({\Omega }_{n-1}-\overline{\Omega })-{\gamma }_1\overline{\Omega }{e}^{-(\overline{\Omega }+\overline{\varpi })}({\varpi }_n-\overline{\varpi })$$

$$+{\gamma }_1\overline{\Omega }{e}^{-(\overline{\Omega }+\overline{\varpi })}(O\left({({\Omega }_n-\overline{\Omega })}^2\right)+O\left({({\varpi }_n-\overline{\varpi })}^2\right)-1)$$

$$\approx ({\beta }_1-{\gamma }_1\overline{\Omega }{e}^{-(\overline{\Omega }+\overline{\varpi })})({\Omega }_n-\overline{\Omega })+{\gamma }_1{e}^{-({\Omega }_n+{\varpi }_n)}({\Omega }_{n-1}-\overline{\Omega })-{\gamma }_1\overline{\Omega }{e}^{-(\overline{\Omega }+\overline{\varpi })}({\varpi }_n-\overline{\varpi }).$$

Similarly, ${\varpi }_{n+1}-\overline{\varpi }\approx$

$$\begin{array}{c}\hfill ({\beta }_2-{\gamma }_2\overline{\varpi }{e}^{-(\overline{\Omega }+\overline{\varpi })})({\varpi }_n-\overline{\varpi })+{\gamma }_2{e}^{-({\Omega }_n+{\varpi }_n)}({\varpi }_{n-1}-\overline{\varpi })-{\gamma }_2\overline{\varpi }{e}^{-(\overline{\Omega }+\overline{\varpi })}({\Omega }_n-\overline{\Omega }).\end{array}$$

Set $e_n^1={\Omega }_n-\overline{\Omega }$ and $e_n^2={\varpi }_n-\overline{\varpi }$, one has

$$e_{n+1}^1=A_n{e}_n^1+B_n{e}_n^2+C_n{e}_{n-1}^1+D_n{e}_{n-1}^2,$$

$$e_{n+1}^2=E_n{e}_n^1+F_n{e}_n^2+G_n{e}_{n-1}^1+H_n{e}_{n-1}^2,$$

where

$$A_n={\beta }_1-{\gamma }_1\overline{\Omega }{e}^{-(\overline{\Omega }+\overline{\varpi })},B_n=-{\gamma }_1\overline{\Omega }{e}^{-(\overline{\Omega }+\overline{\varpi })},C_n={\gamma }_1{e}^{-({\Omega }_n+{\varpi }_n)},D_n=0.$$

$$E_n=-{\gamma }_2\overline{\varpi }{e}^{-(\overline{\Omega }+\overline{\varpi })},F_n=({\beta }_2-{\gamma }_2\overline{\varpi }{e}^{-(\overline{\Omega }+\overline{\varpi })}),G_n=0,H_n={\gamma }_2{e}^{-({\Omega }_n+{\varpi }_n)}.$$

We obtain

$$\underset{n\to \infty }{lim}{A}_n={\beta }_1-{\gamma }_1\overline{\Omega }{e}^{-(\overline{\Omega }+\overline{\varpi })},\underset{n\to \infty }{lim}{B}_n=-{\gamma }_1\overline{\Omega }{e}^{-(\overline{\Omega }+\overline{\varpi })},\underset{n\to \infty }{lim}{C}_n={\gamma }_1{e}^{-(\overline{\Omega }+\overline{\varpi })},$$

$$\underset{n\to \infty }{lim}{D}_n=0,\underset{n\to \infty }{lim}{E}_n=-{\gamma }_2\overline{\varpi }{e}^{-(\overline{\Omega }+\overline{\varpi })},\underset{n\to \infty }{lim}{F}_n=({\beta }_2-{\gamma }_2\overline{\varpi }{e}^{-(\overline{\Omega }+\overline{\varpi })}),$$

$$\underset{n\to \infty }{lim}{G}_n=0,\underset{n\to \infty }{lim}{H}_n={\gamma }_2{e}^{-(\overline{\Omega }+\overline{\varpi })}.$$

Hence,

$$R_{n+1}=K{R}_n,$$

(28)

where $R_n=\left(\begin{array}{c}{e}_{n+1}^1\\ e_{n+1}^2\\ e_n^1\\ e_n^2\end{array}\right),$ $K=\left(\begin{array}{cccc}{\beta }_1-{\gamma }_1\overline{\Omega }{e}^{-(\overline{\Omega }+\overline{\varpi })}& -{\gamma }_1\overline{\Omega }{e}^{-(\overline{\Omega }+\overline{\varpi })}& {\gamma }_1{e}^{-(\overline{\Omega }+\overline{\varpi })}& 0\\ -{\gamma }_2\overline{\varpi }{e}^{-(\overline{\Omega }+\overline{\varpi })}& {\beta }_2-{\gamma }_2\overline{\varpi }{e}^{-(\overline{\Omega }+\overline{\varpi })}& 0& {\gamma }_2{e}^{-(\overline{\Omega }+\overline{\varpi })}\\ 1& 0& 0& 0\\ 0& 1& 0& 0\end{array}\right),$ by using proposition (1), we get the following theorem.

