Introduction

The theory of discrete dynamical system has many applications in applied sciences. Mathematical modeling of a physical, biological or ecological problem mostly leads to a nonlinear difference system. (See [1,2,3,4,5,6,7,8,9,10].)

In [4], Papachinopoulos et al. proposed a system of equation with exponents as

$$\begin{aligned} f_{n+1}= a + b f_{n-1} e ^{-g_n},g_{n+1}= c + d g_{n-1} e ^{-f_n}, \quad n=0,1,2,\ldots , \end{aligned}$$
(1)

where a, b, c, d and the initial conditions \(f_{-1},f_0, g_{-1}, g_0\) are positive real values. They studied the existence, boundedness and asymptotic behavior of the positive solutions of (1).

In [5], G.Papaschinopoulos and C.J.Schinas together modified the system as

$$\begin{aligned} f_{ n + 1}&= a + bg_{n - 1} e^{ - f_n} ,g_{n + 1} = c + {\mathrm{d}}f_{n - 1} e^{- g_n} ,\nonumber \\ f_{n + 1}&= a + bg_{n - 1} e^{- g_n} , g_{n + 1} = c + {\mathrm{d}}f_{n - 1} e^{- f_n} , \end{aligned}$$
(2)

and put forward conditions for the positive solutions to be asymptotic.

In [11], authors multiplied \(f_n\) and \(g_n\) with a and c, respectively, in (2) and formed a new system of difference equations

$$\begin{aligned} f_{n+1}= af_n+bg_{n-1}e^{-f_n},g_{n+1}= cg_n+ {\mathrm{d}}f_{n-1}e^{-g_n}, n=0,1,\ldots \end{aligned}$$

and described the existence of a unique positive equilibrium, the boundedness, persistence and global attractivity of the positive solutions.

Parallelly in [12], the authors worked on the asymptotic behavior of the positive solutions of a similar difference system

$$\begin{aligned} f_{n+1}= ag_n+bf_{n-1}e^{-g_n},g_{n+1}= cf_n+ {\mathrm{d}}g_{n-1}e^{-f_n}, n=0,1,\ldots . \end{aligned}$$

N.Psarros and G.Papaschinopoulos in [13] proposed a new first-order model

$$\begin{aligned} f_{n+1}= ag_n+bf_{n}e^{-f_n-g_n},g_{n+1}= cf_n+ {\mathrm{d}}g_{n}e^{-f_n-g_n}, \end{aligned}$$

and studied the asymptotic behavior of the positive solutions of the system.

Motivated by the above research articles, we propose a new second order difference system

$$\begin{aligned} x_{n+1}&= \alpha _1 + a e ^{-x_{n-1}} + b y_{n} e ^{-y_{n-1}},\\ y_{n+1}&= \alpha _2 +c e ^{-y_{n-1}}+ d x_{n} e ^{-x_{n-1}} \quad n=0,1,2,\ldots \end{aligned}$$
(3)

where \(\alpha _1, \alpha _2, a, b , c,d\) are positive real numbers and the initial conditions \(x_{-1},x_0, y_{-1}, y_0\) are arbitrary nonnegative numbers, and investigate the persistence, boundedness, convergence, invariance, and global asymptotic behavior of the positive solutions of the system.

Methods

We use Theorem 1.16 of [14] to prove the lemma which we use to derive a condition for the existence, uniqueness of equilibrium solutions and the convergence of positive solutions to the equilibrium solution. We also use Remark 1.3.1 of [15] to obtain conditions for global asymptotic stability of the unique equilibrium point.

Results and discussion

The following theorem proposes conditions for persistence and boundedness for the positive solution \((x_n,y_n)\) of (3).

Theorem 1

Every positive solution \((x_n,y_n)\) of (3) is bounded and persists whenever \(bde^{-\alpha _1-\alpha _2}<1\).

Proof

\(x_n \ge \alpha _1, y_n \ge \alpha _2\), \(n=3,4,\ldots .\)

Hence, \((x_n,y_n)\) of system (3) persists.

