  We study self-regulating processes modeling biological transportation
networks as presented in \cite{portaro2023}. In particular, we focus on the 1D
setting for Dirichlet and Neumann boundary conditions. We prove an existence
and uniqueness result under the assumption of positivity of the diffusivity
$D$. We explore systematically various scenarios and gain insights into the
behavior of $D$ and its impact on the studied system. This involves analyzing
the system with a signed measure distribution of sources and sinks. Finally, we
perform several numerical tests in which the solution $D$ touches zero,
confirming the previous hints of local existence in particular cases.
