The Classification on the Global Phase Portraits of Two-dimensional Lotka–Volterra System

Abstract

This paper provides a complete classification on the global phase portraits of quadratic Lotka–Volterra system \({\dot{x}=x(a_{0}+a_{1}x+a_{2}y),\, \dot{y}=y(b_{0}+b_{1}x+b_{2}y)}\) . There are 143 different global phase portraits in the Poincare disc up to a reversal of sense of all orbits. Firstly the sufficient and necessary conditions of the system with closed orbits are given. Then the properties on infinite and finite critical points are discussed. All global phase portraits are finally classified by combining all local information.