Linear multiplicative noise destroys a two-dimensional attractive compact manifold of three-dimensional Kolmogorov systems

TL;DR

This work identifies cubic three-dimensional Kolmogorov systems that admit a two-dimensional attractive invariant sphere $S^2$ in $R^3$ and derives a global attraction criterion on $R^3\setminus\{O\}$. Using Darboux theory and Lyapunov methods, it characterizes when such a sphere exists and reduces the system to an explicit cubic form with positive parameters $\alpha_i$, revealing two global-dynamics regimes on $S^2$ (periodic or nonperiodic orbits) depending on parameter signs. It then studies stochastic perturbations by linear multiplicative noise and proves a sharp noise threshold $\sigma_0=\sqrt{2\max\{\alpha_i\}}$ at which the invariant sphere is destroyed, accompanied by bifurcations in stationary measures ranging from multiple measures at small/noise to a unique measure at large noise. The results illuminate how multiplicative noise can destroy invariant manifolds and induce stochastic phase transitions in ecological-like nonlinear systems.

Abstract

In the paper we first characterize three-dimensional Kolmogorov systems possessing a two-dimensional invariant sphere in $\mathbb{R}^3$, then establish a global attracting criterion for this invariant sphere in $\mathbb{R}^3$ except the origin, and give global dynamics with isolated equilibria on the sphere. Finally, we consider the persistence of the attractive invariant sphere under the perturbation induced by linear multiplicative Wiener noise. It is shown that suitable noise intensity can destroy the sphere and lead to bifurcation of stationary measures.

Figures (1)

• Figure 2.1: The global dynamics of system \ref{['sys0']} with isolated equilibria

Theorems & Definitions (13)

• Proposition 2.1
• proof
• Proposition 2.2
• proof
• Corollary 2.3
• Theorem 2.4
• proof
• Lemma 2.5: Existence of isolated equilibria
• Theorem 2.6: Global dynamics
• proof