Advances in Difference Equations
Volume 2009 (2009), Article ID 132802, 30 pages
doi:10.1155/2009/132802

Global dynamics of a competitive system of rational difference equations in the plane

S. Kalabušić1 , M.R.S. Kulenović2 and E. Pilav1

1Department of Mathematics, University of Sarajevo, 71 000 Sarajevo, Bosnia and Herzegovina
 2Department of Mathematics, University of Rhode Island, Kingston, RI 02881-0816, USA

Abstract

We investigate global dynamics of the following systems of difference equations xn+1=(α1+β1xn)/yn, yn+1=(α2+γ2yn)/(A2+xn), n=0,1,2,…, where the parameters α1, β1, α2, γ2, and A2 are positive numbers and initial conditions x0 and y0 are arbitrary nonnegative numbers such that y0>0. 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.