Global dynamic scenarios for competitive maps in the plane

Abstract

Abstract In this paper we present some global dynamic scenarios for general competitive maps in the plane. We apply these results to the class of second-order autonomous difference equations whose transition functions are decreasing in the variable xn $x_{n}$ and increasing in the variable xn−1 $x_{n-1}$. We illustrate our results with the application to the difference equation xn+1=Cxn−12+Exn−1axn2+dxn+f,n=0,1,…, $$ x_{n+1}=\frac{Cx_{n-1}^{2}+Ex_{n-1}}{a x_{n}^{2}+d x_{n}+f},\quad n=0,1,\ldots, $$ where the initial conditions x−1 $x_{-1}$ and x0 $x_{0}$ are arbitrary nonnegative numbers such that the solution is defined and the parameters satisfy C,E,a,d,f≥0 $C,E,a,d,f\geq0$, C+E>0 $C+E>0$, a+C>0 $a+C>0$, and a+d>0 $a+d>0$. We characterize the global dynamics of this equation with the basins of attraction of its equilibria and periodic solutions.

Keywords

• Competitive
• Asymptotic stability
• Attractivity
• Difference equation
• Global
• Local stability