Theme
organizers: PietroLuciano Buono (UOIT) and Michael Ward (UBC)
This
theme is focussed on the mathematics and application of swarming
behaviour. This topic will link modelers, PDE experts, and math
biologists. Mathematically, even models of swarming in particle
systems can lead to novel phenomena which are still in the process
of being understood. Possible subthemes include pattern formation,
blowup solutions and concentration phenomena, criticality and phase
transitions and second order models.
Speakers:
Andrew
Bernoff, Harvey Mudd College
Hermann Eberl, University of Guelph
Raluca Eftimie, University of Dundee
Razvan Fetecau, Simon Fraser University
Cristian Huepe, Northwestern
Zachary Kilpatrick, University of Pittsburgh
Theodore Kolokolnikov, Dalhousie University
Alan Lindsay, U.Arizona
Ryan Lukeman, St. Francis Xavier University
Noam Miller, Princeton
Nessy Tania, Smith College
Chad Topaz, Macalester College
Colin Torney, Princeton University
Justin Tzou, Northwestern
David Uminsky, UCLA
Michael Ward, University of British Columbia
Contributed
Talks

Andrew J. Bernoff
Department of Mathematics, Harvey Mudd College, Claremont, CA 91711
We
study equilibrium configurations of swarming biological organisms
subject to exogenous and pairwise endogenous forces. Equilibrium
solutions are extrema of an energy functional, and satisfy a Fredholm
integral equation. In one spatial dimension, for a variety of
exogenous forces, endogenous forces, and domain configurations,
we find exact analytical expressions for the equilibria. The equilibria
typically are compactly supported and may contain concentrations
or jump discontinuities at the edge of the support. In two dimensions
we show that the Morse Potential and other "pointy"
potentials lead to inverse squareroot singularities in the density
at the edge of the swarm support.
Joint with: Louis Ryan (lryan<at>hmc.edu),Department of
Mathematics,
Harvey Mudd College, Claremont, CA 91711 and
Chad M. Topaz (ctopaz<at>macalester.edu), Macalester College
St. Paul, MN 55105.
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Hermann
Eberl, University of Guelph
Coauthors: Kazi Rahman (Guelph)
Two derivations for a class of nonnegativity preserving crossdiffusion
models
Starting
from a discrete lattice master equation or from continuum mechanical
principles we derive a class of crossdiffusion equations. We
show that the members of this class automatically satisfy a necessary
and sufficient condition to preserve nonnegativity. We demonstrate
that this class of models includes certain adhoc models that
have been used in spatial population dynamics, such as the wellknown
ShigesadaKawasakiTeramoto equation.
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Raluca
Eftimie
Dept. of Mathematics,Univ. of Dundee
Mathematical mechanisms behind pattern formation in a class
of nonlocal hyperbolic models for selforganized biological aggregations
Pattern formation is one of most studied aspects of animal communities.
Here, we discuss a class of nonlocal hyperbolic models derived
to reproduce and further investigate some of the aggregation patterns
observed in various species. These models display a variety of
spatial and spatiotemporal patterns: from simple stationary and
travelling pulses, to more complex patterns such as ripples, zigzags
and breathers.
The mathematical mechanisms behind the simpler patterns can be
investigated using weakly nonlinear analysis or travelling wave
analysis. However, the complex patterns (which usually arise through
bifurcations from simpler patterns) are more difficult to investigate.
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Razvan
C. Fetecau, Dept. of Mathematics, Simon Fraser Univesity, B.C.
A mathematical model for flight guidance in honeybee swarms
When
a colony of honeybees relocates to a new nest site, less than
5% of the bees (the scout bees) know the location of the new nest.
