## 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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### References:

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