Integrable Deformations and Dynamical Properties of Systems with Constant Population

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.

Preface to the Special Issue on “Advances in Differential Dynamical Systems with Applications to Economics and Biology”

“…The paper of Lǎzureanu [16] considers systems of three autonomous first-order differential equations such that the sum of the three variables is constant in time. Hamilton-Poisson formulations and integrable deformations are presented, and the case of Kolmogorov systems is also analyzed.…”