Stochastic Bifurcations of a Three-Dimensional Stochastic Kolmogorov System

Abstract

In this paper we systematically investigate the stochastic bifurcations of both ergodic stationary measures and stochastic dynamics for a stochastic Kolmogorov differential system by the change of the sign of Lyapunov exponents. It is derived that there exists a threshold \(\sigma _0\) such that, if the noise intensity \(\sigma \ge \sigma _0\), the noise destroys all bifurcations of the deterministic system and the corresponding stochastic Kolmogorov system is uniquely ergodic. On the other hand, when the noise intensity \(0<\sigma <\sigma _0\), there exist further bifurcation thresholds such that the stochastic system undergoes bifurcations from the unique ergodic stationary measure to three different kinds of ergodic measures: (I) finitely many ergodic measures supported on rays, (II) infinitely many ergodic measures supported on rays, (III) infinitely many ergodic measures supported on invariant cones. Correspondingly, the system undergoes stochastic bifurcations of stochastic dynamics, which even displays infinitely many Crauel random periodic solutions in the sense of Engel and Kuehn (Comm Math Phys 386(3): 1603–1641, 2021). Furthermore, we prove that as \(\sigma \) tends to zero, the ergodic stationary measures converge to either Dirac measures supported on equilibria, or to Haar measures supported on non-trivial deterministic periodic orbits.