Joep H. M. Evers

Anisotropic interactions in a first-order aggregation model

We extend a well studied ODE model for collective behaviour by considering anisotropic interactions among individuals. Anisotropy is modelled by limited sensorial perception of individuals, that depends on their current direction of motion. Consequently, the first-order model becomes implicit, and new analytical issues, such as non-uniqueness and jump discontinuities in velocities, are raised. We study the well-posedness of the anisotropic model and discuss its modes of breakdown. To extend solutions beyond breakdown we propose a relaxation system containing a small parameter e, which can be interpreted as a small amount of inertia or response time. We show that the limit e → 0 can be used as a jump criterion to select the physically correct velocities. In smooth regimes, the convergence of the relaxation system as e → 0 is guaranteed by a theorem due to Tikhonov. We illustrate the results with numerical simulations in two dimensions.

Cognitive distance, absorptive capacity and group rationality: a simulation study.

We report the results of a simulation study in which we explore the joint effect of group absorptive capacity (as the average individual rationality of the group members) and cognitive distance (as the distance between the most rational group member and the rest of the group) on the emergence of collective rationality in groups. We start from empirical results reported in the literature on group rationality as collective group level competence and use data on real-life groups of four and five to validate a mathematical model. We then use this mathematical model to predict group level scores from a variety of possible group configurations (varying both in cognitive distance and average individual rationality). Our results show that both group competence and cognitive distance are necessary conditions for emergent group rationality. Group configurations, in which the groups become more rational than the most rational group member, are groups scoring low on cognitive distance and scoring high on absorptive capacity.

Crowds reaching targets by maximizing entropy : a Clausius-Duhem inequality approach

In this paper we propose the use of concepts from thermodynamics in the study of crowd dynamics. Our continuous model consists of the continuity equation for the density of the crowd and a kinetic equation for the velocity field. The latter includes a nonlocal term that models interactions between individuals. To support our modelling assumptions, we introduce an inequality that resembles the Second Law of Thermodynamics, containing an entropy-like functional. We show that its time derivative equals a positive dissipation term minus a corrector term. The latter term should be small for the time derivative of the entropy to be positive. In case of isotropic interactions the corrector term is absent. For the anisotropic case, we support the claim that the corrector term is small by simulations for the corresponding particle system. They reveal that this term is sufficiently small for the entropy still to increase. Moreover, we show that the entropy converges in time towards a limit value.

SMALL INERTIA REGULARIZATION OF AN ANISOTROPIC AGGREGATION MODEL

We consider an anisotropic first-order ODE aggregation model and its approximation by a second-order relaxation system. The relaxation model contains a small parameter

Modelling with measures : approximation of a mass-emitting object by a point source

\varepsilon

Metastable States for an Aggregation Model with Noise

, which can be interpreted as inertia or response time. We examine rigorously the limit

Anisotropic interactions in a first-order aggregation model : a proof of concept

\varepsilon \to 0

Equilibria for an Aggregation Model with Two Species

of solutions to the relaxation system. Of major interest is how discontinuous (in velocities) solutions to the first-order model are captured in the zero-inertia limit. We find that near such discontinuities, solutions to the second-order model perform fast transitions within a time layer of size

The effect of perception anisotropy on particle systems describing pedestrian flows in corridors

\mathcal{O}(\varepsilon^{2/3})

Ring states in swarmalator systems

. We validate this scale with numerical simulations.