Discrete Dynamics in Nature and Society

Volume 2018, Article ID 8362837, 35 pages

https://doi.org/10.1155/2018/8362837

## 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 [1–14] 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.