Emergent behaviour in multi-particle systems with non-local interactions (12w5041)

Objectives

The purpose of this workshop is to disseminate recent advances in
theory and applications of multi-particle models in mathematics,
biology, engineering and other fields. One of the goals is to bring
together leading experts representing the diverse groups that are
working on multi-particle models, both from the point of view of
modelling and mathematical techniques. To accomplish this, we seek
active participation not only from mathematicians but equally from
experimentalists, particularly in biology and robotics.

The use of nonlocal models is becoming widespread in both modelling
and theory. From biological point of view, identifying the proper
mechanisms for swarm formations among the many possible scenarios is
a challenging problem. Equally important is to identify the benefits
that the swarming behaviour confers on induvidual members or the
species as a whole. Mathematically, even the simplest models can
lead to novel phenomena which are still in the process of being
understood. Below we list the recent directions and some of the
outstanding questions in this field.

1. Pattern formation for biological fitness/optimality.

Many animals swarm and form complex spatio-temporal patterns.
Examples include streaming bacteria, foraging ants, rolling swarms
of locusts, highly aligned schools of fish, flocks of birds, and
migrating ungulates. Emergent swarming patterns arise on large
spatial scales that are far beyond those for individual
interactions. There is a rich history of mathematical modeling.
To date many models have focused on translating from local
interactions to emergent patterns. Models include stochastic
processes and nonlinear partial different equations (both
parabolic and hyperbolic), and integrodifferential equations.
However, the emergent patterns can also confer a fitness
advantages to individuals, whether improved foraging and
searching, defence against predators, or enhanced social
interactions. This coupling of emergent pattern formation to
individual fitness is a new and rich area of research. We
anticipate that the BIRS workshop will provide an opportunity to
develop this theme in new mathematical directions.

2. Blowup solutions and concentration phenomena:

Blowup typically refers to concentration of mass in finite or
infinite time and is important in many applications. For example
black hole formation is described by a finite time singularity in
the equations describing gravitational collapse. Convergence of
particles and/or their velocities is often referred to as
"consensus" in the control theory literature and this behavior has
received alot of attention in the last five years. In the case of
kinematic aggregation problems, smooth solutions exhibit finite
time blowup with self-similar structure "of the second kind"
meaning that scaling behavior can not be predicted from
dimensional analysis. This poses some interesting open problems
regarding the asymptotics of blowup. Related questions arise in
models with diffusion - not only dynamics of blowup but also
critical exponents for such problems.

Another type of blowup occurs when solutions concentrate on sets
of lower dimension, such as curves in 2D. This is observed for
example in fish schools which tend to form milling formations,
consisting of individual fish rotating along a circle. Some recent
results about the local stability of ring solutions helps to
classify some of the observed patterns, but many open questions
remain. Questions like global stability and boundedness of steady
states are poorly understood, and only some partial results are
available so far.

3. Criticality, and phase transitions.

An intriguiging phenomenon is the state transitions that swarms
may experience when system parameters are gradually changed. For
example, noise-induced transitions may bring a swarm from a
translational to a rotational state. The study of the role of
noise in swarm models is currently an active area or research.
Recent works showed that intrinsic noise within a swarm can
facilitate coherence in a collective motion, helping explain for
instance the spontanteneous direction switching of the whole
swarm. Several participants in the proposed BIRS workshop have
worked on this topic and will disseminate their knowledge to the
others.

4. Second order models.

Microscopic or individual-based models are typically described by
large systems of ordinary differential equations that govern the
evolution of each individual as it interacts with the other
members of the group. Most of such models are second-order, as
they involve the acceleration of the particles through the
Newton's second law. Macroscopic/continuum models cast the
problem as a partial differential equation for the dynamics of the
population density field. The macroscopic and microscopic
approaches can be connected through a mesoscopic description based
on kinetic equations. The latter approach has recently started to
include various realistic aspects of aggregation such as
orientation, attraction and repulsion and it has grown into an
actively researched topic. At the proposed workshop we aim to have
a diverse group of participants, representing all the three
descriptions of aggregation modeling, and have a fruitful exchange
of ideas regarding the interplay between them, their advantages
and disadvatanges over each other, their ability to capture
biologically realistic aggregations, etc.

In addition to traditional talks, we anticipate to have a series of
lessons by well-known experts in the field. There are multiple groups
working on related topics across several disciplines; as such we
request the full 42 slots for this workshop. We have already
contacted about 30 collegues who have indicated their desire to
participate.

We encourage a strong participation from younger researchers,
including graduate students and postdocs. The list of speakers
reflects this; about a third of the participants fit in that
category.