On the convergence of the Escalator Boxcar Train

TL;DR

This paper provides a rigorous proof of convergence for the Escalator Boxcar Train (EBT) method, a numerical technique used in structured population models, thereby validating its reliability and facilitating broader application.

Contribution

The paper formally proves the convergence of the EBT method using measure-valued solutions, addressing a long-standing gap in its theoretical foundation.

Findings

EBT sequence converges weakly to the true solution measure

Convergence holds under weak conditions on model parameters

Results enable wider acceptance and integration of EBT in numerical analysis

Abstract

The Escalator Boxcar Train (EBT) is a numerical method that is widely used in theoretical biology to investigate the dynamics of physiologically structured population models, i.e., models in which individuals differ by size or other physiological characteristics. The method was developed more than two decades ago, but has so far resisted attempts to give a formal proof of convergence. Using a modern framework of measure-valued solutions, we investigate the EBT method and show that the sequence of approximating solution measures generated by the EBT method converges weakly to the true solution measure under weak conditions on the growth rate, birth rate, and mortality rate. In rigorously establishing the convergence of the EBT method, our results pave the way for wider acceptance of the EBT method beyond theoretical biology and constitutes an important step towards integration with established numerical schemes.