Also, (3) becomes

$$\begin{aligned} x_{n+1}&\le \alpha _1 + a e ^{-\alpha _1} + b e^{-\alpha _2} [\alpha _2 +{\mathrm{d}}x_{n-1}e^{-x_{n-2}}+ce^{-y_{n-2}}].\nonumber \\&\le A + bdx_{n-1}e^{-\alpha _1-\alpha _2} \end{aligned}$$
(4)

where \(A= \alpha _1 + a e ^{-\alpha _1} + b \alpha _2e^{-\alpha _2}+ bce^{-\alpha _2-\alpha _2}\).

Similarly,

$$\begin{aligned} y_{n+1} \le C + bdy_{n-1}e^{-\alpha _1-\alpha _2} \end{aligned}$$
(5)

where \(C= \alpha _2 + c e ^{-\alpha _2} + d \alpha _1e^{-\alpha _1}+ ade^{-\alpha _1-\alpha _1}\).

Now, consider the difference equations

$$\begin{aligned} z_{n+1}&= A + Bz_{n-1}.\nonumber \\ v_{n+1}&= C + Dv_{n-1}, \end{aligned}$$
(6)

where \(B=D=bde^{-\alpha _1-\alpha _2}<1\). Therefore, an arbitrary solution \((z_n, v_n)\) of (6) can be written as

$$\begin{aligned} z_n&= r_1B^{n/2} + r_2(-1)^nB^{n/2}+\frac{A}{1-B} , \quad n=0,1,2,\ldots \end{aligned}$$
(7)
$$\begin{aligned} v_n&= s_1B^{n/2} + s_2(-1)^nB^{n/2}+\frac{C}{1-B} , \quad n=0,1,2,\ldots \end{aligned}$$
(8)

where \(r_1\), \(r_2\) rely on the initial conditions \(z_{-1}\), \(z_0\) and \(s_1\), \(s_2\) rely on the initial conditions \(v_{-1}\), \(v_0\). Hence, \((z_n, v_n)\) is bounded.

Let us examine the solution \((z_n, v_n)\) such that \(z_{-1}=x_{-1}, z_0=x_0,v_{-1}=y_{-1}, v_0=y_0.\)

Hence by induction, \(x_n \le z_n\) and \(y_n \le v_n, n=0,1,2,\ldots\).

Therefore, we get \((x_n, y_n)\) is bounded. \(\square\)

The following two theorems confirm the existence of invariant boxes of (3).

Theorem 2

Let \(bde^{-\alpha _1-\alpha _2}<1\). Let \((x_n, y_n)\) denote a positive solution of (3). Then \(\displaystyle [\alpha _1,\frac{\alpha _1 + a e ^{-\alpha _1} + b \alpha _2e^{-\alpha _2}+ bce^{-\alpha _2-\alpha _2}}{(1-bde^{-\alpha _1-\alpha _2})}]\) \(\displaystyle \times [\alpha _2,\frac{\alpha _2 + c e ^{-\alpha _2} + d \alpha _1e^{-\alpha _1}+ ade^{-\alpha _1-\alpha _1}}{(1-bde^{-\alpha _1-\alpha _2})}]\) is an invariant set for (3).

Proof

Let \(I_1=\displaystyle [\alpha _1,\frac{\alpha _1 + a e ^{-\alpha _1} + b \alpha _2e^{-\alpha _2}+ bce^{-\alpha _2-\alpha _2}}{(1-bde^{-\alpha _1-\alpha _2})}]\) and \(\displaystyle I_2=[\alpha _2,\frac{\alpha _2 + c e ^{-\alpha _2} + d \alpha _1e^{-\alpha _1}+ ade^{-\alpha _1-\alpha _1}}{(1-bde^{-\alpha _1-\alpha _2})}]\).

Let \(x_{-1}, x_{0} \in I_1\) and \(y_{-1}, y_{0} \in I_2.\)

Then

$$\begin{aligned} x_{1}&\le \alpha _1+ ae ^{-\alpha _1} + be^{-\alpha _2}y_0\\&\le \alpha _1+ ae ^{-\alpha _1} + be^{-\alpha _2}\displaystyle \left[ \frac{\alpha _2 + c e ^{-\alpha _2} + d \alpha _1e^{-\alpha _1}+ ade^{-\alpha _1-\alpha _1}}{1-bde^{-\alpha _1-\alpha _2}}\right] . \end{aligned}$$

