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 portaro2022emergence. 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 astuto2024nodal. An interesting aspect emerges when the solution is very articulated and the branches occupy a wide region of the domain.
Astuto et al. (Sun,) studied this question.