Limit cycles bifurcating of Kolmogorov systems in R² and in R³

In this work we consider the Kolmogorov system of degree 3 in R² and R³ having an equilibrium point in the positive quadrant and octant, respectively. We provide sufficient conditions in order that the equilibrium point will be a Hopf point for the planar case and a zero-Hopf point for the spatial one. We study the limit cycles bifurcating from these equilibria using averaging theory of second and first order, respectively. We note that the equilibrium point is located in the quadrant or octant where the Kolmogorov systems have biological meaning.