We study the boundedness character and persistence, existence and uniqueness of positive equilibrium, local and global behavior, and rate of convergence of positive solutions of the following system of exponential difference equations: , , where the parameters α_i, β_i, γ_i, a_i, b_i, and c_i for i ∈ {1,2} and initial conditions x₀, x₋₁, y₀, and y₋₁ are positive real numbers. Furthermore, by constructing a discrete Lyapunov function, we obtain the global asymptotic stability of the positive equilibrium. Some numerical examples are given to verify our theoretical results.

1. Introduction

Many population models are governed by exponential difference equations. We refer to [1–6] and the references therein. Systems of nonlinear difference equations of higher-order are of paramount importance in applications. Such equations also appear naturally as discrete analogues and as numerical solutions of systems differential and delay differential equations which model diverse phenomena in biology, ecology, physiology, physics, engineering, and economics. For applications and basic theory of rational difference equations we refer to [7–9]. In [10–17] applications of difference equations in mathematical biology are given. It is very interesting to investigate the behavior of solutions of a system of nonlinear difference equations and to discuss the local asymptotic stability of their equilibrium points.

El-Metwally et al. [1] investigated boundedness character, asymptotic behavior, periodicity nature of the positive solutions, and stability of equilibrium point of the following population model:

()

Papaschinopoulos et al. [2] studied the boundedness, the persistence, and the asymptotic behavior of positive solutions of the following two directional interactive and invasive species models:

()

Papaschinopoulos et al. [3] investigated the asymptotic behavior of the solutions of the following three systems of difference equations of exponential form:

()

Recently, Papaschinopoulos and Schinas [4] studied the asymptotic behavior of the positive solutions of the systems of the two difference equations:

()

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

()

where the parameters α_i, β_i, γ_i, a_i, b_i, and c_i for i ∈ {1,2} and initial conditions x₀, x₋₁, y₀, and y₋₁ are positive real numbers.

More precisely, we investigate the boundedness character and 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 (5) which converge to its unique positive equilibrium point. Some special cases of system (5) can be treated as population models of two species [3].

2. Main Results

The following theorem shows that every solution of (5) is bounded and persists.

Theorem 1. Every positive solution {(x_n, y_n)} of system (5) is bounded and persists.

Proof. For any positive solution {(x_n, y_n)} of system (5), one has

()

Furthermore, from systems (5) and (6) we obtain that

()

From (6) and (7), it follows that

()

Hence, the theorem is proved.

Lemma 2. Let {(x_n, y_n)} be a positive solution of system (5). Then, [L₁, U₁] × [L₂, U₂] is invariant set for system (5).

Proof. The proof follows by induction.

2.1. Stability Analysis

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

()

where f : I² × J² → I and g : I² × J² → J are continuously differentiable functions and I, J are some intervals of real numbers. Furthermore, a solution

of system (9) is uniquely determined by initial conditions (x_i, y_i) ∈ I × J for i ∈ {−1,0}. Along with system (9) we consider the corresponding vector map F = (f, x_n, g, y_n). An equilibrium point of (9) is a point

that satisfies

()

The point

is also called a fixed point of the vector map F.

Definition 3. Let be an equilibrium point of the system (9).

(i)
An equilibrium point is said to be stable if for every ɛ > 0 there exists δ > 0 such that for every initial condition (x_i, y_i), i ∈ {−1,0}, if implies for all n > 0, where ∥·∥ is usual Euclidian norm in .
(ii)
An equilibrium point is said to be unstable if it is not stable.
(iii)
An equilibrium point is said to be asymptotically stable if there exists η > 0 such that and as n → ∞.
(iv)
An equilibrium point is called global attractor if as n → ∞.
(v)
An equilibrium point is called asymptotic global attractor if it is a global attractor and stable.

Definition 4. Let be an equilibrium point of a map F = (f, x_n, g, y_n), where f and g are continuously differentiable functions at . The linearized system of (9) about the equilibrium point is

()

where

and F_J is Jacobian matrix of system (9) about the equilibrium point

To construct corresponding linearized form of system (5) we consider the following transformation:

()

where f = x_n+1, g = y_n+1, f₁ = x_n, and g₁ = y_n. The linearized system of (5) about

is given by

()

where

and the Jacobian matrix about the fixed point

under transformation (12) is given by

()

where

()

Lemma 5 (see [9].)Assume that X_n+1 = F(X_n), n = 0,1, …, is a system of difference equations such that is a fixed point of F. If all eigenvalues of the Jacobian matrix J_F about lie inside the open unit disk |λ | < 1, then is locally asymptotically stable. If one of them has a modulus greater than one, then is unstable.