Hence, we get \(x_1 \le \displaystyle \frac{\alpha _1 + a e ^{-\alpha _1} + b \alpha _2e^{-\alpha _2}+ bce^{-\alpha _2-\alpha _2}}{1-bde^{-\alpha _1-\alpha _2}}.\), i.e., \(x_{1} \in I_1\). Similarly, we get \(y_1 \in I_2.\)

Hence, the proof follows by applying the method of induction. \(\square\)

Theorem 3

Let \(bde^{-\alpha _1-\alpha _2}<1\). Consider the intervals

$$\begin{aligned} I_3= \displaystyle \left[ \alpha _1,\frac{\alpha _1 + a e ^{-\alpha _1} + b \alpha _2e^{-\alpha _2}+ bce^{-\alpha _2-\alpha _2}+\epsilon }{1-bde^{-\alpha _1-\alpha _2}}\right] \end{aligned}$$

and

$$\begin{aligned} I_4=\displaystyle \left[ \alpha _2,\frac{\alpha _2 + c e ^{-\alpha _2} + d \alpha _1e^{-\alpha _1}+ ade^{-\alpha _1-\alpha _1}+\epsilon }{1-bde^{-\alpha _1-\alpha _2}}\right] \end{aligned}$$

where \(\epsilon\) is an arbitrary positive number. If \((x_n,y_n)\) is any arbitrary solution of (3), then there exists an \(N \in {\mathbb {N}}\) such that \(x_n \in I_3\) and \(y_n \in I_4, n \ge N\).

Proof

Let \((x_n,y_n)\) denote an arbitrary solution of (3).

Then by Theorem 1, \(\limsup _{n \rightarrow \infty }{x_n}=M< \infty\) and \(\limsup _{n \rightarrow \infty }{y_n}=L< \infty\).

Hence from Theorem 1, \(x_{n+1} \le A + bdx_{n-1}e^{-\alpha _1-\alpha _2}\) and \(y_{n+1}\le C + bdy_{n-1}e^{-\alpha _1-\alpha _2}\)

Hence \(\displaystyle M \le \frac{A}{1-bde^{-\alpha _1-\alpha _2}}\), and \(\displaystyle L \le \frac{C}{1-bde^{-\alpha _1-\alpha _2}}\).

Hence, there exists an \(N \in {\mathbb {N}}\) such that the theorem holds. \(\square\)

Now we prove a lemma which is an alteration of Theorem 1.16 of [14].

Lemma 4

Let [a, b] and [c, d] denote intervals of real numbers. Let \(f:[a,b]\times [c,d]\times [c,d] \rightarrow [a,b]\) and \(g:[a,b]\times [a,b]\times [c,d] \rightarrow [c,d]\) be continuous functions. Consider the difference system

$$\begin{aligned} x_{n+1}&= f(x_{n-1},y_n,y_{n-1}),\nonumber \\ y_{n+1}&= g(x_n,x_{n-1},y_{n-1}), \quad n=0,1,2,\ldots \end{aligned}$$
(9)

such that the initial values \(x_{-1},x_0 \in [a,b]\) and \(y_{-1}, y_0 \in [c,d]\). (or \(x_{n_0},x_{n_0+1} \in [a,b],\) \(y_{n_0},y_{n_0+1} \in [c,d], n_0 \in {\mathbb {N}}\)). Suppose the following are true.

  1. 1.

    If f(x, y, z) is nonincreasing in x, f(x, y, z) is nondecreasing in y and f(x, y, z) is nonincreasing in z.

  2. 2.

    If g(x, y, z) is nondecreasing in x, g(x, y, z) is nonincreasing in y and g(x, y, z) is nonincreasing in z.

  3. 3.

    If \((m_1,M_1,m_2,M_2) \in [a,b]^2\times [c,d]^2\) satisfies the systems \(m_1=f(M_1,m_2,M_2),\) \(M_1=f(m_1,M_2,m_2)\) and \(m_2=g(m_1,M_1,M_2), M_2=g(M_1,m_1,m_2)\) then \(M_1=m_1\) and \(M_2=m_2\),

then there exists a unique equilibrium solution \(({\bar{x}},{\bar{y}})\) of (9) with \({\bar{x}} \in [a,b]\), \({\bar{y}} \in [c,d]\). Also every solution of (9) converges to \(({\bar{x}},{\bar{y}})\).

