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 that
there exist at most seven small amplitude limit cycles restricted to one
center manifold and Hopf cyclicity 8 at the positive equilibrium.
Secondly, by using the normal form algorithm, we discuss the existence
of stable periodic solution 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
is rarely considered. Finally, some related illustrations are given by
means of numerical simulation.