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

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.