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

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.

References in corpus (3)

• Limit Cycle Oscillations, response time and the time-dependent solution to the Lotka-Volterra Predator-Prey model
• Analysis of a class of Kolmogorov systems
• Invariant circles and phase portraits of cubic vector fields on the sphere