Self-regulated biological transportation structures with general entropy dissipation: 2D case and leaf-shaped domain

¹ Department of Mathematics and Computer Sciences, University of Catania, Catania 95125, Italy
² Mathematical and Computer Sciences and Engineering Division, King Abdullah University of Science and Technology, Thuwal 23955-6900, Kingdom of Saudi Arabia
³ Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, Vienna 1090, Austria

^* Corresponding author: This email address is being protected from spambots. You need JavaScript enabled to view it.

Received: 20 October 2024
Accepted: 27 January 2026

Abstract

In recent years, the study of biological transportation networks has attracted significant interest, focusing on their self-regulating, demand-driven nature. This paper examines a mathematical model for these networks, featuring nonlinear elliptic equations for pressure and an auxiliary variable, and a reaction-diffusion parabolic equation for the conductivity tensor, introduced in Haskovec et al. [Discrete Contin. Dyn. Syst. 43 (2022) 1499–1515]. The model, based on an energy functional with diffusive and metabolic terms, allows for various entropy generating functions, facilitating its application to different biological scenarios. We proved a local well-posedness result for the problem in Hölder spaces employing Schauder and semigroup theory. Then, after a suitable parameter reduction through scaling, we computed the numerical solution for the proposed system using a recently developed ghost nodal finite element method in Astuto et al. [Comput. Methods Appl. Mech. Eng. 443 (2025) 118041]. An interesting aspect emerges when the solution is very articulated and the branches occupy a wide region of the domain.