22-27 January 2012.

Click here for participant list.

**Abstracts: click on links below
to see the PDF of the talk (if available) **

Martial Agueh
*A refined flocking and swarming model of Cucker-Smale type*
The Cucker-Smale model is a mathematical model of swarming that leads to
(unrealistic!) unconditional flocking of all the birds in the swarm. In this
work, we generalize this model to scenarios where a typical bird is subject to
a friction force driving it to fly at optimal speed, a repulsive short-range
force to avoid collisions, an attractive "flocking" force which takes into
account a cone of vision of the bird, and a boundary force to bring the bird
back inside the swarm if it is on the edge flying outward. We present the
particle model system, derive the associated kinetic equation, and study its
well-posedness. Finally, we show some numerical simulations of the model. Our
simulations confirm that breakup of a swarm does occur, contrarily to the
standard Cucker-Smale model.
This is a joint work with R. Illner and A. Richardson.

Daniel Balague
*Stationary states for the aggregation equation with power law
attractive-repulsive potentials*
We analyze the stability of a uniform distribution on a sphere as stationary
solution of the aggregation equation with power law attractive-repulsive
potentials. We give a sharp condition that establishes its stability under
radial perturbations.

Alethia Barbaro
* A phase transition in a kinetic Cucker-Smale model with friction*
We consider the steady-states of a kinetic Cucker-Smale model with friction. As we vary the noise and friction coefficients, we find that a phase transition occurs. For small values of the noise coefficient, there are three steady-state solutions while for larger values, there is only one.

*Global Existence and Finite Time Blow-Up for Critical
Patlak-Keller-Segel Models with Inhomogeneous Diffusion*
The $L^{1}$-critical parabolic-elliptic Patlak-Keller-Segel system is a
classical model of chemotactic aggregation in micro-organisms well-known to
have critical mass phenomena. In this paper we study this critical mass
phenomenon in the context of Patlak-Keller-Segel models with spatially varying
diffusivity of the chemo-attractant in three dimensions and higher. The
critical mass is identified to depend only on the local value of the
diffusivity and finite time blow-up results show it to be sharp under certain
conditions. The methods also provide new blow-up results for homogeneous
problems, showing that there exist blow-up solutions with arbitrarily large
(positive) initial free energy.

*A Primer of Swarm Equilibria.*
We study equilibrium configurations of swarming biological organisms subject
to exogenous and pairwise endogenous forces. Beginning with a discrete
dynamical model, we derive a variational description of the continuum
population density. Equilibrium solutions are extrema of an energy functional,
and satisfy a Fredholm integral equation. We find conditions for the extrema
to be local minimizers, global minimizers, and minimizers with respect to
infinitesimal Lagrangian displacements of mass. In one spatial dimension, for
a variety of exogenous forces, endogenous forces, and domain configurations,
we find exact analytical expressions for the equilibria. These agree closely
with numerical simulations of the underlying discrete model. The exact
solutions provide a sampling of the wide variety of equilibrium configurations
possible within our general swarm modeling framework. The equilibria typically
are compactly supported and may contain d-concentrations or jump
discontinuities at the edge of the support. In two-dimensions we show that the
Morse Potential and other "pointy" potentials can generically lead to inverse
square-root singularities in the density at the boundary of the swarm support.
Joint work with Louis Ryan and Chad M. Topaz.

*Swarming and Aggregation Equations*
Abstract: This lecture is an introduction to the interesting phenomena of
swarming and to the open problems in this area. I will review numerical and
analytical results for both kinematic and dynamic aggregation equations. I
will discuss how models are constructed and the emergence of phenomenological
behavior for different types of models including flocking, milling, and other
patterns. I will also review some results on well-posedness of aggregation
equations including a sharp condition on blowup from smooth initial data. This
talk will include joint work with Bedrossian, Brandman, Carrillo, Laurent,
Rodriguez, and Slepcev.