Nevertheless, the small minority of informed bees manages to provide
guidance to the rest and the entire swarm is able to fly to the
new nest intact. The streaker bee hypothesis, one of the several
theories proposed to explain the guidance mechanism in bee swarms,
seems to be supported by recent experimental observations. Originally
proposed by Lindauer in 1955, the theory suggests that the informed
bees make highspeed flights through the swarm in the direction
of the new nest, hence conspicuously pointing to the desired direction
of travel. Once they reach the front of the swarm, they return
at low speeds to the back, by flying along the edges of the swarm,
where they are less visible to the rest of the bees. This work
presents a mathematical model of flight guidance in bee swarms
based on the streaker bee hypothesis. Numerical experiments, parameter
studies and
comparison with experimental data will be presented.
Joint work with Angela Guo, Dept. of Mathematics, Simon Fraser
University.
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Cristián
Huepe
Dept. of Engineering Science and Applied Math, Northwestern University,
Evanston, Illinois.
Variable speed and attractiverepulsive interactions in swarming
systems
Animal
groups, such as bird flocks, fish schools, or insect swarms, often
exhibit complex, coordinated collective dynamics resulting from
individual interactions. Despite recent progress in characterizing
them, many currentdescriptions are still based on minimal models
with fixedspeed selfpropelled individuals that align.
In this talk, I will present two simple models extending standard
agentbased swarming algorithms to include variable speed and
strong attractionrepulsion forces. The first one considers a
speeding rule, inspired on experimental observations, that leads
to nontrivial density fluctuation and cluster formation. The second
unveils a new elasticitydriven mechanism that can also lead to
collective motion.
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Zachary
Kilpatrick
Dept. of Mathematics, Univ. of Pittsburgh, U.S.A.
Wandering
and transitions of pulses in stochastic neural fields
Localized solutions of spatially extended neural fields provide
avaluable framework for studying coherent activity in the brain.
We examine the dynamics of standing pulses (bumps) and traveling
pulses in spatially extended neural fields with noise. Bumps are
shown to diffusively wander about the spatial domain. We can calculate
the associated diffusion coefficient using a small noise
expansion. Multiplicative noise can even shift bifurcations that
indicate the disappearance or destabilization of the bump. Finally,
it is shown noise can switch the direction of propagation of traveling
pulses. We study these switches as transitions between two branches
of a pitchfork bifurcation.
Joint work with Bard Ermentrout, Dept. of Mathematics, Univ. of
Pittsburgh
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Theodore
Kolokolnikov
Dept. of Mathematics and Statistics, Dalhousie University, Canada
Asymptotics
of complex patterns in an aggregation model with repulsiveattractive
kernel
The aggregation model with shortrange attraction and longrange
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 nearlyconstant 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 "lowerdimensional" 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.
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Alan
Lindsay, U. Arizona
Optimization of the persistence threshold in spatial envirnoments
with localized patches.
Determining
whether a habitat with fragmented or concentrated resources best
supports a contained population is a natural question to ask in
Ecology. Such fragmentation may occur naturally or as a consequence
of human activities related to development or conservation. In
certain mathematical formulations of this problem, a critical
value known as the persistence threshold indicates the boundary
in parameter space for which the species either persists or becomes
extinct. By assuming simple diffusive logistic dynamics for the
population and accommodating the heterogeneous nature of the landscape
with a spatially varying growth rate, a simple formulation for
the persistence threshold is afforded in terms of an indefinite
weight eigenvalue problem. In this talk I will show that for a
growth rate with strongly localized patches of favorable habitat,
the persistence threshold can be calculated implicitly and minimized
with respect to the location and fragmentation of patches. This
reveals an optimal strategy for minimizing the persistence threshold,
thereby allowing the species to persist for the largest range
of physical parameters. The techniques developed can be extended
to study the effects of heterogeneity in a variety of ecological
models.
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Ryan
Lukeman
Dept. of Mathematics Saint Francis Xavier University
Temporal dynamics in collective animal motion
Collective
animal motion data generally involves many individuals moving
simultaneously, interacting with others the group. To obtain a
clear, meaningful signal, data is often averaged across time and
individuals. However, by studying the dynamics of the collective
through time, properties of the individual and collective can
be more clearly linked. In this talk, I present results on flock
dynamics of
surf scoters, an aquatic duck, showing correlations between group
polarity and individual speed. These observations are in turn
used as a blueprint for model selection and parameterization.
