Mathematical Analysis of a PDE System for Biological Network Formation

Motivated by recent physics papers describing rules for natural network formation, we study an elliptic-parabolic system of partial differential equations proposed by Hu and Cai 13 Hu , D. (2013). Optimization, Adaptation, and Initialization of Biological Transport Networks. Workshop on multiscale problems from physics, biology and material sciences. May 28–31st, 2014, Shanghai. Available at: http://math.sjtu.edu.cn/conference/multiscale2014/ (accessed 6 January 2015). Google Scholar], 15 Hu , D. , Cai , D. ( 2013 ). Adaptation and optimization of biological transport networks . Phys. Rev. Lett. 111 : 138701 .Crossref], PubMed], Web of Science ®] , Google Scholar]]. The model describes the pressure field thanks to Darcy's type equation and the dynamics of the conductance network under pressure force effects with a diffusion rate D ≥ 0 representing randomness in the material structure. We prove the existence of global weak solutions and of local mild solutions and study their long term behavior. It turns out that, by energy dissipation, steady states play a central role to understand the network formation capacity of the system. We show that for a large diffusion coefficient D, the zero steady state is stable, while network formation occurs for small values of D due to the instability of the zero steady state, and the borderline case D = 0 exhibits a large class of dynamically stable (in the linearized sense) steady states.