Pedestrians moving in the dark: Balancing measures and playing games on lattices

Abstract

We present two conceptually new modeling approaches aimed at describing the motion of pedestrians in obscured corridors:(i) a Becker-Döring-type dynamics and (ii) a probabilistic cellular automaton model. In both models the group formation is affected by a threshold. The pedestrians are supposed to have very limited knowledge about their current position and their neighborhood; they can form groups up to a certain size and they can leave them. Their main goal is to find the exit of the corridor.Although being of mathematically different character, the discussion of both models shows that it seems to be a disadvantage for the individual to adhere to larger groups.We illustrate this effect numerically by solving both model systems. Finally we list some of our main open questions and conjectures.

Free to move or trapped in your group: Mathematical modeling of information overload and coordination in crowded populations

“…The existence of the underlying measures is a matter that will be investigated at a later stage, in a different context. As in [34], we can point out a direct connection with a suitable Becker-Döring-like model for colloidal interactions (cf. e.g.…”

“…The mass (finite) measures, which are responsible for handling the information pool, are a closed subset in this Banach space. Much of the theory of ordinary differential equations applies directly to dynamics in Banach spaces and therefore to measure-valued differential equations; see [30] for measure-valued equations offering descriptions of the evolution of an unstructured language or our recent work on cccm-ps-prep.tex crowd dynamics [22,23,34], where the mass measures evolve in a bounded environment where boundary interactions are allowed. A nice account of measure-theoretical models in the context of crowd dynamics and traffic flows can be found in [18].…”

“…We intend to provide here a general modeling framework for building "features"-based continuity-like equations for size-ordered interacting populations, which should be seen as an alternative of working with local balance laws using densities. The presentation recalls from [34]. In this context, the "location in the corridor" and the "group size" constitute the two "features"; we refer the reader to [21] for a related reading.…”

“…This leads to a loss for the groups of size i (modeled by −α i u i ) as well as to a gain for the groups of size i − 1 and a gain for the groups with i = 1 (modeled by N i=2 α i u i ). In simple crowd simulations [34], the coefficients α i , β i ≥ 0 and D i > 0 can be assumed to be constant. It is interesting to note that the fragmentation terms express a flux rather than a volume source or sink.…”

A lattice model for active-passive pedestrian dynamics: a quest for drafting effects

“…Such scenarios are encountered for instance when catastrophic situations occur in urban environments (e.g., in large office spaces), in tunnels, within underground spaces and/or in forests in fire. We refer the reader, for instance, to [2,27,28,33,34] and references cited therein, as well as to our previous results [8,[11][12][13][14]29].…”