Tensor PDE model of biological network formation

We study an elliptic-parabolic system of partial differential equations describing formation biological network structures. The model takes into consideration the evolution permeability tensor under influence a diffusion term, representing randomness in material structure, decay term metabolic cost and pressure force. A Darcy's law type equation describes field. In spatially two-dimensional setting, we present constructive, formal derivation PDE from discrete refinement limit sequence unstructured triangulations. Moreover, show that is $L^2$-gradient flow energy functional with interpretation, its convexity properties. For case when convex, construct unique global weak solutions system. Finally, steady state one- multi-dimensional settings discuss their stability