Integrability of a class of N–dimensional Lotka–Volterra and Kolmogorov systems

Abstract

We study the integrability of an N–dimensional differential Kolmogorov systems of the formx˙j=xj(aj+∑k=1Najkxk)+xjΨ(x1,…,xN),j=1,…,N, where aj, and ajk are constants for j,k=1,…,N and Ψ(x1,…,xN) is a homogeneous polynomial of degree n>2, with either one additional invariant hyperplane, or with one exponential factor. We also study the integrability of the N–dimensional classical Lotka-Volterra systems (when Ψ(x1,…,xN)=0). In particular we consider the integrability of the asymmetric May–Leonard systems.