Dynamics of Some Three-Dimensional Lotka–Volterra Systems

Abstract

We characterize the dynamics of the following two Lotka-Volterra differential systemṡWe analyze the biological meaning of the dynamics of these Lotka-Volterra systems.

Three-Dimensional Lotka–Volterra Systems with 3:−1:2-Resonance

“…Over many years these systems have been used in many branches of sciences like laser physics [27], neural networks [40], plasma physics [28], chemical kinetics [39], etc. The dynamical behaviour of these models has been widely studied, see for instance [1,14,8,45,49,34] among others. The integrability of some Lotka-Voltera families was considered by several authors, see for example [15,16,18,31,34,36].…”

“…The dynamical behaviour of these models has been widely studied, see for instance [1,14,8,45,49,34] among others. The integrability of some Lotka-Voltera families was considered by several authors, see for example [15,16,18,31,34,36]. Similar technics applied to some more general families like the ones in [5,23].…”

Global dynamics of the May-Leonard system with a Darboux invariant

“…were introduce by Lotka and Volterra in the 1920's for describing the interaction among species, see [8,12,17]. Since their appearance, these systems and related ones have been intensively investigated because they are also important for describing different phenomena in biology, ecology and chemistry; see for instance [3,5,6,7,9,11,13,15] and the references therein. Because of its simplicity, special attention has attracted the so-called classical May-Leonard system introduced by May and Leonard in [14] which again describes a competition between three species but is a simple model with a rich dynamical behavior.…”

Integrable Deformations and Dynamical Properties of Systems with Constant Population

“…In the next result we prove that there are periodic orbits of the considered system around the stable equilibrium states. In [40], the existence of periodic orbits of a similar system has been proven by another approach. Proof.…”

“…By geometrical point of view, the heteroclinic orbits given in Proposition 13 are open line segments with endpoints given by the unstable equilibrium states. Another approach in the study of the existence of heteroclinic orbits of a similar system has been used in [40].…”