Emergence of biological transportation networks as a self-regulated process

Abstract

We study self-regulating processes modeling biological transportation networks. Firstly, we write the formal L 2 -gradient flow for the symmetric tensor valued diffusivity D of a broad class of entropy dissipations associated with a purely diffusive model. The introduction of a prescribed electric potential leads to the Fokker-Planck equation, for whose entropy dissipations we also investigate the formal L 2 -gradient flow. We derive an integral formula for the second variation of the dissipation functional, proving convexity (in dependence of diffusivity tensor) for a quadratic entropy density modeling Joule heating. Finally, we couple in the Poisson equation for the electric potential obtaining the Poisson-Nernst-Planck system. The formal gradient flow of the associated entropy loss functional is derived, giving an evolution equation for D coupled with two auxiliary elliptic PDEs.

Comparison of Two Aspects of a PDE Model for Biological Network Formation

“…Assuming the validity of Darcy's law for slow flow in porous media (see, for instance, [4,5]), the energy functional is constrained by a Poisson equation for the fluid pressure. We use two modes of description of the network conductivity: first, in terms of a conductance vector m, and, second, in terms of a symmetric positive definite conductance tensor C. Taking the L 2 -gradient flow with respect to m (see, for instance, [6][7][8][9][10]) and, resp., with respect to C (see [11]), leads to two structurally similar elliptic-parabolic PDE systems.…”