Self-regulated biological transportation structures with general entropy dissipations, part I: the 1D case

TL;DR

This paper investigates self-regulating biological transportation networks in one dimension, establishing existence and uniqueness results, analyzing the influence of diffusivity, and exploring scenarios with measure-based sources and sinks.

Contribution

It provides the first rigorous analysis of 1D biological transportation models with general entropy dissipations, including existence, uniqueness, and numerical insights into diffusivity behavior.

Findings

Existence and uniqueness of solutions under positive diffusivity D.

Numerical evidence of solutions where D approaches zero.

Insights into the impact of source-sink distributions on system behavior.

Abstract

We study self-regulating processes modeling biological transportation networks as presented in \cite{portaro2023}. In particular, we focus on the 1D setting for Dirichlet and Neumann boundary conditions. We prove an existence and uniqueness result under the assumption of positivity of the diffusivity $D$. We explore systematically various scenarios and gain insights into the behavior of $D$ and its impact on the studied system. This involves analyzing the system with a signed measure distribution of sources and sinks. Finally, we perform several numerical tests in which the solution $D$ touches zero, confirming the previous hints of local existence in particular cases.