Modeling Still Matters: A Surprising Instance of Catastrophic Floating Point Errors in Mathematical Biology and Numerical Methods for ODEs

Abstract

SIAM Review, Volume 67, Issue 3, Page 624-641, August 2025.
Abstract.We guide the reader on a journey through mathematical modeling and numerical analysis, emphasizing the crucial interplay of both disciplines. Targeting undergraduate students with basic knowledge of dynamical systems and numerical methods for ordinary differential equations, we explore a model from mathematical biology where numerical methods fail badly due to catastrophic floating point errors. We analyze the reasons for this behavior by studying the steady states of the model and use the theory of invariants to develop an alternative model suited for numerical simulations. Our story is intended to motivate the combining of analytical knowledge and numerical knowledge, even in those cases where the world looks fine at first sight. We have set up an online repository containing an interactive notebook with all the numerical experiments in this article to make this study fully reproducible and useful for classroom teaching.