Two-dimensional heteroclinic connections in the generalized Lotka–Volterra system

Abstract

We consider a three-dimensional generalized Lotka–Volterra (GLV) system assuming that it has equilibria on each of the coordinate axes, stable along the respective directions, and heteroclinic trajectories, and , that belong to coordinate planes. For such a system we give a complete classification of possible types of dynamics, characterized by the existence or non-existence of various two-dimensional heteroclinic connections. For each of these classes, we derive inequalities satisfied by coefficients of the system. The results can be used for the construction of GLV systems possessing various heteroclinic cycles or networks.