Global dynamic scenarios for competitive maps in the plane

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 x_{n} and increasing in the variable x_{n-1}. We illustrate our results with the application to the difference equation \t\t\txn+1=Cxn−12+Exn−1axn2+dxn+f,n=0,1,…,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$ x_{n+1}=\\frac{Cx_{n-1}^{2}+Ex_{n-1}}{a x_{n}^{2}+d x_{n}+f},\\quad n=0,1,\\ldots, $$\\end{document} where the initial conditions x_{-1} and x_{0} are arbitrary nonnegative numbers such that the solution is defined and the parameters satisfy C,E,a,d,fgeq0, C+E>0, a+C>0, and a+d>0. We characterize the global dynamics of this equation with the basins of attraction of its equilibria and periodic solutions.