Structured Population Models on Polish spaces: A unified Approach including Graphs, Riemannian Manifolds and Measure Spaces to describe Dynamics of Heterogeneous Populations

TL;DR

This paper introduces a unified mathematical framework for modeling heterogeneous population dynamics on abstract metric spaces, enabling analysis of complex systems like crowd behavior, tissue growth, and coagulation processes.

Contribution

It develops a general approach using measure evolution on metric spaces, including graphs and manifolds, to address nonlinear structured population models in a unified way.

Findings

Framework accommodates non-conservative problems.

Applicable to infinite-dimensional and hybrid discrete-continuous spaces.

Facilitates modeling of biological and social systems.

Abstract

This paper presents a mathematical framework for modeling the dynamics of heterogeneous populations. Models describing local and non-local growth and transport processes, dependent on dynamically changing population structures, appear in a variety of applications such as crowd dynamics, tissue regeneration, cancer development, and coagulation-fragmentation processes. The current body of literature regarding mathematical modeling presents common challenges to mathematicians due to the multiscale nature of the structures that underlie self-organisation and control within complex, heterogeneous systems. In various applications, similar, abstract mathematical concepts arise through problem formulation and the assimilation of mathematical depictions into the language of measure evolution on a multi-faceted state space. In view of the above observations, we propose an overarching mathematical framework for nonlinear structured population models on abstract metric spaces, which are only assumed to be separable and complete. To achieve this, we exploit the structure of the space of non-negative Radon measures under the dual bounded Lipschitz distance (flat metric), a generalization of the Wasserstein distance that is capable of addressing non-conservative problems. The formulation of models on generic metric spaces facilitates the study on infinite-dimensional state spaces or graphs with a combination of discrete and continuous structures. This opens up exciting possibilities for modeling single cell data, crowd dynamics or coagulation-fragmentation processes.