Bifurcation and Global Dynamics of a Leslie-Gower Type Competitive System of Rational Difference Equations with Quadratic Terms

Abstract

We investigate global dynamics of the following systems of difference equations $x_{n+\mathrm{1}}=x_n/\left(A_{\mathrm{1}}+B_{\mathrm{1}}{x}_n+C_{\mathrm{1}}{y}_n\right)$, $y_{n+\mathrm{1}}=y_n^{\mathrm{2}}/\left(A_{\mathrm{2}}+B_{\mathrm{2}}{x}_n+C_{\mathrm{2}}{y}_n^{\mathrm{2}}\right)$, $n=\mathrm{0,1},\dots$, where the parameters $A_{\mathrm{1}}$, $A_{\mathrm{2}}$, $B_{\mathrm{1}}$, $B_{\mathrm{2}}$, $C_{\mathrm{1}}$, and $C_{\mathrm{2}}$ are positive numbers and the initial conditions $x_{\mathrm{0}}$ and $y_{\mathrm{0}}$ 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.