Bjorn Birnir
* Dynamic Energy Budget Theory and the Environment*
Dynamics Energy Budget Theory (DEB) can be used to model the physiology of
animals and how it influences their interactions with the environment. We
explain with the example of the Icelandic capelin how changes in physiology
can trigger entirely different group behavior influencing migration patterns
over large distances.

Jose Antonio Carrillo * A review of 2nd order models for
swarming*
I will present several examples of the derivation by means of kinetic
theory arguments of kinetic equations for swarming. All of them arise
from individual based models in the recent literature. For instance, the
system of interacting, self-propelled discrete particles by D'Orsogna
etal. Starting from the particle model, one can construct solutions to a
Vlasov-like kinetic equation for the single particle probability
distribution function using distances between measures. Another example
is the continuous kinetic version of flocking by Cucker and Smale.
The large-time behavior of the distribution in phase space is
subsequently studied by means of particle approximations and a stability
property in distances between measures. A continuous analogue of the
theorems of Cucker-Smale will be shown to hold for the solutions on the
kinetic model. More precisely, the solutions concentrate exponentially
fast their velocity to their mean while in space they will converge
towards a translational flocking solution. The mean field limit
with/without noise for these models will also be discussed. Issues
related to particular solutions such as flocks and mills will also be
investigated. Hydrodynamic systems will also be formally derived.

*Formation of Animal Groups: The Importance of Communication*
We investigate the formation and movement of self-organizing collectives of
individuals in homogeneous environments. We review a hyperbolic system of
conservation laws based on the assumption that the interactions governing
movement depend not only on distance between individuals, but also on whether
neighbours move towards or away from the reference individual. The inclusion
of direction-dependent communication mechanisms significantly enriches the
model behavior; the model exhibits classical patterns such as stationary
pulses and traveling trains, but also novel patterns such as zigzag pulses,
breathers, and feathers. The same enrichment of model behavior is observed
when we include direction-dependent communication mechanisms in
individual-based models.

Maria D'Orsogna
*Stochastic nucleation and growth of particle clusters*
The binding of individual components to form composite structures is a
ubiquitous phenomenon within the sciences. Within heterogeneous nucleation,
particles may be attracted to an initial exogenous site: the formation of
droplets, aerosols and crystals usually begins around impurities or
boundaries. Homogeneous nucleation on the other hand describes identical
particles spontaneously clustering upon contact.
Given their ubiquity in physics, chemistry and material sciences, nucleation
and growth have been extensively studied in the past decades, often assuming
infinitely large numbers of building blocks and unbounded cluster sizes. These
assumptions also led to the use of mass-action, mean field descriptions such
as the well known Becker Doering equations.
In cellular biology, however, nucleation events often take place in confined
spaces, with a finite number of components, so that discreteness and
stochastic effects must be taken into account.
In this talk we examine finite sized homogeneous nucleation by considering a
fully stochastic master equation, solved via Monte-Carlo simulations and via
analytical insight. We find striking differences between the mean cluster
sizes obtained from our discrete, stochastic treatment and those predicted by
mean field treatments.
We also consider heterogeneous nucleation stochastic treatments, first passage
time results and possible applications to prion unfolding and clustering dynamics.

Baldvin Einarsson
*Noise Driven Solutions of Schooling Fish and A Cellular Automata Model
for Biofilm Growth with Surface Flow*
The talk will be in two parts. In the first part, we will describe the model
from Birnir:2007 and some of the solutions. We will then show that the average
velocity of the particles tends to zero under most conditions. We therefore
propose to add noise to the model and investigate whether some of the
described structure of the solutions is maintained.
For the second part, we describe a two dimensional cellular automata model for
biofilm growth in a rectangular tube with nonlaminar surface flow. Nutrient
levels and structure of the biofilm determine the probability of the following
mechanisms: i) Cell division and spreading ii) Cell erosion due to sheer
stress iii) Production of EPS (extracellular polymeric substances) iv) Influx
of cells which adhere to the surface and biofilm. We describe these mechanisms
and the numerical code used to simulate the model. We then show how the model
reproduces biofilm development in the form of flat biofilms, ripples,
streamers, towers, \textquotedblright mushroom\textquotedblright\ growth etc.
Finally, we briefly describe the effect of rugosity and an extension to three dimensions.

