Electronic Journal of Differential Equations, 
Vol. 2021 (2021), No. 35, pp. 1-38.

Title: Phase portraits of a family of Kolmogorov systems depending on six parameters
       
Authors: Erika Diz-Pita (Univ. de Santiago de Compostela, Spain)
         Jaume Llibre  (Univ. Autonoma de Barcelona, Spain)
         M. Victoria Otero-Espinar (Univ. de Santiago de Compostela, Spain)

Abstract: 
 We consider a general 3-dimensional Lotka-Volterra system with a rational first
 integral of degree two of the form H=x<sup>I y<sup>j</sup z<up>k</sup>.
 The restriction of this Lotka-Volterra system to each surface H(x,y,z)=h varying
 h in R provide Kolmogorov systems. With the additional assumption that
 they have a Darboux invariant of the form x<ups>l</sup y<sup>m</sup> e<sup>st</sup>
 they reduce to the
 Kolmogorov systems
 $$\displaylines{
 \dot{x}=x\big(a_0- \mu (c_1 x + c_2 z^2 + c_3 z)\big),\cr
 \dot{z}=z\big(c_0+ c_1 x + c_2 z^2 + c_3 z\big).
 }$$
 We classify the phase portraits in the Poincare disc of all these Kolmogorov systems
 which depend on six parameters.

Submitted June 1, 2020. Published May 03, 2021.
Math Subject Classifications: 34C05.
Key Words: Kolmogorov system; Lotka-Volterra system; phase portrait; Poincare disc