Behavior of an Exponential System of Difference Equations

A Q Khan,M N Qureshi

doi:10.1155/2014/607281

Abstract

We study the qualitative behavior of the following exponential system of rational difference equations:xn+1 = αe-yn+βe-yn-1/γ+αxn+βxn-1, yn+1 = α1e-xn+β1e-xn-1/γ1+α1yn+β1yn-1, n = 0,1,…,whereα,β,γ,α1,β1, andγ1and initial conditionsx0, x-1, yo, and y-1are positive real numbers. More precisely, we investigate the boundedness character and persistence, existence and uniqueness of positive equilibrium, local and global behavior, and rate of convergence of positive solutions that converges to unique positive equilibrium point of the system. Some numerical examples are given to verify our theoretical results.

Highlights

Mathematical models of population dynamics have created great interest in the field of difference equations
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
We have shown that unique positive equilibrium point of system (4) is locally as well as globally asymptotically stable

Summary

Introduction

Mathematical models of population dynamics have created great interest in the field of difference equations. Exponential difference equations made their appearance in population dynamics Though their analysis is hard, it is very interesting at the same time. 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: xn+1 = α + βxn−1e−xn , n = 0, 1, . Ozturk et al [7] have investigated the boundedness, asymptotic behavior, periodicity, and stability of the positive solutions of the following difference equation: yn+1. Bozkurt [8] has investigated the local and global behavior of positive solutions of the following difference equation: αe−yn + βe−yn−1 yn+1. 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

Linearized Stability

Global Character

Rate of Convergence

Examples

Conclusion