Theorem 7.

Let $\left\{({\Omega }_n,{\varpi }_n)\right\}$ be a solution of (1) provided that ${lim}_{n\to \infty }{\Omega }_n=\overline{\Omega }$, and ${lim}_{n\to \infty }{\varpi }_n=\overline{\varpi }$, with $(\overline{\Omega },\overline{\varpi })$ ∈ $\left[{\alpha }_1,\frac{{\alpha }_1}{1-{\beta }_1-{\gamma }_1\rho }\right]\times \left[{\alpha }_2,\frac{{\alpha }_2}{1-{\beta }_2-{\gamma }_2\rho }\right]$. Then, the error vector $R_n$ of each solution for (1) satisfies

$$\underset{n\to \infty }{lim}(\parallel R_n{\parallel )}^{\frac{1}{n}}=max\left\{{\chi }_{1,2,3,4}{F}_J(\overline{\Omega },\overline{\varpi })\right\},\underset{n\to \infty }{lim}\frac{\parallel R_{n+1}\parallel }{\parallel R_n\parallel }=max\left\{{\chi }_{1,2,3,4}{F}_J(\overline{\Omega },\overline{\varpi })\right\}$$

with ${\chi }_{1,2,3,4}{F}_J(\overline{\Omega },\overline{\varpi })$ are the characteristic roots of $F_J(\overline{\Omega },\overline{\varpi })$ at $(\overline{\Omega },\overline{\varpi })$.

7. Unbounded Solution’s Existence

Our goal in the current part is scrutinizing the unbounded solutions for (1).

Theorem 8.

Suppose that

(i) 

${\alpha }_1{\gamma }_2>{\alpha }_2{\gamma }_1.$

(ii) 

${\gamma }_1(1-{\beta }_2)\ne {\gamma }_2(1-{\beta }_1).$

(iii) 

$\frac{{\alpha }_1{\gamma }_2-{\alpha }_2{\gamma }_1}{{\gamma }_2(1-{\beta }_1)-{\gamma }_1(1-{\beta }_2)}>\frac{{\alpha }_1}{1-{\beta }_1}.$

(iv) 

$\frac{{\alpha }_1{\gamma }_2-{\alpha }_2{\gamma }_1}{{\gamma }_2(1-{\beta }_1)-{\gamma }_1(1-{\beta }_2)}>\frac{{\alpha }_2}{1-{\beta }_2}.$

Then

$$\underset{n\to \infty }{lim}{\Omega }_n=\underset{n\to \infty }{lim}{\varpi }_n=\infty .$$

Proof. 

Consider a solution $\left\{({\Omega }_n,{\varpi }_n)\right\}$ of (1) with intial values ${\Omega }_0,{\Omega }_{-1},{\varpi }_0,{\varpi }_{-1}$ such that ${\Omega }_{-1}<m$, ${\Omega }_0<m$, ${\varpi }_{-1}<m$, ${\varpi }_0<m$, ${\Omega }_0>M$ and ${\varpi }_0>M$ with

$$m=\frac{{\alpha }_1{\gamma }_2-{\alpha }_2{\gamma }_1}{{\gamma }_2(1-{\beta }_1)-{\gamma }_1(1-{\beta }_2)},M=\frac{{\alpha }_1+{\alpha }_2}{2}.$$

Then

$${\Omega }_1={\alpha }_1+{\beta }_1{\Omega }_0+{\gamma }_1{\Omega }_{-1}{e}^{-({\Omega }_0+{\varpi }_0)}<{\alpha }_1+{\beta }_{1}m+{\gamma }_{1}m{e}^{-\left(2M\right)}=m.$$

Similarly,

$${\varpi }_1<{\alpha }_2+{\beta }_{2}m+{\gamma }_{2}m{e}^{-\left(2M\right)}=m.$$

$${\Omega }_2=$$

$${\alpha }_1+{\beta }_1{\Omega }_1+{\gamma }_1{\Omega }_0{e}^{-({\Omega }_1+{\varpi }_1)}<{\alpha }_1+{\beta }_1{\Omega }_1+{\gamma }_1{\Omega }_0{e}^{-({\alpha }_1+{\alpha }_2)}<{\alpha }_1+{\beta }_{1}m+{\gamma }_{1}m{e}^{-\left(2M\right)}$$

$$=m.$$

Likewise,

$${\varpi }_2<{\alpha }_2+{\beta }_{2}m+{\gamma }_{2}m{e}^{-\left(2M\right)}=m.$$

