Dynamics, integrability and topology for some classes of Kolmogorov Hamiltonian systems in R₊⁴

In this paper we first give the sufficient and necessary conditions in order that two classes of polynomial Kolmogorov systems in R₊⁴ are Hamiltonian systems. Then we study the integrability of these Hamiltonian systems in the Liouville sense. Finally, we investigate the global dynamics of the completely integrable Lotka–Volterra Hamiltonian systems in R₊⁴. As an application of the invariant subsets of these systems, we obtain topological classifications of the 3-submanifolds in R₊⁴ defined by the hypersurfaces axy+bzw+cx²y+dxy²+ez²w+fzw²=h, where a,b,c,d,e,f,w and h are real constants.