Simulating stochastic population dynamics: The Linear Noise Approximation can capture non-linear phenomena

There is an abundance of highly stochastic, non-linear phenomena present in population dynamics across various fields, including molecular biology, epidemiology, and ecology. Oscillations and multi-stability are particularly prevalent in gene regulation systems. None of the currently available stochastic models for population dynamics are accurate and computationally efficient for long-term predictions. A prominent model in this field, known as the Linear Noise Approximation (LNA), is computationally efficient for tasks such as simulation, sensitivity analysis, and parameter estimation, still, 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 indeed accurately capture non-linear dynamics in population dynamics. We introduce a new framework that employs centre manifold theory, a classical concept in non-linear dynamical systems. This approach allows us to identify simple, specialised modifications to the LNA tailored to classes of qualitatively similar non-linear dynamical systems. With these modifications, the LNA can achieve accurate long-term predictions without compromising its computational efficiency. We apply our methodology to a class of oscillatory systems and a class of bi-stable systems, and we provide multiple examples in molecular population dynamics that showcase accurate long-term predictions, and significant improvements in computational efficiency.