Hopf bifurcation in Kolmogorov systems of degree 4 in R2 and R3

We consider the Kolmogorov system of degree 4 in R2 having an equilibrium point in the positive quadrant and the Kolmogorov system of degree 4 in R3 having an equilibrium point in the positive octant. Note that from a biological point of view the two equilibria considered must be in the positive quadrant and octant. We give the conditions under which the equilibrium points of the planar and spatial systems will be a Hopf point and a zero-Hopf point, respectively. The explicit conditions for the existence of limit cycles bifurcating from these two equilibria are obtained using the averaging theory. Moreover, we characterize the stability or instability of these limit cycles.