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Discrete Dynamics in Nature and Society
Volume 2018, Article ID 8362837, 35 pages
https://doi.org/10.1155/2018/8362837
Research Article

Global Dynamics of Some 3 × 6 Systems of Exponential Difference Equations

Department of Mathematics, University of Azad Jammu & Kashmir, Muzaffarabad 13100, Pakistan

Correspondence should be addressed to A. Q. Khan; moc.liamg@1nahkreedaqludba

Received 15 February 2018; Revised 23 May 2018; Accepted 4 July 2018; Published 2 September 2018

Academic Editor: Victor S. Kozyakin

Copyright © 2018 A. Q. Khan and A. Sharif. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We study the global dynamics of a list of following systems of exponential difference equations: , , , , , , , , , , , , , , where and the initial conditions are arbitrary nonnegative real numbers. This proposed work is considerably extended and improve some existing results in the literature. Some numerical examples are given to not only verify our theoretical results but also provide different types of qualitative behavior of solution of these systems.

1. Introduction

Discrete dynamical systems described by difference equations are more appropriate for population dynamics as compared to continuous ones. Population models involve exponential difference equations and their stability analysis though complicated, but interesting. The beginning of 2 century has witnessed an increasing interest in the population dynamics. Consequently, many works have been appeared on difference equations or systems of difference equations involving exponential terms (see [114] and reference cited therein). For instance, Metwally et al. [1] have investigated the dynamics of the following second-order difference equation:which is the solution of the following logistic equation with piecewise constant arguments:where and initial conditions are arbitrary nonnegative real numbers. Equation (1) may be viewed as a model in Mathematical Biology where is immigration rate and is the population growth rate. Moreover it is also pointed out in [2] that this model is suggested by the people from the Harvard school of public health, studying the population dynamics of single-species . Further Papaschinopoulos et al. [2] and Papaschinopoulos and Schinas [3] produced nice results in this direction by investigating the dynamical properties like boundedness and persistence of positive solutions, existence of the unique positive equilibrium, and local and global asymptotic stability of two-species model described by systems of difference equations, which is natural extension of single-species population model depicted in (1).

In [4], Grove et al. have investigated the global dynamics of the positive solution of the following difference equations:where and initial conditions are arbitrary nonnegative real numbers. This equation can be considered as a biological model, since it arises from models studying the amount of litter in perennial grassland (see [6]). After that Papaschinopoulos et al. [5, 6] have studied the asymptotic behavior of the positive solution of two-species model which is also natural extension of single-species model represented in (3). In 2016, Wang and Feng [7] have investigated the dynamics of positive solution for the following difference equation which is naturally a new form of single-species model depicted in (1):where and initial conditions are arbitrary nonnegative real numbers. In biological point of view is immigration rate and is population growth rate.

Ozturk et al. [8] have investigated the global asymptotic stability, boundedness and periodic nature of the following -order exponential difference equation:where α, β, γ and are arbitrary nonnegative numbers. Equation (5) is also viewed as a model in Mathematical Biology where is immigration rate, is population growth rate and is the carrying capacity. Later Papaschinopoulos et al. [9] have investigated boundedness and persistence and local and global asymptotic behavior of two-species model which is natural extension of single-species model (5), represented by the following exponential systems of difference equations: where and initial conditions are nonnegative real numbers. Motivated from aforementioned studies, our aim in this paper is to investigate the global dynamics of three-species models, which is natural extension of single-species model studied by Ozturk et al. [8], represented by the following list of systems of exponential difference equations:where and the initial conditions are arbitrary nonnegative real numbers. Equations (7) to (11) may be viewed as discrete-time population models in Mathematical Biology and biological parameters can be interpretations as shown in Table 1.

Table 1: Parameters of the discrete-time model (7) to (11) along their Biological interpretations.

The rest of the paper is organized as follows: In Section 2, we investigate the global dynamics and rate of convergence of system (7). This includes boundeness and persistence, existence and uniqueness of positive equilibrium point, local and global stability about the unique positive equilibrium point and rate of convergence that converge to its unique positive equilibrium point of the system. Sections 3, 4, 5 and 6 include same analysis for system (8), (9), (10) and (11), respectively. Section 7 is about the numerical simulations that verify our theoretical results. A brief conclusion is given in Section 8.

2. On the System: , ,

In this section we will study the main results for system (7).

2.1. Boundedness and Persistence

The following theorem shows that every positive solution of system (7) is bounded and persists.

Theorem 1. Every positive solution of system (7) is bounded and persists.

Proof. If is an arbitrary solution of (7), thenIn addition from (7) and (12), one getsHence from (12) and (13), one gets

2.2. Existence of Invariant Set for Solutions

Theorem 2. If is a positive solution of system (7), then is invariant set for system (7), where , , , , and .

Proof. For any positive solution of system (7) with initial conditions and, one has from (7)and AlsoandFinally, andHence and . Similarly one can show that if and , then and .

2.3. Existence and Uniqueness of Positive Equilibrium Point

In the following we will study the existence and uniqueness of positive equilibrium point of system (7).

Theorem 3. If where then system (7) has a unique positive equilibrium point (.

Proof. Consider the following system of equations:From (23) set and Denote whereand . We claim that has a unique solution in From (26) and (27), one getsIf is a solution of then from (26) and (27), one gets where In view of (29), (30), and (31), (28) becomesAfter some manipulation, equation (32) reduces into the following form:where . Now assuming that (21) along with (22) hold, then from (33) we get Hence has a unique positive solution in .

2.4. Local and Global Asymptotic Stability

Theorem 4. For local stability about of system (7), the following statements hold:
(i) Equilibrium of system (7) is locally asymptotically stable if (ii) Equilibrium of system (7) is an unstable if

Proof. If ( is equilibrium point of system (7), thenIn order to construct the corresponding linearized form of system (7), we consider the following transformation:whereThe Jacobian matrix about ( under the transformation (37) is given bywhereThe characteristic equation of about ( is given bywhereNowAssuming condition (34) holds, then from (43) one gets and hence Theorem 1.5 of [15] implies that equilibrium of system (7) is locally asymptotically stable.
Proof (ii). Using same manipulations as in the proof of (i), we haveAssume that (35) hold, then from (44) we get Hence by Theorem 1.5 of [15], of (7) is unstable.

Hereafter we will study the global dynamics of system (7) about the unique positive equilibrium point . This is main component in the theory of dynamical systems based on the knowledge of its present state. In recent years it is challenging task to determine the global dynamics for higher-order difference equations or systems of difference equations. So, in this paper we study the global dynamics of (7) about the by Lyapunov stability theory because semiconjugacy and weak contraction cannot be used to analyze global dynamics of system (7). Here we construct the discrete-time Lyapunov function: whose nonnegativity follows from inequality: (see [1214]).

Theorem 5. Equilibrium of (7) is globally asymptotically stable if

Proof. Consider the following discrete-time Lyapunov function:where the nonnegativity of follows from the following inequality:Moreover, we have Now,Using (48) in (49) and assume that (45) hold then one getsfor all n . It follows that Hence, we obtain that . Therefore, of (7) is globally asymptotically stable.

In the following we will investigate the rate of convergence that converges to of system (7) by employing the method [16, 17].

2.5. Rate of Convergence

Theorem 6. If is a positive solution of (7) such thatwherethen the error vector of every solution of (7) satisfies both the following asymptotic relations: where are the characteristic root of .

Proof. If is a positive solution of system (7) such that (51) along with (52) holds. In order to find the error terms, one has from (7)