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AbstractWe present an overview of recent analytical and numerical results for the elliptic–parabolic system of partial differential equations proposed by Hu and Cai, which models the formation of biological transportation networks. The model describes the pressure field using a Darcy type equation and the dynamics of the conductance network under pressure force effects. Randomness in the material structure is represented by a linear diffusion term and conductance relaxation by an algebraic decay term. We first introduce micro- and mesoscopic models and show how they are connected to the macroscopic PDE system. Then, we provide an overview of analytical results for the PDE model, focusing mainly on the existence of weak and mild solutions and analysis of the steady states. The analytical part is complemented by extensive numerical simulations. We propose a discretization based on finite elements and study the qualitative properties of network structures for various parameter values.
CitationAlbi G, Burger M, Haskovec J, Markowich P, Schlottbom M (2017) Continuum Modeling of Biological Network Formation. Modeling and Simulation in Science, Engineering and Technology: 1–48. Available: http://dx.doi.org/10.1007/978-3-319-49996-3_1.
SponsorsMB and MS acknowledge support by ERC via Grant EU FP 7 - ERC Con- solidator Grant 615216 LifeInverse. MB acknowledges support by the German Science Foundation DFG via EXC 1003 Cells in Motion Cluster of Excellence, Münster, Germany. GA acknowledges the ERC-Starting Grant project High-Dimensional Sparse Optimal Control (HDSPCONTR).
PublisherSpringer International Publishing