Proof

Set \(m_1^{-1}=a, m_1^{0}=a,m_2^{-1}=c, m_2^{0}=c.\)

$$\begin{aligned} M_1^{-1}=b, M_1^{0}=b,M_2^{-1}=d, M_2^{0}=d. \end{aligned}$$

For each \(i \ge 0\), let \(m_1^{i+1}=f(M_1^{i-1},m_2^i,M_2^{i-1}), M_1^{i+1}=f(m_1^{i-1},M_2^i,m_2^{i-1})\) and

$$\begin{aligned} m_2^{i+1}=g(M_1^i,m_1^{i-1},m_2^{i-1}), M_2^{i+1}=g(m_1^i,M_1^{i-1},M_2^{i-1}). \end{aligned}$$

Hence \(m_1^1= f(M_1^{-1},m_2^0,M_2^{-1}) \le f(m_1^{-1},M_2^0,m_2^{-1})= M_1^{1} ,\) and

$$\begin{aligned} m_2^{1}=g(m_1^0,M_1^{-1}, M_2^{-1}) \le g(M_1^0,m_1^{-1},m_2^{-1})= M_2^1. \end{aligned}$$

Therefore,

$$\begin{aligned} M_1^{-1}\ge & {} M_1^0 \ge M_1^1 \ge m_1^1 \ge m_1^0 \ge m_1^{-1} \quad {\hbox {and}}\\ M_2^{-1}\ge & {} M_2^0 \ge M_2^1 \ge m_2^1 \ge m_2^0 \ge m_2^{-1}. \end{aligned}$$

Also \(m_1^{0}=a \le x_n \le b =M_1^0, n\ge 0\) and \(m_2^{0}=c \le y_n \le d =M_2^0, n\ge 0\).

For all \(n\ge 0\), we have

$$\begin{aligned} m_1^{1}&= f(M_1^{-1},m_2^0,M_2^{-1}) \le f(x_{n-1},y_n,y_{n-1}) \le f(m_1^{-1},M_2^0,m_1^{-1}) = M_1^1.\\ m_2^{1}&= g(m_1^0,M_1^{-1},M_2^{-1}) \le g(x_n,x_{n-1},y_{n-1}) \le g(M_1^0,m_1^{-1},M_2^0) =M_2^{1}. \end{aligned}$$

Hence \(m_1^{1} \le x_n \le M_1^1, n\ge 1\) and \(m_2^{1} \le y_n \le M_2^1, n\ge 1\).

We then obtain by induction that for \(i \ge 0\), the following are true.

  1. 1.

    \(a=m_1^{-1}\le m_1^0 \le m_1^1 \ldots \le m_1^{i-1}\le m_1^{i}\le M_1^{i}\ldots \le M_1^1 \le M_1^0 \le M_1^{-1}=b\).

  2. 2.

    \(c=m_2^{-1}\le m_2^0 \le m_2^1 \ldots \le m_2^{i-1}\le m_2^{i}\le M_2^{i}\ldots \le M_2^1 \le M_2^0 \le M_2^{-1}=d\).

  3. 3.

    \(m_1^{i} \le x_n \le M_1^i, n\ge 1\) and \(m_2^{i} \le y_n \le M_2^i, n\ge 1\).

Set \(m_1= \lim _{i \rightarrow \infty } m_1^{i} , m_2= \lim _{i \rightarrow \infty } m_2^{i}\) and \(M_1= \lim _{i \rightarrow \infty } M_1^{i} , M_2= \lim _{i \rightarrow \infty } M_2^{i}\).

Since f and g are continuous, we get \(m_1=f(M_1,m_2,M_2), M_1=f(m_1,M_2,m_2)\) and \(m_2=g(m_1,M_1,M_2), M_2=g(M_1,m_1,m_2).\)

Hence \(M_1=m_1={\bar{x}}\) and \(M_2=m_2={\bar{y}}\), from which we get the proof. \(\square\)

The following theorem proposes conditions for the convergence of the equilibrium solution of (3).

