Emergence of biological transportation networks as a self-regulated process

TL;DR

This paper investigates the self-regulating mechanisms of biological transportation networks through gradient flows of entropy dissipations, coupling diffusion, electric potential, and Poisson equations to model network emergence.

Contribution

It introduces a formal $L^2$-gradient flow framework for modeling the evolution of diffusivity in biological networks, incorporating electric potential and deriving coupled PDE systems.

Findings

Derived convexity of the entropy dissipation functional.

Formulated gradient flows for diffusivity and electric potential.

Established a coupled PDE system modeling network self-organization.

Abstract

We study self-regulating processes modeling biological transportation networks. Firstly, we write the formal $L^2$-gradient flow for the symmetric tensor valued diffusivity $D$ of a broad class of entropy dissipations associated with a purely diffusive model. The introduction of a prescribed electric potential leads to the Fokker-Planck equation, for whose entropy dissipations we also investigate the formal $L^2$-gradient flow. We derive an integral formula for the second variation of the dissipation functional, proving convexity (in dependence of diffusivity tensor) for a quadratic entropy density modeling Joule heating. Finally, we couple in the Poisson equation for the electric potential obtaining the Poisson-Nernst-Planck system. The formal gradient flow of the associated entropy loss functional is derived, giving an evolution equation for $D$ coupled with two auxiliary elliptic PDEs.