Two species competitive model with the Allee effect
We consider the following system of difference equations: xn+1=axn21+xn2+cyn, yn+1=byn21+yn2+dxn, n=0,1,… , where a, b, c, d are positive constants and x0,y0≥0 are initial conditions. This system has interesting dynamics and can have up to nine equilibrium points. The most complex and perhaps most interesting case is the case of nine equilibrium points, four of which are local attractors, four of which are saddle points, and one of which is a repeller. Using recent results of Kulenović and Merino we are able to characterize the basins of attractions of all local attractors and thus to describe the global dynamics of this system. This case can be considered as a two-dimensional version of the Allee effect for competitive systems. MSC:39A10, 39A30, 37G35.