Multiscale modelling of birth-death processes.

Authors

Tom Kimpson, Domenic P J Germano, Jennifer A Flegg, Mark B Flegg

Published in

Journal of mathematical biology. Volume 93. Issue 2. Aug 03, 2026. Epub Aug 03, 2026.

Abstract

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 $\Omega$ . In such methods, the choice of $\Omega$ controls the trade-off between computational cost and accuracy, but is typically made by trial-and-error: there is no principled way to choose $\Omega$ 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 $\Omega$ 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 simulation scheme.

PMID:
42545502
Bibliographic data and abstract were imported from PubMed on 03 Aug 2026.

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