Bifurcation of Limit Cycles and Isochronous Centers on Center Manifolds for a Class of Cubic Kolmogorov Systems in R3

Our work is concerned with the number of limit cycles and isochronous center conditions for a class of three-dimensional cubic Kolmogorov systems with an equilibrium point in the positive octant where the system has biological meaning. A formal series method of computing singular point values (equivalent to focal values) is applied to investigate the Hopf bifurcation and center problem on center manifolds for the Kolmogorov system. Using this we derive two sets of conditions for the equilibrium point to be a center on a center manifold for the system, and prove that at most seven small-amplitude limit cycles can be bifurcated from an isolated positive equilibrium point. We prove that seven limit cycles can be created in this way, obtaining a new result on the number of limit cycles in three-dimensional cubic Kolmogorov systems. Moreover, two sets of necessary conditions for the existence of an isochronous center on the center manifold for such systems are obtained by the computation of period constants. The Darboux theory of linearizability is applied to show the sufficiency of the conditions. Mathematics Subject Classification (2010). 34C05, 34C07.