Noise-Induced Dynamics and Stochastic Slow Manifolds in the Bistable Xiong-Ferrell Model

In biology many phenomena involve noise and multiple time scales. Noise may be constructive and induce dynamics not seen in the deterministic version of the system of interest. The aim of this work is to give an explanation to the noise-induce dynamics observed in a polynomial simplification of the Xiong-Ferrell model as the noise variance (γ) and stimulation intensity (S) are changed. To understand this behaviour, the Stochastic Differential Equation (SDE) system is analysed with the use of a slow manifold statistical description and converted into an Ordinary Differential Equation (ODE) system via the G method, to investigate the location of saddle-node bifurcations in the (S, γ) plane. Our results explain why, due to and depending on the delocalization of these latter bifurcations, the system might present fluctuations around a stable equilibrium or exhibit noise-induced transitions and oscillations.