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：我们研究《portaro2023》中提出的模拟生物输运网络的自调节过程。特别地，我们聚焦于Dirichlet和Neumann边界条件下的一维情形。在扩散率$D$为正的假设下，我们证明了解的存在唯一性结果。我们系统地探讨了各种情景，深入分析了$D$的行为及其对研究系统的影响，包括对带有符号测度源汇分布的系统分析。最后，我们进行了若干数值实验，其中解$D$触及零值，验证了先前在特定情形下局部存在的迹象。