Seminár z kvalitatívnej teórie diferenciálnych rovníc

Seminar on Qualitative Theory of Differential Equations

Thursday 14.12.2017 at 14:00 Lecture room M-223

Jan Haškovec (KAUST):

Discrete and continuum modeling of biological network formation

Abstract:

Motivated by recent papers describing rules for natural network formation

in discrete settings, we propose an elliptic-parabolic system

of partial differential equations. The model describes the pressure field

due to Darcyâs type equation and the dynamics of the conductance network

under pressure force effects with a diffusion rate representing randomness

in the material structure. After a short overview of the principles

of discrete network modeling, we will show how to derive the corresponding

macroscopic (continuum) description. The highly unusual structure

of the resulting PDE system induces several interesting challenges

for its mathematical analysis. We give a short overview of the tools

and tricks that can be used to overcome them. In particular,

we present results regarding the existence of weak solutions of the system,

based on recent results on elliptic regularity theory. Moreover, we study

the structure and stability properties of steady states that play

a central role to understand the pattern capacity of the system.

We present results of systematic numerical simulations of the system that

provide further insights into the properties of the network-type solutions.