Splitting-Particle Methods for Structured Population Models: Convergence and Applications

TL;DR

This paper introduces a new particle-based numerical scheme for structured population models that achieves convergence and is flexible enough to handle complex models beyond traditional methods.

Contribution

A novel operator splitting particle method for structured population models with proven convergence and broader applicability than existing techniques.

Findings

The scheme converges in the space of finite Radon measures.

Validation through test cases confirms theoretical convergence.

The method effectively handles models where traditional EBT methods fail.

Abstract

We propose a new numerical scheme designed for a wide class of structured population models based on the idea of operator splitting and particle approximations. This scheme is related to the Escalator Boxcar Train (EBT) method commonly used in biology, which is in essence an analogue of particle methods used in physics. Our method exploits the split-up technique, thanks to which the transport step and the nonlocal integral terms in the equation can be separately considered. The order of convergence of the proposed method is obtained in the natural space of finite nonnegative Radon measures equipped with the flat metric. This convergence is studied even adding reconstruction and approximation steps in the particle simulation to keep the number of approximation particles under control. We validate our scheme in several test cases showing the theoretical convergence error. Finally, we use the scheme in situations in which the EBT method does not apply showing the flexibility of this new method to cope with the different terms in general structured population models.