Theorem 6. System (5) has a unique positive equilibrium point , if the following condition is satisfied:

()

where

()

Proof. Consider the following system of equations:

()

From (18), it follows that

()

Set

()

where f(x) = (α₁ + (β₁ + γ₁)e^−x − a₁x)/((b₁ + c₁)x) and x ∈ [L₁, U₁]. Then, it follows that

()

Furthermore, it is easy to see that

()

From (22) it follows that

()

Let

. Then F(L₁) can be expressed as

()

Suppose that a₂ + L₁(b₂ + c₂) < (α₂ + (β₂ + γ₂)e^−K)/K; then, it follows that F(L₁) > 0. Furthermore, we have

()

Then it is easy to see that

()

which gives that F(U₁) < 0. Hence, F(x) = 0 has at least one positive solution in [L₁, U₁]. Moreover, we obtain that

()

where

()

Then, from (28) it follows that f(U₁) < f(x) < f(L₁) and using (26) we obtain

()

Hence, F(x) = 0 has a unique positive solution in [L₁, U₁]. The proof is therefore completed.

Theorem 7. The unique positive equilibrium point of system (5) is locally asymptotically stable under the following condition:

()

Proof. The characteristic polynomial of Jacobian matrix about is given by

()

Clearly, one root of P(λ) is 0. To check the behavior of the other three roots of P(λ), we let Φ(λ) = λ³ and Ψ(λ) = (A₁ + B₂)λ² + (A₂B₁ − A₁B₂ + B₃ + B₄)λ + (A₃ + A₄)B₁ − (B₃ + B₄)A₁. Assume that (30) holds and |λ | = 1; then, one has

()

Then, by Rouche’s Theorem, Φ(λ) and Φ(λ) − Ψ(λ) have the same number of zeroes in an open unit disk |λ | < 1. Hence, all the roots of (31) satisfy |λ | < 1, and it follows from Lemma 5 that the unique positive equilibrium point

of the system (5) is locally asymptotically stable.

Theorem 8. The unique positive equilibrium point of system (5) is globally asymptotically stable, if the following condition is satisfied:

()

Proof. Arguing as in [18], we consider the following discrete time analogue of Lyapunov function:

()

Then nonnegativity of V_n follows from the following inequality:

()

Furthermore, we have

()

Assume that (33) holds true; then, it follows that

()

for all n ≥ 0 so that V_n ≥ 0 is monotonically decreasing sequence. It follows that lim⁡_n→∞V_n ≥ 0. Hence, we obtain that

()

Then it follows that

and

. Furthermore, V_n ≤ V₀ for all n ≥ 0, which gives that

is uniformly stable. Hence, unique positive equilibrium point

of system (5) is globally asymptotically stable.

2.2. 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 (5).

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

()

where X_n is an m-dimensional vector, A ∈ C^m×m is a constant matrix, and

is a matrix function satisfying

()

as n → ∞, where ∥·∥ denotes any matrix norm which is associated with the vector norm

()

Proposition 9 ((Perron’s Theorem) [19]). Suppose that condition (40) holds. If X_n is a solution of (39), then either X_n = 0 for all large n or

()

exists and is equal to the modulus of one of the eigenvalues of matrix A.

Proposition 10 (see [19].)Suppose that condition (40) holds. If X_n is a solution of (39), then either X_n = 0 for all large n or

()

exists and is equal to the modulus of one of the eigenvalues of matrix A.

Let {(x_n, y_n)} be any solution of the system (5) such that

and

, where

and

. To find the error terms, one has from the system (5)

()

Let

and

; then, one has

()

where

()

Moreover,

()

Now the limiting system of error terms can be written as

()

which is similar to linearized system of (5) about the equilibrium point

Using Proposition 9, one has the following result.

Theorem 11. Assume that {(x_n, y_n)} is a positive solution of the system (5) such that , and , where and . Then, the error vector of every solution of (5) satisfies both of the following asymptotic relations:

()

where

are the characteristic roots of Jacobian matrix

2.3. 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 (5). The first and last examples show that positive equilibrium of system (5) is unstable with suitable parametric choices. Moreover, from the remaining examples it is clear that unique positive equilibrium point of system (5) is globally asymptotically stable with different parametric values.

