Basins of attraction for two-species competitive model with quadratic terms and the singular Allee effect

Abstract

We consider the following system of difference equations: xn+1=xn2/B1xn2+C1yn2, yn+1=yn2/A2+B2xn2+C2yn2, n=0, 1,., where B1, C1, A2, B2, C2 are positive constants and x0, y0≥0 are initial conditions. This system has interesting dynamics and it can have up to seven equilibrium points as well as a singular point at (0,0), which always possesses a basin of attraction. We characterize the basins of attractions of all equilibrium points as well as the singular point at (0,0) and thus describe the global dynamics of this system. Since the singular point at (0,0) always possesses a basin of attraction this system exhibits Allee's effect.