Multiscale Modelling of Birth-Death Processes
Many biological systems exhibit multiscale dynamics, where some species occur in high copy numbers while others remain rare. This heterogeneity necessitates hybrid modelling approaches: deterministic models are computationally efficient but inaccurate for low-count species, while fully stochastic simulations are accurate but prohibitively expensive. Threshold-based hybrid methods, such as the Jump--Switch--Flow (JSF) algorithm, address this by simulating low-count species stochastically and high-count species deterministically, switching at a user-chosen threshold $Ω$. In such methods, the choice of $Ω$ controls the trade-off between computational cost and accuracy, but is typically made by trial-and-error: there is no principled way to choose $Ω$ a priori for a given observable of interest. We close this gap for extinction probability. Our contribution is a computable, method-agnostic error bound that quantifies the discrepancy introduced by the threshold and yields an explicit rule for selecting $Ω$ to meet a user-specified error tolerance. We formalise JSF as a piecewise-deterministic Markov process and derive backward equations for extinction under exact and hybrid dynamics. Near extinction boundaries, the complex nonlinear dynamics reduce to tractable time-inhomogeneous linear birth--death processes; this structure yields a rigorous error decomposition into early and late excursions, whose dominant term becomes a fast, actionable heuristic requiring only the solution of a scalar Riccati equation. Monte Carlo studies on a stochastic Lotka--Volterra model confirm that the heuristic reliably upper-bounds the empirical error in extinction probability across a wide parameter range. The framework depends only on the birth and death rates near extinction, not on the specific simulation method, and therefore applies beyond JSF to any threshold-based hybrid scheme.