Global Dynamics of a 3 × 6 System of Difference Equations

S M Qureshi,A Q Khan

doi:10.1155/2019/9797242

Abstract

In the proposed work, global dynamics of a 3×6 system of rational difference equations has been studied in the interior of R+3. It is proved that system has at least one and at most seven boundary equilibria and a unique +ve equilibrium under certain parametric conditions. By utilizing method of Linearization, local dynamical properties about equilibria have been investigated. It is shown that every +ve solution of the system is bounded, and equilibrium P0 becomes a globally asymptotically stable if α1<α2,α4<α5, α7<α8. It is also shown that every +ve solution of the system converges to P0. Finally theoretical results are verified numerically.

Highlights

The importance of difference equations cannot be overemphasized
In recent years several authors have explored the behavior of solution of such difference equations or system of difference equations by studying equilibrium point, local and global dynamics about equilibria, boundedness and persistence, periodicity nature, prime period 2-solution, semicycle analysis, forbidden set, and many more
This proposed work is about the global dynamical properties of a 3 × 6 system of difference equations, which is our key finding towards discrete dynamical system

Summary

Introduction

The importance of difference equations cannot be overemphasized These equations model discrete physical phenomena on one hand and on the other hand these integral parts of numerical schemes used to solve differential equations. Amleh et al [22, 23] have explored boundedness, periodicity nature, and global dynamical properties of following difference equation: α1 α4. Zhang et al [27] have extended the work studied by several authors [21, 25, 26], to explore the dynamical properties of following system of difference equations: xn−2 ∏2i=0 yn−i. Motivated from aforementioned studies, our aim is to extend the work studied by [21, 25,26,27], to explore the global dynamical properties of the following 3 × 6 difference equations system: α1xn−1 α3∏1i=0yn−i α5.

Existence of Equilibria

Global Stability about the Equilibrium P0

Discussion and Numerical