Therefore, by induction, we have

$${\Omega }_n<m,{\varpi }_n<m.$$

Now

$${\Omega }_{n+1}={\alpha }_1+{\beta }_1{\Omega }_n+{\gamma }_1{\Omega }_{n-1}{e}^{-({\Omega }_n+{\varpi }_n)}>{\alpha }_1+{\beta }_1{\Omega }_n+{\gamma }_1{\Omega }_{n-1}{e}^{-\left(2m\right)},$$

$${\varpi }_{n+1}={\alpha }_2+{\beta }_2{\Omega }_n+{\gamma }_2{\varpi }_{n-1}{e}^{-({\Omega }_n+{\varpi }_n)}>{\alpha }_2+{\beta }_2{\Omega }_n+{\gamma }_2{\varpi }_{n-1}{e}^{-\left(2m\right)}.$$

Consider

$${\Omega }_{n+1}=a_1+a_2{\Omega }_n+a_3{\Omega }_{n-1},$$

(29)

$${\varpi }_{n+1}=b_1+b_2{\varpi }_n+b_3{\varpi }_{n-1}.$$

(30)

The solution of Equation (29)

$${\Omega }_n=\frac{a_1}{1-a_2-a_3}+r_1{2}^{-n}{\{a_2-\sqrt{a_2^2+4a_3}\}}^n+r_2{2}^{-n}{\{a_2+\sqrt{a_2^2+4a_3}\}}^n.$$

The solution of Equation (30)

$${\varpi }_n=\frac{b_1}{1-b_2-b_3}+r_3{2}^{-n}{\{b_2-\sqrt{b_2^2+4b_3}\}}^n+r_4{2}^{-n}{\{b_2+\sqrt{b_2^2+4b_3}\}}^n.$$

□

We note that ${\Omega }_n,{\varpi }_n\to \infty$ as $n\to \infty$. So ${\Omega }_n,{\varpi }_n\to \infty$ as $n\to \infty$.

8. Numerical Examples

We provide some examples in this final section to validate the conclusions of our previous parts.

Example 1.

Let ${\alpha }_1=1,{\alpha }_2=1,{\beta }_1=1/10,{\beta }_2=1/9,{\gamma }_1=-2,{\gamma }_2=-3$, then (1) with ${\Omega }_{-1}=0.1,{\Omega }_0=0.8,{\varpi }_{-1}=1.2,{\varpi }_0=0.7$ can be written as:

$${\Omega }_{n+1}=1+(1/10){\Omega }_n-2{\Omega }_{n-1}{e}^{-({\Omega }_n+{\varpi }_n)},$$

$${\varpi }_{n+1}=1+(1/9){\Omega }_n-3{\varpi }_{n-1}{e}^{-({\Omega }_n+{\varpi }_n)},n=0,1,\cdots$$

(31)

So the equilibrium (0.679366, 0.572465) of (31) is asymptotically globally stable. (see Figure 1).

Example 2.

Let ${\alpha }_1=3,{\alpha }_2=5,{\beta }_1=1,{\beta }_2=1/5,{\gamma }_1=3,{\gamma }_2=2$, then (1) with ${\Omega }_{-1}=0.2,{\Omega }_0=0.1,{\varpi }_{-1}=0.8,{\varpi }_0=1.2$ can be written as:

$${\Omega }_{n+1}=2+{\Omega }_n+3{\Omega }_{n-1}{e}^{-({\Omega }_n+{\varpi }_n)},{\varpi }_{n+1}=3+{\Omega }_n+2{\varpi }_{n-1}{e}^{-({\Omega }_n+{\varpi }_n)},n=0,1,\cdots$$

(32)

Therefore, the equilibrium of (32) will be unstable. (see Figure 2).

9. Conclusions

We have analyzed the behavior of a following system of recursive equations in this paper:

$${\Omega }_{n+1}={\alpha }_1+{\beta }_1{\Omega }_n+{\gamma }_1{\Omega }_{n-1}{e}^{-({\Omega }_n+{\varpi }_n)},$$

$${\varpi }_{n+1}={\alpha }_2+{\beta }_2{\varpi }_n+{\gamma }_2{\varpi }_{n-1}{e}^{-({\Omega }_n+{\varpi }_n)},$$

$n=0,1,\cdots$, whereby initial values ${\Omega }_{-1},{\varpi }_{-1},{\Omega }_0,{\varpi }_0$ and parameters ${\alpha }_1,{\alpha }_2$ are non-negative real numbers and ${\beta }_1,{\beta }_2\in (0,1)$ and ${\gamma }_1,{\gamma }_2\le 0$. We showed that each non-negative solution $\left\{({\Omega }_n,{\varpi }_n)\right\}$ of this system persists and is bounded. Besides, the uniqueness and existence of the equilibrium, global and local stability, and convergence rate for non-negative solutions that merge to just one non-negative point, as well as the unboundedness, have been demonstrated.