*The role of communication mechanisms on the movement of self-organised
aggregations*
The formation, persistence and movement of self-organised biological
aggregations are mediated by signals (e.g., visual, acoustic or chemical) that
organisms use to communicate with each other. To investigate the effect that
different communication mechanisms have on the movement of biological
aggregations, we use a class of nonlocal hyperbolic models that incorporate
social interactions. We calculate analytically the maximum speed for
left-moving and right-moving groups, and show numerically that the travelling
pulses exhibited by the nonlocal hyperbolic models actually travel at this
maximum speed. We also show that the way organisms communicate with each other
influences the magnitude of the speed of newly formed groups. However, it does
not influence the magnitude of the speed of groups that have travelled for a
long time. Finally, we discuss the role of communication mechanisms and social
interactions on the choice of movement direction of travelling groups.

*Modelling multi-particle systems in cellular and molecular biology*
I will discuss methods for spatio-temporal modelling in cellular and molecular
biology. Three classes of models will be analysed in detail: (i) microscopic
(molecular-based) models which are based on the simulation of trajectories of
individual molecules and their reactive collisions; (ii) mesoscopic
(lattice-based) models which divide the computational domain into a finite
number of compartments and simulate the time evolution of the numbers of
molecules in each compartment; and (iii) hybrid (multiscale) algorithms which
use models with a different level of detail in different parts of the
computational domain. All three classes of models (i)-(iii) are stochastic.
The connections between these models and the deterministic models (based on
reaction-diffusion-advection partial differential equations) will be
presented. If time permits, I will also discuss hybrid modelling of chemotaxis
where an individual-based model of cells is coupled with PDEs for
extracellular chemical signals. Travelling waves in these hybrid models will
be investigated.

*Kinetic theory two-species coagulation*
We will present results concerning the stochastic process of two-species
coagulation. Analytically, we derive a kinetic theory that approximately
describes the process dynamics and determine its asymptotic behavior. We
compare our analytical results with direct numerical simulations of the
stochastic process and both corroborate its predictions and check its limitations.

Klemens Fellner
*Aggregation patterns in non-local equations: discrete stochastic and
continuum modelling.*
Non-local evolution equations featuring interaction of individuals due to a
repulsive-aggregative potential are observed to produce a rich dynamical
behaviour, which leads to a multitude of stationary pattern. For interaction
potential with suitable attractive singularities convergence to measure
solutions is deduced from a gradient flow structure in Wasserstein metric.
Alternatively propagate singular repulsive interaction potential regular
solutions. However, the case of interaction potential with both singular and
repulsive singularities remains an open problem, for which we present an
interesting comparison of numerical results with a stochastic lattice model.
Joint work with E. Hackett-Jones, B. Hughes, K. Landman, University of Melbourne.

Razvan Fetecau
*Swarm dynamics and equilibria for a nonlocal aggregation model*
We consider the aggregation equation $\rho_{t}-\nabla\cdot\left( \rho\nabla
K\ast\rho\right) =0$ in $
\mathbb{R}
^{n}$ where the interaction potential $K$ models short-range repulsion and
long-range attraction. We study a family of interaction potentials with
repulsion given by a Newtonian potential and attraction in the form of a power
law. We show global well-posedness of solutions and investigate analytically
and numerically the equilibria and their global stability. The equilibria have
biologically relevant features, such as finite densities and compact support
with sharp boundaries. This is joint work with Yanghong Huang and Theodore Kolokolnikov.

Eric Forgoston
*Coherent Pattern Prediction in Swarms of Delay-Coupled Agents*
We consider a general swarm model of self-propelling agents interacting
through a pairwise potential in the presence of noise and communication time
delay. In previous work [Phys. Rev. E 77, 035203(R) (2008)], we showed that a
communication time delay in the swarm induces a bifurcation that depends on
the size of the coupling amplitude. We extend these results by unfolding the
bifurcation structure of the mean field approximation. Our analysis reveals a
direct correspondence between the different dynamical behaviors found in
different regions of the coupling-time delay plane with the different classes
of simulated coherent swarm patterns. We derive the spatio-temporal scales of
these swarm structures, and also demonstrate how the complicated interplay of
coupling strength, time delay, noise intensity, and choice of initial
conditions can affect the swarm.

