Murray's law for discrete and continuum models of biological networks

Rigorous Continuum Limit for the Discrete Network Formation Problem

Motivated by recent physics papers describing the formation of biological transport networks we study a discrete model proposed by Hu and Cai consisting of an energy consumption function constrained by a linear system on a graph. For the spatially two-dimensional rectangular setting we prove the rigorous continuum limit of the constrained energy functional as the number of nodes of the underlying graph tends to infinity and the edge lengths shrink to zero uniformly. The proof is based on reformulating the discrete energy functional as a sequence of integral functionals and proving their $Γ$-co…

Global existence of strong solutions to a biological network formulation model in $2+1$ dimensions

In this paper we study the initial boundary value problem for the system\\ $-\mbox{div}\left[(I+\mathbf{m} \mathbf{m}^T)\nabla p\right]=s(x),\ \ \mathbf{m}_t-α^2Δ\mathbf{m}+|\mathbf{m}|^{2(γ-1)}\mathbf{m}=β^2(\mathbf{m}\cdot\nabla p)\nabla p$ in two space dimensions. This problem has been proposed as a continuum model for biological transportation networks. The mathematical challenge is due to the presence of cubic nonlinearities, also known as trilinear forms, in the system. We obtain a weak solution $(\mathbf{m},p) $ with both $|\nabla p|$ and $|\nabla\mathbf{m}|$ being bounded. The result i…

基于微分方程和偏微分方程的生物运输网络建模

ODE and PDE based modeling of biological transportation networks

本文研究了基于图的离散（ODE基于）模型中解的全局存在的问题，提出了一种适应性模型，使其可以得到宏观（PDE基于）系统的正式连续极限，并证明了宏观PDE模型弱解的全局存在性，最后通过离散模型的数值模拟展示了向稳态收敛、稳态的非唯一性以及其对初始数据和模型参数的依赖性。

源自生物网络模型的 $p$-Laplacian 梯度流：一种新的动力学松弛方法

Gradient Flows for the $p$-Laplacian Arising from Biological Network Models: A Novel Dynamical Relaxation Approach

本文从生物运输网络的离散图模型出发，推导连续能量泛函，提出一种动力学松弛方案，高效求解 $p$-Laplacian 方程并再现生物网络模式。

The semilinear heat equation on sparse random graphs

Using the theory of $L^p$-graphons (Borgs et al, 2014), we derive and rigorously justify the continuum limit for systems of differential equations on sparse random graphs. Specifically, we show that the solutions of the initial value problems for the discrete models can be approximated by those of an appropriate nonlocal diffusion equation. Our results apply to a range of spatially extended dynamical models of different physical, biological, social, and economic networks. Importantly, our assumptions cover network topologies featured in many important real-world networks. In particular, we der…

排斥-吸引幂律相互作用能的全局极小元

On global minimizers of repulsive-attractive power-law interaction energies

我们认为，在许多生物和物理的情况下发生的幂律排斥吸引相互作用能的最小化。我们证明了在离散情形下全局极小解的存在性，并在一定的指数范围内得到了它们的支持度与Dirac Delta数无关的界。这些全局离散极小值对应于群集模型中群体模式的稳定空间分布。利用紧性得到了连续问题的全局极小解。我们还通过数值模拟来说明我们的结果。