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Bifurcation and global dynamics of a Leslie-Gower type competitive system of rational difference equations with quadratic terms. (English) Zbl 1470.39035

Summary: We investigate global dynamics of the following systems of difference equations \(x_{n + 1} = x_n / \left(A_1 + B_1 x_n + C_1 y_n\right)\), \(y_{n + 1} = y_n^2 / \left(A_2 + B_2 x_n + C_2 y_n^2\right)\), \(n = 0,1, \ldots\), where the parameters \(A_1\), \(A_2\), \(B_1\), \(B_2\), \(C_1\), and \(C_2\) are positive numbers and the initial conditions \(x_0\) and \(y_0\) are arbitrary nonnegative numbers. This system is a version of the Leslie-Gower competition model for two species. We show that this system has rich dynamics which depends on the part of parametric space.

MSC:

39A30 Stability theory for difference equations
92D25 Population dynamics (general)
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