2,523 research outputs found
A Model for Collective Dynamics in Ant Raids
Ant raiding, the process of identifying and returning food to the nest or
bivouac, is a fascinating example of collective motion in nature. During such
raids ants lay pheromones to form trails for others to find a food source. In
this work a coupled PDE/ODE model is introduced to study ant dynamics and
pheromone concentration. The key idea is the introduction of two forms of ant
dynamics: foraging and returning, each governed by different environmental and
social cues. The model accounts for all aspects of the raiding cycle including
local collisional interactions, the laying of pheromone along a trail, and the
transition from one class of ants to another. Through analysis of an order
parameter measuring the orientational order in the system, the model shows
self-organization into a collective state consisting of lanes of ants moving in
opposite directions as well as the transition back to the individual state once
the food source is depleted matching prior experimental results. This indicates
that in the absence of direct communication ants naturally form an efficient
method for transporting food to the nest/bivouac. The model exhibits a
continuous kinetic phase transition in the order parameter as a function of
certain system parameters. The associated critical exponents are found,
shedding light on the behavior of the system near the transition.Comment: Preprint Version, 30 pgs., 18 figures, complete version with
supplementary movies to appear in Journal of Mathematical Biology (Springer
A Compartmental Model for Traffic Networks and its Dynamical Behavior
We propose a macroscopic traffic network flow model suitable for analysis as
a dynamical system, and we qualitatively analyze equilibrium flows as well as
convergence. Flows at a junction are determined by downstream supply of
capacity as well as upstream demand of traffic wishing to flow through the
junction. This approach is rooted in the celebrated Cell Transmission Model for
freeway traffic flow. Unlike related results which rely on certain system
cooperativity properties, our model generally does not possess these
properties. We show that the lack of cooperativity is in fact a useful feature
that allows traffic control methods, such as ramp metering, to be effective.
Finally, we leverage the results of the paper to develop a linear program for
optimal ramp metering
On a Small Elliptic Perturbation of a Backward-Forward Parabolic Problem, with Applications to Stochastic Models
We consider an elliptic PDE in two variables. As one parameter approaches
zero, this PDE collapses to a parabolic one, that is forward parabolic in a
part of the domain and backward parabolic in the remainder. Such problems arise
naturally in various stochastic models, such as fluid models for data-handling
systems and Markov-modulated queues. We employ singular perturbation methods to
study the problem for small values of the parameter
Entropy solutions for a traffic model with phase transitions
In this paper, we consider the two phases macroscopic traffic model
introduced in [P. Goatin, The Aw-Rascle vehicular traffic flow with phase
transitions, Mathematical and Computer Modeling 44 (2006) 287-303]. We first
apply the wave-front tracking method to prove existence and a priori bounds for
weak solutions. Then, in the case the characteristic field corresponding to the
free phase is linearly degenerate, we prove that the obtained weak solutions
are in fact entropy solutions \`a la Kruzhkov. The case of solutions attaining
values at the vacuum is considered. We also present an explicit numerical
example to describe some qualitative features of the solutions
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