English

Multiscale Modelling of Birth-Death Processes

Populations and Evolution 2026-01-21 v1

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. Hybrid methods like the Jump-Switch-Flow (JSF) algorithm address this by simulating low-count species stochastically and high-count species deterministically. However, selecting regime-switching thresholds to control errors for specific observables remains an open challenge. We develop a principled framework for threshold selection targeting extinction probability. We formalise JSF as a piecewise-deterministic Markov process and derive backward equations for extinction under exact and hybrid dynamics. Near extinction boundaries, complex nonlinear dynamics reduce to tractable time-inhomogeneous linear birth-death processes. This structure yields a rigorous error decomposition based on early and late excursions. Isolating the dominant error term motivates a fast, actionable heuristic. We demonstrate via Monte Carlo studies on a stochastic Lotka-Volterra model that our heuristic reliably upper-bounds empirical errors in extinction probability. This enables users to select the smallest threshold that satisfies a target error tolerance. This work paves the way for principled, efficient multiscale modelling and simulation in stochastic biological systems.

Keywords

Cite

@article{arxiv.2601.12712,
  title  = {Multiscale Modelling of Birth-Death Processes},
  author = {Tom Kimpson and Domenic P. J. Germano and Jennifer A. Flegg and Mark B. Flegg},
  journal= {arXiv preprint arXiv:2601.12712},
  year   = {2026}
}

Comments

25 pages, 5 figures. Submitted to the Journal of Mathematical Biology

R2 v1 2026-07-01T09:09:59.435Z