Theorem 5

Suppose

$$\begin{aligned} bde^{-\alpha _1-\alpha _2}<1, ce^{-\alpha _2}<1, ae^{-\alpha _1}<1 \end{aligned}$$
(10)

and

$$\begin{aligned}& \frac{bde^{-\alpha _1-\alpha _2}}{[1-bde^{-\alpha _1-\alpha _2}]^2} \frac{[1-bde^{-\alpha _1-\alpha _2}+ \alpha _2 +ce^{-\alpha _2}+{\mathrm{d}}\alpha _1e^{-\alpha _1} +ade^{-\alpha _1-\alpha _1}]}{[1-ae^{-\alpha _1}]} \\&\quad \times \frac{[1-bde^{-\alpha _1-\alpha _2}+ \alpha _1 +ae^{-\alpha _1}+b\alpha _2e^{-\alpha _2} +bce^{-\alpha _2-\alpha _2}]}{ [1-ce^{-\alpha _2}]}<1. \end{aligned}$$
(11)

Then (3) has a unique positive equilibrium \(E({\bar{x}},{\bar{y}})\). Also, every solution of (3) converges to \(E({\bar{x}},{\bar{y}})\).

Proof

Let \(f: {\mathbb {R}}^+ \times {\mathbb {R}}^+ \times {\mathbb {R}}^+ \rightarrow {\mathbb {R}}^+,g: {\mathbb {R}}^+ \times {\mathbb {R}}^+ \times {\mathbb {R}}^+ \rightarrow {\mathbb {R}}^+\) be continuous functions such that \(f(x,y,z)= \alpha _1 + ae^{-x} +bye^{-z}\), \(g(x,y,z)=\alpha _2 + ce^{-z} +dxe^{-y}\).

Let \(M_1,m_1,M_2,m_2\) be positive real numbers satisfying

$$\begin{aligned} m_1&= \alpha _1 +ae^{-M_1} + bm_2e^{-M_2}, M_1=\alpha _1 +ae^{-m_1} + bM_2e^{-m_2} \end{aligned}$$

and

$$\begin{aligned} m_2=\alpha _2 +ce^{-M_2}+ {\mathrm{d}}m_1e^{-M_1} , M_2=\alpha _2 +ce^{-m_2}+{\mathrm{d}}M_1e^{-m_1}. \end{aligned}$$
(12)

Therefore, \(M_1-m_1=a[e^{-m_1}-e^{-M_1}] + b[M_2e^{-m_2}-m_2e^{-M_2}].\)

$$\begin{aligned} M_1-m_1=a[e^{-m_1}-e^{-M_1}] + be^{-m_2-M_2}[M_2e^{M_2}-m_2e^{m_2}]. \end{aligned}$$
(13)

Also, there exists a \(\zeta\) , \(m_2 \le \zeta \le M_2\) satisfying

$$\begin{aligned} M_2e^{M_2}-m_2e^{m_2}= (1+\zeta ) e^\zeta (M_2-m_2). \end{aligned}$$
(14)

From (13) and (14), we get

$$\begin{aligned} M_1-m_1=a[e^{-m_1}-e^{-M_1}] + be^{-m_2-M_2+\zeta }(1+\zeta )[M_2-m_2]. \end{aligned}$$
(15)

Now, \(a[e^{-m_1}-e^{-M_1}] = ae^{-m_1-M_1}[e^{M_1}-e^{m_1}].\)

Also there exists a \(\lambda\), \(m_1 \le \lambda \le M_1\) satisfying

$$\begin{aligned} a[e^{-m_1}-e^{-M_1}] = a e^{{-m_1-M_1+\lambda }}[M_1-m_1]. \end{aligned}$$
(16)

Since \(M_1,m_1 \ge \alpha _1\) and \(\lambda \le M_1,\)

$$\begin{aligned} a[e^{-m_1}-e^{-M_1}] \le ae^{-\alpha _1}[M_1-m_1]. \end{aligned}$$
(17)

Thus, from (15) and (17) we get,

$$\begin{aligned} M_1-m_1 \le ae^{-\alpha _1}[M_1-m_1] + be^{-m_2-M_2+\zeta }(1+\zeta )[M_2-m_2]. \end{aligned}$$
(18)

Since \(M_2,m_2 \ge \alpha _2\) and \(\zeta \le M_2\), (18) becomes

$$\begin{aligned} M_1-m_1 \le ae^{-\alpha _1}[M_1-m_1] + be^{-\alpha _2}(1+\zeta )[M_2-m_2]. \end{aligned}$$
(19)

, i.e.,

$$\begin{aligned}{}[1-ae^{-\alpha _1}][M_1-m_1] \le be^{-\alpha _2}(1+\zeta )[M_2-m_2]. \end{aligned}$$
(20)

