Integrable Deformations and Dynamical Properties of Systems with Constant Population

Cristian Lăzureanu

doi:10.3390/math9121378

Abstract

In this paper we consider systems of three autonomous first-order differential equations x˙=f(x),x=(x,y,z),f=(f1,f2,f3) such that x(t)+y(t)+z(t) is constant for all t. We present some Hamilton–Poisson formulations and integrable deformations. We also analyze the case of Kolmogorov systems. We study from some standard and nonstandard Poisson geometry points of view the three-dimensional Lotka–Volterra system with constant population.

Highlights

IntroductionLotka–Volterra systems with constant population [4,5]
We deduce the general form of a Lotka–Volterra system with constant population and we present a Hamilton–Poisson formulation of it
We study the properties of the energy-Casimir mapping associated with this system and their connections with the dynamics of the system

Summary

Introduction

Lotka–Volterra systems with constant population [4,5]. In the three-dimensional case, if the system has a second constant of motion, it admits a Hamilton–Poisson formulation. We deduce the general form of a Lotka–Volterra system with constant population and we present a Hamilton–Poisson formulation of it. We obtain the general form of a polynomial Kolmogorov system of degree 3 with constant population and a particular version of it, which is Hamilton–Poisson. We study the properties of the energy-Casimir mapping associated with this system and their connections with the dynamics of the system

Hamilton–Poisson Formulations and Integrable Deformations

A Particular Case

Dynamical Properties of the Three-Dimensional Lotka–Volterra System with

Conclusions