Multiple Attractors for a Competitive System of Rational Difference Equations in the Plane

Abstract

We investigate global dynamics of the following systems of difference equations xn+1=β1xn/(B1xn+yn), yn+1=(α2+γ2yn)/(A2+xn), n=0,1,2,…, where the parameters β1, B1, β2, α2, γ2, A2 are positive numbers, and initial conditions x0 and y0 are arbitrary nonnegative numbers such that x0+y0>0. We show that this system has up to three equilibrium points with various dynamics which depends on the part of parametric space. We show that the basins of attractions of different locally asymptotically stable equilibrium points or nonhyperbolic equilibrium points are separated by the global stable manifolds of either saddle points or of nonhyperbolic equilibrium points. We give an example of globally attractive nonhyperbolic equilibrium point and semistable non-hyperbolic equilibrium point.