Notes on a PDE System for Biological Network Formation

Jan Haskovec (1) , Peter Markowich (1) , Benoît Perthame (2) , Matthias Schlottbom (3)

1 MCSE - Division of Mathematical and Computer Sciences and Engineering
2 MAMBA - Modelling and Analysis for Medical and Biological Applications
3 Numerische und Angewandte Mathematik - Institute for Computational and Applied Mathematics [Münster]

We present new analytical and numerical results for the elliptic-parabolic system of partial differential equations proposed by Hu and Cai [8, 10], which models the formation of biological transport networks. The model describes the pressure field using a Darcy's 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. The analytical part extends the results of [7] regarding the existence of weak and mild solutions to the whole range of meaningful relaxation exponents. Moreover, we prove finite time extinction or breakdown of solutions in the spatially one-dimensional setting for certain ranges of the relaxation exponent. We also construct stationary solutions for the case of vanishing diffusion and critical value of the relaxation exponent, using a variational formulation and a penalty method. The analytical part is complemented by extensive numerical examples. We propose a discretization based on mixed finite elements and study the qualitative properties of network structures for various parameters values. Furthermore, we indicate numerically that some analytical results proved for the spatially one-dimensional setting are likely to be valid also in several space dimensions.

Mots clés: Network formation; Weak solutions; Stability; Penalty method

Domaines: Equations aux dérivées partielles [math.AP]; Adaptation et Systèmes auto-organisés [nlin.AO]

Jan Haskovec, Peter Markowich, Benoît Perthame, Matthias Schlottbom. Notes on a PDE System for Biological Network Formation. Nonlinear Analysis: Real World Applications, 2016, Nonlinear Partial Differential Equations, in honor of Juan Luis Vázquez for his 70th birthday, 138, pp.127-155. ⟨10.1016/j.na.2015.12.018⟩. ⟨hal-01232080⟩