Also, (12) can be written as

$$\begin{aligned} M_2&= \alpha _2 +ce^{-m_2}+{\mathrm{d}}[\alpha _1 +ae^{-m_1}+ bM_2e^{-m_2}]e^{-m_1}. \end{aligned}$$
(21)
$$\begin{aligned} M_2&\le \frac{\alpha _2 +ce^{-\alpha _2}+ {\mathrm{d}}\alpha _1e^{-\alpha _1}+ ade^{-\alpha _1-\alpha _1}}{1-bde^{-\alpha _1-\alpha _2}}. \end{aligned}$$
(22)

Since \(\zeta \le M_2\) we get,

$$\begin{aligned} \zeta \le \frac{\alpha _2 +ce^{-\alpha _2}+ {\mathrm{d}}\alpha _1e^{-\alpha _1}+ade^{-\alpha _1-\alpha _1}}{1-bde^{-\alpha _1-\alpha _2}}. \end{aligned}$$
(23)

Therefore, (20) becomes

$$\begin{aligned}&[1-ae^{-\alpha _1}][M_1-m_1]\nonumber \\&\quad \le be^{-\alpha _2}\left[ \frac{1-bde^{-\alpha _1-\alpha _2}+ \alpha _2 +ce^{-\alpha _2}+{\mathrm{d}}\alpha _1e^{-\alpha _1} +ade^{-\alpha _1-\alpha _1}] }{1-bde^{-\alpha _1-\alpha _2}}\right] [M_2-m_2]. \end{aligned}$$
(24)

Similarly, we get

$$\begin{aligned}&[1-ce^{-\alpha _2}][M_2-m_2]\nonumber \\&\quad \le {\mathrm{d}}e^{-\alpha _1}\left[ \frac{1-bde^{-\alpha _1-\alpha _2}+ \alpha _1 +ae^{-\alpha _1}+b\alpha _2e^{-\alpha _2} +bce^{-\alpha _2-\alpha _2}] }{1-bde^{-\alpha _1-\alpha _2}}\right] [M_1-m_1]. \end{aligned}$$
(25)

From (24) and (25), we get

$$\begin{aligned}&\displaystyle [M_1-m_1]\nonumber \\&\quad \displaystyle \le \frac{bde^{-\alpha _1-\alpha _2}}{[1-(bde^{-\alpha _1-\alpha _2)}]^2} \frac{[1-bde^{-\alpha _1-\alpha _2}+ \alpha _2 +ce^{-\alpha _2}+{\mathrm{d}}\alpha _1e^{-\alpha _1} +ade^{-\alpha _1-\alpha _1}]}{[1-ae^{-\alpha _1}]}\nonumber \\&\qquad \times \frac{[1-bde^{-\alpha _1-\alpha _2}+ \alpha _1 +ae^{-\alpha _1}+b\alpha _2e^{-\alpha _2} +bce^{-\alpha _2-\alpha _2}]}{ [1-ce^{-\alpha _2}]}[M_1-m_1]. \end{aligned}$$
(26)

Therefore from (11) and (26), we get \(M_1=m_1\) and \(M_2=m_2\).

Therefore by applying Lemma 4, the result is obtained. \(\square\)

In the next theorem, we derive conditions for the global asymptotic stability of the equilibrium solution of (3).

Theorem 6

Assume (10) and (11) holds.

  1. 1.

    Let \((a+ac+c)<1\). If \((1+{\bar{x}})(1+{\bar{y}})< \displaystyle \frac{1-(a+ac+c)}{bd}\), then the unique equilibrium \(E({\bar{x}},{\bar{y}})\) is globally asymptotically stable.

  2. 2.

    If \((a+c+ac+bd)+ bd[\frac{A}{1-B}+\frac{C}{1-B}+\frac{AC}{(1-B)^2}]<1\), where A, B and C are defined as in (4) and (5), then the unique equilibrium \(E({\bar{x}},{\bar{y}})\) is globally asymptotically stable.

