Behavior of an Exponential System of Difference Equations

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.

Global dynamics of an exponential system by a discrete-time Lyapunov function

“…where α s (s = 1, 2, 3) and x s (s = −1, 0) are +ve real numbers. Khan and Qureshi [6] have explored the global dynamics of the following exponential system:…”

“…, 6) and x s (s = −1, 0) are +ve real numbers. Motivation from above existing studies, here our purpose is to explore the global dynamical properties of the following three-directional exponential system, that is, the extension of the work of Bozkurt [5] and Khan and Qureshi [6]:…”

Global dynamics of (1,2)- type systems of difference equations

“…This made the study of qualitative behavior of difference equations an active area of research (see [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17] and references cited therein). For instance, Touafek and Elsayed [18,19] investigated the behavior of following systems of difference equations:…”

Stability and Neimark–Sacker bifurcation of a ratio‐dependence predator–prey model

“…Yang and Li [10] studied the permanence of species for a delayed discrete ratio-dependent predator-prey model with monotonic functional response. Study of discrete dynamical behavior of systems is usually focussed on boundedness and persistence, existence and uniqueness of equilibria, periodicity, and there local and global stability (see for example, [11][12][13][14][15][16][17][18][19][20][21][22][23][24][25]), but there are few articles that discuss the dynamical behavior of discrete-time predator-prey models for exploring the possibility of bifurcation and chaos phenomena [26][27][28][29][30].…”