Global Dynamics of a Competitive System of Rational Difference Equations in the Plane

S Kalabušić,M R S Kulenović,E Pilav

doi:10.1155/2009/132802

Abstract

We investigate global dynamics of the following systems of difference equations , , , where the parameters , , , , and are positive numbers and initial conditions and are arbitrary nonnegative numbers such that . We show that this system has rich dynamics which depend on the part of parametric space. We show that the basins of attractions of different locally asymptotically stable equilibrium points are separated by the global stable manifolds of either saddle points or of nonhyperbolic equilibrium points.

Highlights

Introduction and PreliminariesIn this paper, we study the global dynamics of the following rational system of difference equations: xn 1 α1 β1xn yn, n 1.1 yn 1 α2 A2 γ2yn xn, where the parameters α1, β1, α2, γ2, and A2 are positive numbers and initial conditions x0 ≥ 0 and y0 > 0 are arbitrary numbers
We study the global dynamics of the following rational system of difference equations: xn 1
System 1.1 was mentioned in 1 as a part of Open Problem 3 which asked for a description of global dynamics of three specific competitive systems

Summary

Introduction and Preliminaries

We study the global dynamics of the following rational system of difference equations: xn 1. The following theorems were proved by Kulenovicand Merino 3 for competitive systems in the plane, when one of the eigenvalues of the linearized system at an equilibrium hyperbolic or non-hyperbolic is by absolute value smaller than 1 while the other has an arbitrary value These results are useful for determining basins of attraction of fixed points of competitive maps. Of Theorem 1.5 reduces just to |λ| < 1 This follows from a change of variables 18 that allows the Perron-Frobenius Theorem to be applied to give that, at any point, the Jacobian matrix of a strongly competitive map has two real and distinct eigenvalues, the larger one in absolute value being positive, and that corresponding eigenvectors may be chosen to point in the direction of the second and first quadrants, respectively. Iii For every x ∈ W−, there exists n0 ∈ N such that T n x ∈ int Q2 x for n ≥ n0. iv For every x ∈ W , there exists n0 ∈ N such that T n x ∈ int Q4 x for n ≥ n0

Equilibrium Points

Condition O and Period-Two Solution

Period-two solution satisfies x

10 This is true since

Linearized Stability Analysis

Global Behavior