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Networks & Heterogeneous Media

March 2021 , Volume 16 , Issue 1

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A 2-dimensional shape optimization problem for tree branches
Alberto Bressan and Sondre Tesdal Galtung
2021, 16(1): 1-29 doi: 10.3934/nhm.2020031 +[Abstract](1076) +[HTML](323) +[PDF](681.95KB)
Abstract:

The paper is concerned with a shape optimization problem, where the functional to be maximized describes the total sunlight collected by a distribution of tree leaves, minus the cost for transporting water and nutrient from the base of trunk to all the leaves. In a 2-dimensional setting, the solution is proved to be unique and explicitly determined.

Properties of the LWR model with time delay
Simone Göttlich, Elisa Iacomini and Thomas Jung
2021, 16(1): 31-47 doi: 10.3934/nhm.2020032 +[Abstract](936) +[HTML](192) +[PDF](1281.65KB)
Abstract:

In this article, we investigate theoretical and numerical properties of the first-order Lighthill-Whitham-Richards (LWR) traffic flow model with time delay. Since standard results from the literature are not directly applicable to the delayed model, we mainly focus on the numerical analysis of the proposed finite difference discretization. The simulation results also show that the delay model is able to capture Stop & Go waves.

Pointwise long time behavior for the mixed damped nonlinear wave equation in $ \mathbb{R}^n_+ $
Linglong Du and Min Yang
2021, 16(1): 49-67 doi: 10.3934/nhm.2020033 +[Abstract](764) +[HTML](173) +[PDF](549.73KB)
Abstract:

In this paper, we investigate the long time behavior of the solution for the nonlinear wave equation with frictional and visco-elastic damping terms in \begin{document}$ \mathbb{R}^n_+ $\end{document}. It is shown that for the long time, the frictional damped effect is dominated. The Green's functions for the linear initial boundary value problem can be described in terms of the fundamental solutions for the full space problem and reflected fundamental solutions coupled with the boundary operator. Using the Duhamel's principle, we get the pointwise long time behavior of the solution \begin{document}$ \partial_{{\bf{x}}}^{\alpha}u $\end{document} for \begin{document}$ |\alpha|\le 1 $\end{document}.

A two-dimensional multi-class traffic flow model
Caterina Balzotti and Simone Göttlich
2021, 16(1): 69-90 doi: 10.3934/nhm.2020034 +[Abstract](879) +[HTML](188) +[PDF](5537.77KB)
Abstract:

The aim of this work is to introduce a two-dimensional macroscopic traffic model for multiple populations of vehicles. Starting from the paper [21], where a two-dimensional model for a single class of vehicles is proposed, we extend the dynamics to a multi-class model leading to a coupled system of conservation laws in two space dimensions. Besides the study of the Riemann problems we also present a Lax-Friedrichs type discretization scheme recovering the theoretical results by means of numerical tests. We calibrate the multi-class model with real data and compare the fitted model to the real trajectories. Finally, we test the ability of the model to simulate the overtaking of vehicles.

A new model for the emergence of blood capillary networks
Pedro Aceves-Sanchez, Benjamin Aymard, Diane Peurichard, Pol Kennel, Anne Lorsignol, Franck Plouraboué, Louis Casteilla and Pierre Degond
2021, 16(1): 91-138 doi: 10.3934/nhm.2021001 +[Abstract](935) +[HTML](184) +[PDF](1922.46KB)
Abstract:

We propose a new model for the emergence of blood capillary networks. We assimilate the tissue and extra cellular matrix as a porous medium, using Darcy's law for describing both blood and interstitial fluid flows. Oxygen obeys a convection-diffusion-reaction equation describing advection by the blood, diffusion and consumption by the tissue. Discrete agents named capillary elements and modelling groups of endothelial cells are created or deleted according to different rules involving the oxygen concentration gradient, the blood velocity, the sheer stress or the capillary element density. Once created, a capillary element locally enhances the hydraulic conductivity matrix, contributing to a local increase of the blood velocity and oxygen flow. No connectivity between the capillary elements is imposed. The coupling between blood, oxygen flow and capillary elements provides a positive feedback mechanism which triggers the emergence of a network of channels of high hydraulic conductivity which we identify as new blood capillaries. We provide two different, biologically relevant geometrical settings and numerically analyze the influence of each of the capillary creation mechanism in detail. All mechanisms seem to concur towards a harmonious network but the most important ones are those involving oxygen gradient and sheer stress. A detailed discussion of this model with respect to the literature and its potential future developments concludes the paper.

Periodic consensus in network systems with general distributed processing delays
Yicheng Liu, Yipeng Chen, Jun Wu and Xiao Wang
2021, 16(1): 139-153 doi: 10.3934/nhm.2021002 +[Abstract](742) +[HTML](183) +[PDF](4743.94KB)
Abstract:

How to understand the dynamical consensus patterns in network systems is of particular significance in both theories and applications. In this paper, we are interested in investigating the influences of distributed processing delay on the consensus patterns in a network model. As new observations, we show that the desired network model undergoes both weak consensus and periodic consensus behaviors when the parameters reach a threshold value and the connectedness of the network system may be absent. In results, some criterions of weak consensus and periodic consensus with exponential convergent rate are established by the standard functional differential equations analysis. An analytic formula is given to calculate the asymptotic periodic consensus in terms of model parameters and the initial time interval. Also, we post the threshold values for some typical distributions included uniform distribution and Gamma distribution. Finally, we give the numerical simulation and analyse the influences of different delays on the consensus.

2020 Impact Factor: 1.213
5 Year Impact Factor: 1.384
2020 CiteScore: 1.9

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