  We compare the solutions of two systems of partial differential equations
(PDE), seen as two different interpretations of the same model that describes
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 $\mathbb{C}$ and the pressure $p$. We use finite
differences schemes in a uniform Cartesian grid in the spatially
two-dimensional setting to solve the two systems, where the parabolic equation
is solved by a semi-implicit scheme in time. Since the conductivity vector and
tensor appear also 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 $m\in\mathbb{R}^2$,
and four linear systems in the case of the symmetric conductivity tensor
$\mathbb{C}\in\mathbb{R}^{2\times 2}$, 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 see
the differences in the solutions of the two systems.
