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

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.