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.

翻译：本文研究了文献\cite{portaro2023}中提出的建模生物输运网络的自调节过程。我们特别关注狄利克雷边界条件和诺伊曼边界条件下的一维情形。在扩散系数$D$为正的假设下，我们证明了存在唯一性结果。我们系统性地探讨了多种情景，深入分析了$D$的行为及其对所研究系统的影响。这包括对带有正负源符号测度分布的系统进行分析。最后，我们进行了若干数值实验，其中解$D$触及零值，证实了此前关于特定情形下局部存在的线索。