Amic Frouvelle
*Macroscopic limits of a system of self-propelled particles with phase
transition.*
The Vicsek model, describing alignment and self-organisation in large systems
of self-propelled particles, such as fish schools or flocks of birds, has
attracted a lot of attention with respect to its simplicity and its ability to
reproduce complex phenomena. We consider here a time-continuous version of
this model, in the spirit of the one proposed by P. Degond and S. Motsch, but
where the rate of alignment is proportional to the mean speed of the
neighboring particles. In the hydrodynamic limit, this model undergoes a phase
transition phenomenon between a disordered and an ordered phase, when the
local density crosses a threshold value. We present the two different
macroscopic limits we can obtain under and over this threshold, namely a
nonlinear diffusion equation for the density, and a first-order
non-conservative hydrodynamic system of evolution equations for the local
density and orientation.
Joint work with Pierre Degond and Jian-Guo Liu.

Veysel Gazi
*On the Stability of Swarms with Second Order Agent Dynamics*
In this talk we discuss the overall dynamics and stability of swarms composed
of agents with point mass second order dynamics. The inter-agent interactions
in the individual based swarm model are provided with artificial potential
functions. We discuss the aggregation, social foraging, and formation control problems.

Seung-Yeal Ha
*Asymptotic formation of multi-clusters for the Cucker-Smale and Kuramoto
models*
In this talk, we will discuss asymptotic dynamics of two prototype models for
flocking and synchronization, namely the Cucker-Smale and Kuramoto models. For
these two models, we will present the asymptotic formation of multi-clusters.
In previous literature for the Cucker-Smale model, the mono-cluster
formation(global flocking) has been extensively studied by many researchers.
In this talk, we will derive sufficient conditions for the multi-cluster
formation to the particle and kinetic Cucker-Samle and Kuramoto models.

*Synchronization and aggregation emerging from random processes*
We present two individual based models where social phenomena emerge purely
from random behaviour of the agents, without introducing any deterministic
"social force" that would push the system towards its organized phase.
Instead, organization on the global level results merely from reducing the
individual noise level in response to local organization, which is induced by
stochastic fluctuations. The first model describes the recently experimentally
observed collective motion of locust nymphs marching in a ring-shaped arena
and is written in terms of coupled velocity jump processes. The second model
was inspired by observations of aggregative behaviour of cockroach nymphs in
homogeneous environments and is based on randomly moving particles with
individual diffusivities depending on the perceived average population density
in their neighbourhood. We show that both models have regimes leading to
global self-organization of the group (synchronization and aggregation).
Moreover, we derive the mean-field limits for both models, leading to PDEs
with nonlocal nonlinearities, perform their mathematical analysis and present
a few interesting numerical examples.

Yanghong Huang
*Steady states and asymptotic limits of a nonlocal aggregation model*
We consider the steady states of the aggregation equation $\rho_{t}
=\nabla\cdot\left( \rho\nabla K\ast\rho\right) $, where the interaction
potential K models short-range singular repulsion and long-range power-law
attraction. We show that there exist unique radially symmetric equilibria
supported on a ball. We perform asymptotic studies for the limiting cases when
the exponent of the power-law attraction approaches infinity and a Newtonian
singularity, respectively. Numerical simulations suggest that equilibria
studied here are global attractors for the dynamics of the aggregation model.

