Darboux first integrals of Kolmogorov systems with invariant $n$-sphere

TL;DR

This paper characterizes polynomial Kolmogorov vector fields with invariant spheres, identifies integrable cases, and explores Darboux first integrals, revealing conditions for complete integrability and non-existence of certain Hamiltonian fields.

Contribution

It provides a complete characterization of polynomial Kolmogorov vector fields with invariant spheres and analyzes their integrability and Darboux first integrals.

Findings

Complete integrability of certain polynomial vector fields on spheres.

Non-existence of cubic Hamiltonian Kolmogorov vector fields with invariant odd-dimensional spheres.

Conditions for Darboux first integrals in cubic Kolmogorov vector fields.

Abstract

In this paper, we characterize all polynomial Kolmogorov vector fields for which the standard $n$-sphere is invariant. We exhibit completely integrable Kolmogorov vector fields of degree $m$ on $\mathbb{S}^n$ for any $m >2$. Then, we show that there is no cubic Hamiltonian Kolmogorov vector field that makes an odd-dimensional sphere invariant. We examine the conditions under which a cubic Kolmogorov vector field has a Darboux first integral. In many cases, we determine whether they constitute necessary and sufficient conditions. Moreover, we study the complete integrability of cubic Kolmogorov vector fields having an invariant $n$-sphere.