An Analytical Framework to Describe the Interactions Between Individuals and a Continuum

TL;DR

This paper develops a comprehensive analytical framework for modeling and analyzing interactions between discrete agents and a continuum, with applications demonstrated through specific examples and numerical simulations.

Contribution

It introduces a well-posedness theory for a coupled system of ODEs and PDEs describing agent-continuum interactions, expanding the mathematical understanding of such models.

Findings

Established existence and uniqueness of solutions for the coupled system.

Provided numerical simulations illustrating the model's behavior.

Applied the framework to predator-prey and herd-driving scenarios.

Abstract

We consider a discrete set of individual agents interacting with a continuum. Examples might be a predator facing a huge group of preys, or a few shepherd dogs driving a herd of sheeps. Analytically, these situations can be described through a system of ordinary differential equations coupled with a scalar conservation law in several space dimensions. This paper provides a complete well posedness theory for the resulting Cauchy problem. A few applications are considered in detail and numerical integrations are provided.