We study the qualitative behavior of the following exponential system of rational difference equations:

Mathematical models of population dynamics have created great interest in the field of difference equations. As pointed out in [

El-Metwally et al. [

Ozturk et al. [

Bozkurt [

Motivated by the above studies, our aim in this paper is to investigate the qualitative behavior of positive solutions of the following exponential system of rational difference equations:

More precisely, we investigate the boundedness character, persistence, existence, and uniqueness of positive steady state, local asymptotic stability and global behavior of unique positive equilibrium point, and rate of convergence of positive solutions of system (

The following theorem shows that every solution of (

Every positive solution

Let

Let

It follows by induction.

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

Let

An equilibrium point

An equilibrium point

An equilibrium point

An equilibrium point

An equilibrium point

Let

Let

For the system

The following theorem shows the existence and uniqueness of positive equilibrium point of system (

If

Consider the following system of algebraic equations:

If

The characteristic equation of the Jacobian matrix

Assuming condition (

Let

If

The unique positive equilibrium point

Define

If condition (

The proof is a direct consequence of Theorems

In this section we will determine the rate of convergence of a solution that converges to the unique positive equilibrium point of the system (

The following result gives the rate of convergence of solutions of a system of difference equations:

Suppose that condition (

Suppose that condition (

Let

Moreover,

Assume that

In order to verify our theoretical results and to support our theoretical discussions, we consider several interesting numerical examples in this section. These examples represent different types of qualitative behavior of solutions to the system of nonlinear difference equations (

Let

The plot of

Plot of

Plot of

An attractor of the system (

Let

The plot of

Plot of

Plot of

An attractor of the system (

Let

The plot of

Plot of

Plot of

An attractor of the system (

This work is related to the qualitative behavior of an exponential system of second-order rational difference equations. We have investigated the existence and uniqueness of positive steady-state of system (

The authors declare that they have no conflict of interests regarding the publication of this paper.

The authors thank the main editor and anonymous referees for their valuable comments and suggestions leading to the improvement of this paper. This work was supported by the Higher Education Commission of Pakistan.