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

Abstract

We compare the solutions of two systems of partial differential equations (PDEs), seen as two different interpretations of the same model which describes the formation of complex biological networks. Both approaches take into account the time evolution of the medium flowing through the network, and we compute the solution of an elliptic–parabolic PDE system for the conductivity vector m, the conductivity tensor C and the pressure p. We use finite differences schemes in a uniform Cartesian grid in a spatially two-dimensional setting to solve the two systems, where the parabolic equation is solved using a semi-implicit scheme in time. Since the conductivity vector and tensor also appear in the Poisson equation for the pressure p, the elliptic equation depends implicitly on time. For this reason, we compute the solution of three linear systems in the case of the conductivity vector m∈R2 and four linear systems in the case of the symmetric conductivity tensor C∈R2×2 at each time step. To accelerate the simulations, we make use of the Alternating Direction Implicit (ADI) method. The role of the parameters is important for obtaining detailed solutions. We provide numerous tests with various values of the parameters involved to determine the differences in the solutions of the two systems.