Example 1. Let α₁ = 6.5, β₁ = 8.5, γ₁ = 12.3, a₁ = 1.6, b₁ = 5.02, c₁ = 1.2, α₂ = 3.5, β₂ = 5.5, γ₂ = 1.3, a₂ = 0.1, b₂ = 0.003, and c₂ = 2.2. Then, system (5) can be written as

()

with initial conditions x₋₁ = 1.2, x₀ = 1.1, y₋₁ = 2.1, and y₀ = 2.2.

In this case the positive equilibrium point of the system (50) is unstable. Moreover, in Figure 1 the plot of x_n is shown in Figure 1(a), the plot of y_n is shown in Figure 1(b), and a phase portrait of the system (50) is shown in Figure 1(c).

Details are in the caption following the image — **Figure 1 (a) Plot of xn for the system (50)**
Open in figure viewer PowerPoint

Plots for the system (50).

Example 2. Let α₁ = 1.5, β₁ = 7.5, γ₁ = 13.5, a₁ = 2.1, b₁ = 7.92, c₁ = 0.95, α₂ = 4.1, β₂ = 7.2, γ₂ = 16.7, a₂ = 0.01, b₂ = 0.2, and c₂ = 2.3. Then, system (5) can be written as

()

with initial conditions x₋₁ = 0.37, x₀ = 0.36, y₋₁ = 4.5, and y₀ = 4.6.

In this case the unique positive equilibrium point of the system (51) is given by . Moreover, in Figure 2 the plot of x_n is shown in Figure 2(a), the plot of y_n is shown in Figure 2(b), and an attractor of the system (51) is shown in Figure 2(c).

Example 3. Let α₁ = 8.98, β₁ = 75, γ₁ = 135, a₁ = 21, b₁ = 79, c₁ = 5, α₂ = 41, β₂ = 71.9, γ₂ = 16, a₂ = 6, b₂ = 2, and c₂ = 30. Then, system (5) can be written as

()

with initial conditions x₋₁ = 0.634, x₀ = 0.63, y₋₁ = 1.99, and y₀ = 2.

In this case the unique positive equilibrium point of the system (52) is given by . Moreover, in Figure 3 the plot of x_n is shown in Figure 3(a), the plot of y_n is shown in Figure 3(b), and an attractor of the system (52) is shown in Figure 3(c).

Example 4. Let α₁ = 23.8, β₁ = 350, γ₁ = 66, a₁ = 55, b₁ = 14, c₁ = 111, α₂ = 66, β₂ = 35, γ₂ = 87, a₂ = 1.7, b₂ = 80, and c₂ = 1.4. Then, system (5) can be written as

()

with initial conditions x₋₁ = 0.82, x₀ = 0.81, y₋₁ = 1.2, and y₀ = 1.3.

In this case the unique positive equilibrium point of the system (53) is given by . Moreover, in Figure 4 the plot of x_n is shown in Figure 4(a), the plot of y_n is shown in Figure 4(b), and an attractor of system (53) is shown in Figure 4(c).

Example 5. Let α₁ = 8, β₁ = 3, γ₁ = 6, a₁ = 0.05, b₁ = 4, c₁ = 16, α₂ = 9, β₂ = 17, γ₂ = 7, a₂ = 0.07, b₂ = 22, and c₂ = 0.2. Then, system (5) can be written as

()

with initial conditions x₋₁ = 0.5, x₀ = 0.4, y₋₁ = 1.2, and y₀ = 1.3.

In this case the positive equilibrium point of the system (54) is unstable. Moreover, in Figure 5 the plot of x_n is shown in Figure 5(a), the plot of y_n is shown in Figure 5(b), and a phase portrait of system (54) is shown in Figure 5(c).

3. Concluding Remarks

In literature several articles are related to qualitative behavior of exponential systems of rational difference equations. It is a very interesting mathematical problem to study the dynamics of such systems because these are closely related to models in population dynamics and biological sciences. This work is related to 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 (5). Under certain parametric conditions the boundedness and persistence of positive solutions are proved. Moreover, we have shown that unique positive equilibrium point of system (5) 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 of determining the possible global behaviors of the system and determining which parametric conditions lead to these long-term behaviors. By constructing a discrete Lyapunov function, we have obtained the global asymptotic stability of the positive equilibrium of (5). Furthermore, rate of convergence of positive solutions of (5) which converge to its unique positive equilibrium point is demonstrated. Finally, some illustrative examples are provided to support our theoretical discussion.

Conflict of Interests

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

Acknowledgments

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

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Stability Analysis of a System of Exponential Difference Equations

Abstract

1. Introduction