Theodore Kolokolnikov
* Asymptotics of complex patterns in an aggregation model with
repulsive-attractive kernel*
The aggregation model with short-range attraction and long-range repulsion can
lead to very complex and intriguing patterns in two or three dimensions.
Depending on the relative strengths of attraction and repulsion, a multitude
of various patterns is observed, from nearly-constant density swarms to
annular solutions, to complex spot patterns that look like "soccer balls". We
show that many of these patterns can be understood in terms of stability
and perturbations of "lower-dimensional" patterns. For example, spots
arise as bifurcations of point clusters [delta concentrations]; annulus
and various triangular shapes are perturbations of a ring. Asymptotic
methods provide a powerful tool to describe the stability, shape and
precise dimensions of these complex patterns.
Joint works with Bertozzi, von Brecht, Fetecau, Huang, Hui, Pavlovsky,
Uminsky.

*Aggregation via Newtonian potential and aggregation patches*
We consider the motion of a density of particles $rho(x,t)$ by a velocity
field $v(x,t)$ obtained by convolving the density of particles with the
gradient of the Newtonian potential, that is $v=-\nabla N\ast\rho$. An
important class of solutions are the ones where the particles are uniformly
distributed on a time evolving domain. We refer to these solutions as
aggregation patches, by analogy to the vortex patch solutions of the 2D
incompressible Euler equations. Numerical simulations as well as some exact
solutions show that the time evolving domain on which the patch is supported
typically collapses on a complex skeleton of codimension one. We also show
that going backward in time, any bounded compactly supported solution
converges as t goes to minus infinity toward a spreading circular patch. We
provide a rate of convergence which is sharp in 2D. This is a joint work with
Bertozzi and Leger.

Joceline Lega
*Coherent behaviors in groups of locally interacting particles*
I will present results of molecular dynamics simulations of disks moving in a
two-dimensional box and interacting through special collisions (J. Lega, SIAM
J. Appl. Dyn. Sys. 10, 1213 -1231 (2011)). Because this work was motivated by
the existence of complex behaviors in colonies of bacteria, the particles also
reorient themselves at random times, thereby simulating bacterial tumbles and
inputing energy into the system. I will show that at low packing fractions
clusters dynamically form and break up and that, as the packing fraction
increases, groups of increasingly larger size are observed, in which the
particles move coherently. Such behaviors are markedly different from those
observed in systems of particles interacting through elastic collisions. Time
permitting, I will also present results on groups of locally interacting rods.

*Mathematical Models for Phototaxis*
Certain organisms undergo phototaxis, that is they migrate toward light. In
this talk we will discuss our recent results on modeling phototaxis in order
to understand the functionality of the cell and how the motion of individual
cells is translated into emerging patterns on macroscopic scales. This is a
joint work with Amanda Galante, Susanne Wisen, Tiago Requeijo, and Devaki Bhaya.

*Quenching properties of a fourth order parabolic non-linear PDE in 2D
geometries.*
This talk will present some recent results on a singularity formation problem
in a nonlinear fourth order PDE modelling a Micro-Elcetro Mechanical Systems
(MEMS) Capacitor. The singularity observed is a divergence in the solution
derivative, rather than the solution itself - a phenomenon known as quenching.
The local structure of singularity is shown to be self-similar in nature.
Additionally, the singularity is observed to form in multiple locations within
the domain with these locations exhibiting an analyzable dependence on the
model parameters and the geometry of the domain. We outline an asymptotic
method which can predict the location(s) where singular solutions form based
on the geometry of the domain and the parameters of the system. The theory is
demonstrated on several examples. This is joint work with J. Lega (Arizona)
and F.J. Sayas ( Delaware).

Stephan Martin
*Explicitly computable flock and mill states of self-propelled particles
systems*
We consider a self-propelled interacting particle system, which has been
frequently used to model complex behavior of swarms such as fish schools or
birds flocks. Typically, the model is equipped with the Morse potential and
patterns such as aligned flocks and rotating mills emerge in particle
simulations. To this day, these stationary states cannot be a priori computed,
except for one-dimensional flocks. We present a class of interaction
potentials, that we call Quasi-Morse potentials, which likewise generate
flocks and rotating mills. However, their stationary states can be explicitly
computed as (affine) linear combinations of up to three elementary functions,
only whose scalar coefficients have to be obtained numerically. This can be
achieved without simulating a time evolution. We present the formulae for the
mill and flock solutions, verify our results by comparing to corresponding
particle simulations and illustrate parameter dependencies.

