Stochastic bifurcation of a three-dimensional stochastic Kolmogorov system

TL;DR

This paper analyzes how stochastic noise influences bifurcations in a three-dimensional stochastic Kolmogorov system, revealing thresholds where noise destroys or induces complex ergodic behaviors and periodic solutions.

Contribution

It systematically characterizes stochastic bifurcations in a 3D Kolmogorov system, identifying thresholds and types of ergodic measures, and describes the convergence of measures as noise diminishes.

Findings

Existence of a noise threshold $\sigma_0$ where bifurcations are destroyed.

Identification of three types of ergodic measures during bifurcations.

Convergence of measures to equilibria or periodic orbits as noise tends to zero.

Abstract

In this paper we systematically investigate the stochastic bifurcations of both ergodic stationary measures and global dynamics for stochastic Kolmogorov differential systems, which relate closely to the change of the sign of Lyapunov exponents. It is derived that there exists a threshold $\sigma_0$ such that, if the noise intensity $\sigma \geq\sigma_0$, 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 $\sigma<\sigma_0$, the stochastic system undergoes bifurcations from the unique ergodic stationary measure to three different types of ergodic stationary 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 global dynamics undergo similar bifurcation phenomena, which even displays infinitely many Crauel random periodic solutions in the sense of \cite{ELR21}. 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.