Global dynamics of a Lotka—Volterra system in ℝ3

Abstract

In this work we consider the Lotka—Volterra system in ℝ3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathop x\\limits^. = - x(x - y - z)\\;\\mathop y\\limits^. \\; = \\; - y( - x + y - z),\\;\\;\\;\\mathop {\\dot z}\\limits^. \\; = \\; - z( - x - y - z)$$\\end{document}introduced recently in [7], and studied also in [8] and [14]. In the first two papers the authors mainly studied the integrability of this differential system, while in the third paper they studied the system as a Hamilton-Poisson system, and also started the analysis of its dynamics. Here we provide the global phase portraits of this 3-dimensional Lotka—Volterra system in the Poincaré ball, that is in R3 adding its extension to the infinity.