Planar Kolmogorov Systems Coming from Spatial Lotka-Volterra Systems

In this paper, we classify the phase portraits in the Poincare disc of all the Kolmogorov systems (y) over dot = y(b(0) + b(1)yz + b(2)y + b(3)z), (z) over dot = z(c(0) - mu(b(1)yz + b(2)y + b(3)z)), which depend on six parameters. We prove that these systems have 52 topologically distinct phase portraits in the Poincare disc. These systems are provided by a general three-dimensional Lotka-Volterra system with a rational first integral of degree two of the form H = x(i)y(j)z(k), restricted to each surface H(x,y,z) = h varying h is an element of Double-struck capital R, with the additional assumption that they have a Darboux invariant of the form y(l)z(m)e(st).

Automated summary

This paper classifies the phase portraits of all Kolmogorov systems in the Poincare disc, showing that there are 52 topologically distinct phase portraits. These systems are derived from a general three-dimensional Lotka-Volterra system and have a specific Darboux invariant.