Mathematical Models and Methods in Applied Sciences

Skeins (one-dimensional queues) of migrating birds show typical fluctuations in swarm length and frequent events of "condensation waves" starting at the leading bird and traveling backward within the moving skein, similar to queuing traffic waves in car files but more smooth.

These dynamical phenomena can be fairly reproduced by stochastic ordinary differential equations for a "multi-particle" system including the individual tendency of birds to attain a preferred speed as well as mutual interaction "forces" between neighbors, induced by distance-dependent attraction or repulsion as well as adjustment of velocities. Such a one-dimensional system constitutes a so-called "stochastic viscoelastic skein."

For the simple case of nearest neighbor interactions we define the density between individualsu = u(t, x) as a step function inversely proportional to the neighbor distance, and the velocity function v = v(t, x) as a standard piecewise linear interpolation between individual velocities. Then, in the limit of infinitely many birds in a skein of finite length, with mean neighbor distance δ converging to zero and after a suitable scaling, we obtain continuum mass and force balance equations that constitute generalized nonlinear compressible Navier–Stokes equations. The resulting density-dependent stress functions and viscosity coefficients are directly derived from the parameter functions in the original model.

We investigate two different sources of additive noise in the force balance equations: (1) independent stochastic accelerations of each bird and (2) exogenous stochastic noise arising from pressure perturbations in the interspace between them. Proper scaling of these noise terms leads, under suitable modeling assumptions, to their maintenance in the continuum limit δ → 0, where they appear as (1) uncorrelated spatiotemporal Gaussian noise or, respectively, (2) certain spatially correlated stochastic integrals. In both cases some a priori estimates are given which support convergence to the resulting stochastic Navier–Stokes system.

Natural conditions at the moving swarm boundaries (along characteristics of the hyperbolic system) appear as singularly perturbed zero-tension Neumann conditions for the velocity function v. Numerical solutions of this free boundary value problem are compared to multi-particle simulations of the original discrete system. By analyzing its linearization around the constant swarm state, we can characterize several properties of swarm dynamics. In particular, we compute approximating values for the averaged speed and length of typical condensation waves.

The discrete Couzin–Vicsek algorithm (CVA), which describes the interactions of individuals among animal societies such as fish schools is considered. In this paper, a kinetic (mean-field) version of the CVA model is proposed and its formal macroscopic limit is provided. The final macroscopic model involves a conservation equation for the density of the individuals and a non-conservative equation for the director of the mean velocity and is proved to be hyperbolic. The derivation is based on the introduction of a non-conventional concept of a collisional invariant of a collision operator.

This paper deals with the mathematical modelling of crowd dynamics within the framework of continuum mechanics. The method uses the mass conservation equation closed by phenomenological models linking the local velocity to density and density gradients. The closures take into account movement in more than one space dimension, presence of obstacles, pedestrian strategies, and modelling of panic conditions. Numerical simulations of the initial-boundary value problems visualize the ability of the models to predict several interesting phenomena related to the complex system under consideration.

Motivated by empirical observations of spatio-temporal clusters of crime across a wide variety of urban settings, we present a model to study the emergence, dynamics, and steady-state properties of crime hotspots. We focus on a two-dimensional lattice model for residential burglary, where each site is characterized by a dynamic attractiveness variable, and where each criminal is represented as a random walker. The dynamics of criminals and of the attractiveness field are coupled to each other via specific biasing and feedback mechanisms. Depending on parameter choices, we observe and describe several regimes of aggregation, including hotspots of high criminal activity. On the basis of the discrete system, we also derive a continuum model; the two are in good quantitative agreement for large system sizes. By means of a linear stability analysis we are able to determine the parameter values that will lead to the creation of stable hotspots. We discuss our model and results in the context of established criminological and sociological findings of criminal behavior.

Recently, Berthelin et al.,⁴ introduced a traffic-flow model describing the formation and the dynamics of traffic jams. This model which consists of a Constrained Pressureless Gas Dynamics system assumes that the maximal density constraint is independent of the velocity. However, in practice, the distribution of vehicles on a highway depends on their velocity. In this paper, a more realistic model namely the Second Order Model with Constraints (in short SOMC) is proposed, derived from the Aw and Rascle model,¹ which takes into account this feature. Moreover, when the maximal density constraint is saturated, the SOMC model "relaxes" to the Lighthill and Whitham model.²⁰ An existence result of weak solutions for this model by means of cluster dynamics is proved in order to construct a sequence of approximations, and the associated Riemann problem is solved completely.

Some models of flow on a network are discussed. Assuming a macroscopic approach on each arc of the network, we consider a system of conservation laws and various possible choices to describe the evolution at vertices are discussed. A general framework proposed in recent literature is presented, then some new solutions for the scalar case are proposed and analyzed.

This paper, that deals with the modelling of crowd dynamics, is the first one of a project finalized to develop a mathematical theory refereing to the modelling of the complex systems constituted by several interacting individuals in bounded and unbounded domains. The first part of the paper is devoted to scaling and related representation problems, then the macroscopic scale is selected and a variety of models are proposed according to different approximations of the pedestrian strategies and interactions. The second part of the paper deals with a qualitative analysis of the models with the aim of analyzing their properties. Finally, a critical analysis is proposed in view of further development of the modelling approach. Additional reasonings are devoted to understanding the conceptual differences between crowd and swarm modelling.