Proof

First we show that \(E({\bar{x}},{\bar{y}})\) is locally asymptotically stable in both the cases. The Jacobian \(JF({\bar{x}},{\bar{y}})\) about the equilibrium point \(E({\bar{x}},{\bar{y}})\) is given by

$$\begin{aligned} \begin{bmatrix} 0 &{}\quad -ae^{-{\bar{x}}} &{}\quad be^{-{\bar{y}}} &{}\quad -b{\bar{y}}e^{-{\bar{y}}}\\ 1 &{}\quad 0 &{}\quad 0&{}\quad 0\\ {\mathrm{d}}e^{-{\bar{x}}} &{}\quad -d{\bar{x}}e^{-{\bar{x}}} &{}\quad 0 &{}\quad -ce^{-{\bar{y}}}\\ 0 &{}\quad 0 &{}\quad 1 &{}\quad 0 \end{bmatrix}. \end{aligned}$$

Hence the characteristic equation of the Jacobian \(JF({\bar{x}},{\bar{y}})\) about the equilibrium point \(E({\bar{x}},{\bar{y}})\) is given by

$$\begin{aligned}&\displaystyle -\lambda ^4 + \lambda ^2(-ce^{-{\bar{y}}} + bde^{-{\bar{x}}}e^{-{\bar{y}}} -ae^{-{\bar{x}}})\\&\quad + \lambda (-bd{\bar{y}}e^{-{\bar{x}}}e^{-{\bar{y}}} -bd{\bar{x}}e^{-{\bar{x}}}e^{-{\bar{y}}}) + bd{\bar{x}}{\bar{y}}e^{-{\bar{x}}}e^{-{\bar{y}}}- ace^{-{\bar{x}}}e^{-{\bar{y}}} =0. \end{aligned}$$

Then

$$\begin{aligned}&|-ce^{-{\bar{y}}}|+ |bde^{-{\bar{x}}}e^{-{\bar{y}}}| + |ae^{-{\bar{x}}}|\\&\quad + |bd{\bar{y}}e^{-{\bar{x}}}e^{-{\bar{y}}}| + |bd{\bar{x}}e^{-{\bar{x}}}e^{-{\bar{y}}}| + |bd{\bar{x}}{\bar{y}}e^{-{\bar{x}}}e^{-{\bar{y}}}| + |ace^{-{\bar{x}}}e^{-{\bar{y}}}| <1 \end{aligned}$$

is satisfied whenever

$$\begin{aligned} |c|+ |bd| + |a| + |bd{\bar{y}}| + |bd{\bar{x}}| + |bd{\bar{x}}{\bar{y}}| + |ac| <1. \end{aligned}$$
(27)
  1. 1.

    From (27), we get

    $$\begin{aligned} (1+{\bar{x}})(1+{\bar{y}})< \displaystyle \frac{1-(a+ac +c)}{bd}. \end{aligned}$$
    (28)

    Hence, by (28) and Remark 1.3.1 of [15], we get the result.

  2. 2.

    Since \(E({\bar{x}},{\bar{y}})\) is the equilibrium point of (3), we get

    $$\begin{aligned} {\bar{x}} \le \alpha _1 + a e ^{-\alpha _1} + b e^{-\alpha _2} [\alpha _2 +d{\bar{x}}e^{-\alpha _1}+ce^{-\alpha _2}]. \end{aligned}$$

    , i.e.,

    $$\begin{aligned} {\bar{x}} \le \frac{A}{(1-bde^{-\alpha _1-\alpha _2})}. \end{aligned}$$
    (29)

    Similarly

    $$\begin{aligned} {\bar{y}} \le \frac{C}{(1-bde^{-\alpha _1-\alpha _2})}. \end{aligned}$$
    (30)

    Substituting (29), (30) in (27), we get

    $$\begin{aligned} (a+c+ac+bd)+ bd\left[ \frac{A}{1-B}+\frac{C}{1-B}+\frac{AC}{(1-B)^2}\right] <1. \end{aligned}$$

    Hence by Remark 1.3.1 of [15], we get the result.

Therefore by using Theorem 5, we obtain the conditions for global asymptotic stability. \(\square\)

Conclusions

In this paper, we analyzed the persistence, boundedness, convergence, invariance and global asymptotic behavior of the positive solutions of a second-order difference system. Here we expressed all the conditions in terms of the parameters occurring in the system. We also obtained two conditions for the occurrence of global stability where in the first one the condition was given in terms of the equilibrium point and in the second one the condition was given in terms of parameters of the system.