Basins of attraction for two-species competitive model with quadratic terms and the singular Allee effect (Q1723545)

Summary: We consider the following system of difference equations: \(x_{n + 1} = x_n^2 / \left(B_1 x_n^2 + C_1 y_n^2\right), y_{n + 1} = y_n^2 / \left(A_2 + B_2 x_n^2 + C_2 y_n^2\right), n = 0, 1, \ldots, \) where \(B_1\), \(C_1\), \(A_2\), \(B_2\), \(C_2\) are positive constants and \(x_0, y_0 \geq 0\) 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.