Sebastien Motsch
*A model of flocking with asymmetric interactions*
In this talk, we introduce a model of flocking which aims at improving the
popular Cucker-Smale (C-S) model. The C-S model relies on a simple rule: the
closer two individuals are, the more they tend to align with each other. In
the new model proposed, the strength of the interaction is also weighted by
the density: the more an agent is surrounded, the less he will be influenced.
As a consequence, interactions between agents are no longer symmetric.
To study the flocking behavior of this dynamics, we base our analysis on a
L\symbol{94}? approach rather than L\symbol{94}2 approach. We find that the
dynamics converges to a flock provided that the interaction function decays
slowly enough.

*Phase transitions in models of Vlasov-McKean type*
I will discuss the problem of non-uniqueness and stability of steady states
for equations of Vlasov-McKean type. These equations provide a mean-field
description of a system of interacting diffusions; particularly we consider
the problem with the spatial variable in a periodic box of size L and
particles interacting through a pairwise potential V. If the Fourier transform
of V has a negative minimum, the system has a critical threshold for the
diffusion constant beyond which the trivial uniform steady state becomes
unstable and the system experiences a phase transition. We show that for a
large class of interactions, when the size of the domain is sufficiently
large, the transition is always discontinuous and is characterized by
coexistence of several stable states in a certain interval of parameter
space.The transition is also shown to occur at a value of the diffusion
constant strictly greater than the critical threshold. I will then briefly
present the results of a numerical study on the character of phase transition
in Vicsek like models of flocking, in which a similar discontinuous transition
is observed.

Mark Pavlovski
* Point cluster and spot patterns in an aggregation model with
repulsive-attractive kernel*
The aggregation model with short-range repulsion and long-range
attraction generates diverse patterns in two or three dimensions. In
this talk we will discuss stability of the point clusters as well as
structure and stability of the spot patterns which arise as
perturbations of the point clusters. This is a joint work with
Theodore Kolokolnikov and Yanghong Huang.

Vakhtang Putkaradze
*Molecular monolayers as interacting rolling balls: crystals, liquid and
vapor*
Molecular monolayers, especially water monolayers, are playing a crucial role
in modern science and technology. In order to derive simplified models of
monolayer dynamics, we consider the set of rolling self-interacting particles
on a plane with an off-set center of mass and a non-isotropic inertia tensor.
To connect with water monolayer dynamics, we assume the properties of the
particles like mass, inertia tensor and dipole moment to be the same as water
molecules. The perfect rolling constraint is considered as a simplified model
of a very strong, but rapidly decaying bond with the surface. Since the
rolling constraint is non-holonomic, it prevents the application of the
standard tools of statistical mechanics: for example the system exhibits two
temperatures -- translational and rotational-- for some degrees of freedom,
and no temperature can be defined for other degrees of freedom.
In spite of apparent simplicity, the behavior of the system is surprisingly
rich. We identify analogies with the regular water by defining crystalline,
liquid and gas states, based on the specific energy of a particle. We show the
existence and nonlinear stability of ordered lattice states,. We also
investigate the effect of rolling on the disturbance propagation through a
crystalline lattice, study the chaotic vibrations of the crystalline states
and identify an interesting phase transition when the crystal is destroyed. We
demonstrate that there are also relatively confined "droplet" states with the
center of mass exhibiting seemingly random walk on the surface. Finally, we
investigate the dynamics of disordered gas states and show that there is a
surprising and robust linear connection between distributions of angular and
linear velocity for both lattice and gas states, allowing to define the
concept of temperature.
Finally, as a first step towards continuous theory, we develop a Vlasov-like
kinetic theory for a gas of rolling balls. Using that framework, we show that
the concept of momentum conservation cannot be borrowed from the classical
fluids, and derive a set of alternative conservation laws for the system.

