Behavior of an Exponential System of Difference Equations

Mathematical models of population dynamics have created great interest in the field of difference equations. As pointed out in [1, 2], to model biological phenomenon, discrete dynamical systems are more appropriate than continuous timemodels, being computationally efficient to get numerical results. Difference equations also appear naturally as discrete analogs of differential and delay differential equations and have applications in finance, biological, physical, and social sciences. Nonlinear difference equations and their stability analysis and global and local behaviors are of great interest on their own. For some interesting results in this regard we refer to [3–6] and the references therein. Exponential difference equations made their appearance in population dynamics. Though their analysis is hard, it is very interesting at the same time. Biologists believe that the equilibrium points and their stability analysis are important to understand the population dynamics. El-Metwally et al. [3] have investigated the boundedness character, asymptotic behavior, periodicity nature of the positive solutions, and stability of equilibrium point of the following population model:


Introduction
Mathematical models of population dynamics have created great interest in the field of difference equations.As pointed out in [1,2], to model biological phenomenon, discrete dynamical systems are more appropriate than continuous time models, being computationally efficient to get numerical results.Difference equations also appear naturally as discrete analogs of differential and delay differential equations and have applications in finance, biological, physical, and social sciences.Nonlinear difference equations and their stability analysis and global and local behaviors are of great interest on their own.For some interesting results in this regard we refer to [3][4][5][6] and the references therein.Exponential difference equations made their appearance in population dynamics.Though their analysis is hard, it is very interesting at the same time.Biologists believe that the equilibrium points and their stability analysis are important to understand the population dynamics.
El-Metwally et al. [3] have investigated the boundedness character, asymptotic behavior, periodicity nature of the positive solutions, and stability of equilibrium point of the following population model: where the parameters ,  are positive numbers and the initial conditions are arbitrary nonnegative real numbers.
Ozturk et al. [7] have investigated the boundedness, asymptotic behavior, periodicity, and stability of the positive solutions of the following difference equation: where the parameters , , and  are positive numbers and the initial conditions are arbitrary nonnegative numbers.
Bozkurt [8] has investigated the local and global behavior of positive solutions of the following difference equation: where the parameters , , and  and the initial conditions are arbitrary positive numbers.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: where the parameters , , ,  1 ,  1 , and  1 are positive numbers and the initial conditions are arbitrary nonnegative real numbers.
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 (4) which converge to its unique positive equilibrium point.

Boundedness and Persistence
The following theorem shows that every solution of ( 4) is bounded and persists.
The point (, ) is also called a fixed point of the vector map .
(ii) An equilibrium point (, ) is said to be unstable if it is not stable. ( where  and  are continuously differentiable functions at (, ).The linearized system of (8) about the equilibrium point (, ) is where   = ( ) and   is the Jacobian matrix of the system (8) about the equilibrium point (, ).
Let (, ) be equilibrium point of the system (4); then To construct corresponding linearized form of system (4) we consider the following transformation: where The Jacobian matrix about the fixed point (, ) under the transformation (13) is given by where Lemma 5 (see [9]).For the system  +1 = (  ),  = 0, 1, .The following theorem shows the existence and uniqueness of positive equilibrium point of system (4).
Proof.The characteristic equation of the Jacobian matrix   (, ) about equilibrium point (, ) is given by where (28) Therefore, inequality (28) and Remark 1.3.1 of reference [10] imply that the unique positive equilibrium point (, ) of the system (4) is locally asymptotically stable.
Theorem 9.The unique positive equilibrium point (, ) of the system (4) is a global attractor.
Corollary 10.If condition (25) of Theorem 7 is satisfied, then the unique positive equilibrium point (, ) of the system (4) is globally asymptotically stable.
Proof.The proof is a direct consequence of Theorems 7 and 9.

Rate of Convergence
In this section we will determine the rate of convergence of a solution that converges to the unique positive equilibrium point of the system (4).
The following result gives the rate of convergence of solutions of a system of difference equations: where   is an -dimensional vector,  ∈  × is a constant matrix, and  : Z + →  × is a matrix function satisfying as  → ∞, where ‖ ⋅ ‖ denotes any matrix norm which is associated with the vector norm     (, ) Proposition 11 (Perron's theorem [12]).Suppose that condition (38) holds.If   is a solution of (37), then either   = 0 for all large  or exists and is equal to the modulus of one of the eigenvalues of matrix .
If   is a solution of (37), then either   = 0 for all large  or exists and is equal to the modulus of one of the eigenvalues of matrix .
Let {(  ,   )} be any solution of the system (4) such that lim  → ∞   =  and lim  → ∞   = .To find the error terms, one has from the system (4) So, Similarly, From ( 43) and (44), we have Let  1  =   −  and  2  =   − .Then system (45) can be represented as where Moreover, lim (48) So, the limiting system of error terms can be written as which is similar to linearized system of (4) about the equilibrium point (, ).Using Proposition 11, one has the following result.

Examples
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 (4).All plots in this section are drawn with Mathematica.

Conclusion
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 (4).Under certain parametric conditions the boundedness and persistence of positive solutions are proved.Moreover, we have shown that unique positive equilibrium point of system (4) is locally as well as globally asymptotically stable.The main objective of dynamical systems theory is to predict the global behavior of a system based on the knowledge of its present state.An approach to this problem consists in determining the possible global behaviors of the system and determining which initial conditions lead to these long-term behaviors.Furthermore, rate of convergence of positive solutions of (4) which converge to its unique positive equilibrium point is demonstrated.Finally, some numerical examples are provided to support our theoretical results.These examples are experimental verification of our theoretical discussions.