Hopf bifurcations in competitive three-dimensional Lotka–Volterra systems

Abstract We study the space of Lotka–Volterra systems modelling three mutually competing species, each of which, in isolation, would exhibit logistic growth. By a theorem of M. W. Hirsch, the compact limit sets of these systems are either fixed points or periodic orbits. We use a geometric analysis of the surfaces ẋ=0 of a system, to define a combinatorial equivalence relation on the space, in terms of simple inequalities on the parameters. We list the 33 stable equivalence classes, and show that in 25 of these classes all the compact limit sets are fixed points, so we can fully describe the dynamics. We study the remaining eight equivalence classes by finding simple algebraic criteria on the parameters, with which we are able to predict the occurrence of Hopf bifurcations and, consequently, isolated periodic orbits.