HOPF AND ZERO-HOPF BIFURCATIONS FOR A CLASS OF CUBIC KOLMOGOROV SYSTEMS IN $ \mathbb R^{3}$

In this paper, Hopf and zero-Hopf bifurcations are investigated for a class of three-dimensional cubic Kolmogorov systems with one positive equilibrium. Firstly, by computing the singular point quantities and figuring out center conditions, we determined that the highest order of the positive equilibrium is eight as a fine focus, which yields Hopf cyclicity eight at the positive equilibrium. Secondly, by extending the normal form method, we discuss the existence of multiple periodic solutions via zero-Hopf bifurcation around the positive equilibrium. At the same time, the relevance between zero-Hopf bifurcation and Hopf bifurcation is analyzed via its special case, which are rarely studied in detail.