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

In the paper we first provide the necessary and sufficient algebraic conditions for the three-dimensional Kolmogorov systems possessing a two-dimensional invariant sphere in [Formula: see text]. Then, we establish a global attracting criterion for this invariant sphere in [Formula: see text] 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 and focus on if stochastic bifurcations occur. It is shown that suitable noise intensity can destroy the sphere and lead to bifurcations of stationary measures.