*Nonlocal Aggregation Equations and Concentration Phenomena*
Nonlocal aggregation equation appear in various fields of physics and biology.
In many situations, the interaction potential presents a singularity at the
origin. This singularity, which can be either attractive or repulsive, has a
significant impact on the qualitative properties of solutions. In this talk, I
will present a qualitative study of those qualitative properties in one and in
several dimensions (although most questions remain open in the latter case).
This work has been done in collaboration with Klemens Fellner, Daniel Balague,
Jose Carrillo and Thomas Laurent.

Nancy Rodriguez
*
Hotspot invasion: Traveling Wave solutions for a Reaction-Diffusion
system for crime patterns. *
We analyze of a reaction-diffusion system of PDE's in order to obtain
insight into propagation of crime patterns. This system of equations can
be divided into three regimes, which lead to one, two, or three
steady-states solutions. In this talk I will first discuss some
preliminary results towards proving that indeed these are the only three
possible regimes. Under the assumption that the parameters are
heterogenous, I will discuss the condition that divide these regimes.
Finally, I will discuss the invasion of hotspots via traveling wave
solutions in one dimension.

* Kinetic Cucker-Smale model of collective behavior*
We'll show the well posedness of the kinetic version of the
Cucker-Smale model for flocking and prove that the unconditional
flocking result that Cucker and Smale showed for the particle model
also holds in the new framework. Then we'll discuss some new models
that can be derived from it.

*Fekete Points, Narrow Escape, and Asymptotics of the Mean First Passage
Time*
The mean first passage time (MFPT) is calculated for a Brownian particle in a
three-dimensional domain that contains $N$ small non-overlapping absorbing
windows on its boundary. The reciprocal of the MFPT of such narrow escape
problems has wide applications in cellular biology where it may be used as an
effective first order rate constant to describe, for example, the nuclear
export of messenger RNA molecules through nuclear pores. In the asymptotic
limit where the absorbing patches have small measure, the method of matched
asymptotic expansions is used together with some detailed analytical
properties of a surface Green's function to calculate a three-term asymptotic
approximation for the MFPT for the unit sphere. The third term in this
expansion depends explicitly on the spatial arrangement of the absorbing
windows on the boundary of the sphere. The MFPT is minimized for particular
trap configurations that minimize a certain discrete variational problem,
which is closely related to the well-known optimization problem of determining
the minimum energy configuration for $N$ repelling Coulomb charges on the unit
sphere. Finally, our three-term asymptotic expansion for the averaged MFPT is
shown to be in very close agreement with full numerical results.
Joint work with: Alexei Cheviakov (U. Saskatchewan) and Ronny Straube
(Max-Planck Institute; Magdeburg).

Bernt Wennberg
*Propagation of chaos in biological swarm models*
We consider two models of biological swarm behavior. In these models, pairs of
particles interact to adjust their velocities one to each other. In the first
process, called 'BDG', they join their average velocity up to some noise. In
the second process, called 'CL', one of the two particles tries to join the
other one's velocity. This paper establishes the master equations and BBGKY
hierarchies of these two processes. It investigates the infinite particle
limit of the hierarchies at large time- scale. It shows that the resulting
kinetic hierarchy for the CL process does not satisfy chaos propagation. We
present numerical simulations that indicate the same behavior for the BDG
model. For the BDG model we also show, by explicit examples, that the
stationary state may not be unique.
Authors: E. Carlen, R. Chatelin, P. Degond and B. Wennberg

Yao Yao
*Degenerate diffusion with nonlocal aggregation: behavior of radial
solutions*
The Patlak-Keller-Segel (PKS) equation models the collective motion of cells
which are attracted by a self-emitted chemical substance. While the global
well-posedness and finite-time blow up criteria are well known, the asymptotic
behaviors of solutions are not completely clear. In this talk I will present
some results on the blow-up behavior and the asymptotic behavior of radial
solutions. Numerically, we show that the solution exhibits three kinds of
blow-up behavior: self-similar with no mass concentrated at the core,
imploding shock solution and near-self-similar blow-up with a fixed amount of
mass concentrated at the core. (joint work with A. Bertozzi) We also present
some theoretical results concerning the asymptotic behavior of radial
solutions when there is global existence. (joint work with I. Kim)