Global Dynamics of a Certain Two-dimensional Competitive System of Rational Difference Equations with Quadratic Terms

We investigate global dynamics of the following system of difference equations {x(n+1) = x(n)(2)/a+y(n)(2) , n = 0,1,2,... y(n+1) = y(n)(2)/b+x(n)(2) where the parameters a, b are positive numbers and the initial conditions x(0), y(0) are arbitrary nonnegative numbers. We show that this system has an interesting dynamics which depends on the part of parametric space. The obtained ynamics is very different than the dynamics of the corresponding linear fractional system. In particular, we show that the system always exhibit Allee's effect.