Simulating Stochastic Population Dynamics: The Linear Noise Approximation Can Capture Nonlinear Phenomena

Abstract.

Population dynamics in fields such as molecular biology, epidemiology, and ecology exhibit highly stochastic and nonlinear behavior. In gene regulatory systems in particular, oscillations and multistability are especially common. Despite this, none of the currently available stochastic models for population dynamics are both accurate and computationally efficient for long-term predictions. A prominent model in this field, the linear noise approximation (LNA), is computationally efficient for tasks such as simulation, sensitivity analysis, and parameter estimation; however, it is only accurate for linear systems and short-time predictions. Other models may achieve greater accuracy across a broader range of systems, but they sacrifice computational efficiency and analytical tractability. This paper demonstrates that, with specific modifications, the LNA can accurately capture nonlinear dynamics in population processes. We introduce a new framework based on center manifold theory, a classical concept from nonlinear dynamical systems. This approach enables the identification of simple, system-specific modifications to the LNA, tailored to classes of qualitatively similar nonlinear dynamical systems. With these modifications, the LNA can achieve accurate long-term simulations without compromising computational efficiency. We apply our methodology to classes of oscillatory and bistable systems and present multiple examples from molecular population dynamics that demonstrate accurate long-term simulations alongside significant improvements in computational efficiency.