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

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.