This paper proposes a survey and critical analysis focused on a variety of chemotaxis models in biology, namely the classical Keller–Segel model and its subsequent modifications, which, in several cases, have been developed to obtain models that prevent the non-physical blow up of solutions. The presentation is organized in three parts. The first part focuses on a survey of some sample models, namely the original model and some of its developments, such as flux limited models, or models derived according to similar concepts. The second part is devoted to the qualitative analysis of analytic problems, such as the existence of solutions, blow-up and asymptotic behavior. The third part deals with the derivation of macroscopic models from the underlying description, delivered by means of kinetic theory methods. This approach leads to the derivation of classical models as well as that of new models, which might deserve attention as far as the related analytic problems are concerned. Finally, an overview of the entire contents leads to suggestions for future research activities.

Toward a mathematical theory of Keller–Segel models of pattern formation in biological tissues

We studied the Brownian motion of isolated ellipsoidal particles in water confined to two dimensions and elucidated the effects of coupling between rotational and translational motion. By using digital video microscopy, we quantified the crossover from short-time anisotropic to long-time isotropic diffusion and directly measured probability distributions functions for displacements. We confirmed and interpreted our measurements by using Langevin theory and numerical simulations. Our theory and observations provide insights into fundamental diffusive processes, which are potentially useful for understanding transport in membranes and for understanding the motions of anisotropic macromolecules.

/pdf/brownian-motion-of-an-ellipsoid-2fpjkl3hr9.pdf

Brownian Motion of an ellipsoid

This paper presents a review and critical analysis on the modeling of the dynamics of vehicular traffic, human crowds and swarms seen as living and, hence, complex systems. It contains a survey of ...

/pdf/vehicular-traffic-crowds-and-swarms-from-kinetic-theory-and-2e3l5tqiwc.pdf

Vehicular traffic, crowds, and swarms: From kinetic theory and multiscale methods to applications and research perspectives

Human behaviours in evacuation crowd dynamics: From modelling to "big data" toward crisis management.

This paper deals with a micro–macro derivation of a variety of cross-diffusion models for a large system of active particles. Some of the models at the macroscopic scale can be viewed as developmen...

From a multiscale derivation of nonlinear cross-diffusion models to Keller–Segel models in a Navier–Stokes fluid

In this paper scalar macroscopic models for traffic and pedestrian flows are coupled and the resulting system is investigated numerically. For the traffic flow the classical Lighthill–Whitham Richards model on a network of roads and for the pedestrian flow the Hughes model are used. These models are coupled via terms in the fundamental diagrams modeling an influence of the traffic and pedestrian flow on the maximal velocities of the corresponding models. Several physical situations, where pedestrians and cars interact, are investigated.

/pdf/coupling-traffic-flow-networks-to-pedestrian-motion-4intozyjdr.pdf

Coupling traffic flow networks to pedestrian motion

Nonlocal conservation laws are used to describe various realistic instances of crowd behaviors. First, a basic analytic framework is established through an ad hoc well-posedness theorem for systems of nonlocal conservation laws in several space dimensions interacting nonlocally with a system of ODEs. Numerical integrations show possible applications to the interaction of different groups of pedestrians and also with other agents.

Differential Equations Modeling Crowd Interactions

Microscopic and macroscopic models for coupled car traffic and pedestrian flow

A hierarchy of models for pedestrian flow with fixed speed is numerically investigated. The starting point is a microscopic model based on a stochastic interacting particle system coupled to an eikonal equation. Starting from this model a nonlocal and nonlinear flux-limited maximum-entropy equation for density and mean velocity is derived via a mean field kinetic equation. Finally, associated classical scalar equations for the density are considered for comparison. These models are compared to each other for different test cases showing the superiority of the flux-limited approach, in particular for situations with smaller values of the stochastic noise.

A numerical investigation of flux-limited approximations for pedestrian dynamics

In the present paper we discuss the coupling of traffic flow with pedestrian motion.
First we review the coupling of the Lighthill-Whitham model for road traffic and the Hughes pedestrian model as presented in [5].
In different numerical examples we investigate the mutual interaction of both dynamics.
A special focus is given on the possible placement of crosswalks to facilitate the passage for the pedestrians over a crowded street.

Anne Meurer

Papers

Coupling traffic flow networks to pedestrian motion

Differential Equations Modeling Crowd Interactions

Microscopic and macroscopic models for coupled car traffic and pedestrian flow

A numerical investigation of flux-limited approximations for pedestrian dynamics

Interaction of road networks and pedestrian motion at crosswalks