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Noam
Miller
Dept. of Ecology and Evolutionary Biology, Princeton University,
Princeton, NJ, USA
Collective learning and optimal consensus decisions in social
animal groups
Groupliving
animals must often make collective decisions, even when individuals
disagree about the best course of action. Additionally, groups
that can share information are provably more accurate than individuals.
A large body of literature exists on optimal voting rules for
various environmental conditions but applying these rules requires
a level of global oversight that is unreasonable in animal groups.
We present a model that requires only that individuals follow
movement
rules common in SPP models (attraction and repulsion) and employ
a wellknown learning rule (based on the RescorlaWagner model).
Our model achieves close to optimal performance under a range
of environmental situations.
Joint work with Albert Kao, Colin Torney, Andrew Hartnett, and
Iain Couzin, Department of Ecology and Evolutionary Biology, Princeton
University, Princeton, NJ, USA
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Martin
Short, Dept. of Mathematics, UCLA
Ecological modeling of gang territories
Like
many territorial species, urban gangs often fight over "turf"
that they find valuable, for social and/or monetary reasons. In
this talk, we will discuss a simple model for gang territoriality,
in the form of diffusive LotkaVolterra competition equations.
We will then compare the results of this model to spatiotemporal
data on intergang violent crimes, finding good agreement between
the predicted spatial distribution of events and the field data.
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Nessy
Tania, Dept. of Mathematics Smith College, Northampton, MA,
U.S.A.
Applied Analysis Theme
Oscillatory
patch formations from social foraging Dynamics of resource patches
and foragers have been widely studied and shown to exhibit pattern
formations. I will show how social interactions among foragers
can create novel spatiotemporal oscillatory patterns. Simple taxis
of foragers towards randomly moving prey is known to exhibit stabilizing
effects and cannot lead to spontaneous formation of patchy environments.
However, a population of foragers with two types of behaviours
can do so. I will also briefly discuss which of these behavior
is more beneficial and how switching between strategies affect
the resulting spatiotemporal patterns.
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Chad
Topaz
Dept. of Mathematics, Statistics, and Computer Science, Macalester
College, St. Paul, MN, U.S.A.
Desert Locust Dynamics: Nonlocal PDEs, Behavioral Phase Change,
and Swarming
The
desert Locust Schistocerca gregaria has two interconvertible phases,
solitary and gregarious. Solitary (gregarious) individuals are
repelled from (attracted to) others, and crowding biases conversion
towards the gregarious form. We construct a nonlinear partial
integrodifferential equation model of the interplay between phase
change and spatial dynamics leading to locust swarms. We
derive conditions for the onset of a locust plague, characterized
by collective transition to the gregarious phase. Via a model
reduction to ODEs describing the bulk dynamics of the two phases,
we calculate the proportion of the population that will gregarize.
Numerical simulations reveal transiently traveling clumps of insects.
joint work with Maria D'Orsogna, Andrew Bernoff, and Leah EdelsteinKeshet.
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Colin
Torney
Dept. of Ecology and Evolutionary Biology,Princeton University,
Princeton
Collective behaviour and social information in mobile animal
groups
In
this talk I will present some of our recent work on information
and its use within swarming systems. I will outline some empirical
results
pertaining to imitation and information sharing in schooling fish,
firstly in dynamic environments, and secondly in situations when
individuals have varying and contradictory personal information,
notably showing how uninformed individuals within the group promote
a majorityrule scenario. Numerical simulations demonstrate how
simple local interactions create this aggregate behaviour, but
still remain largely intractable. I will therefore present some
reduced models based around simple coordination games, that capture
the same qualitative features as the real systems, such as localized
interaction, social influence, and rapid transitions to ordered
states, but which allow some analytical treatment.
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Justin
Tzou, Dept. of Engineering Science and Applied Math, Northwestern
University
The Stability of Pulses in a Singularly Perturbed Brusselator
Model
In
a onedimensional domain, the stability of localized steadystate
and quasisteadystate spike patterns is analyzed for a singularly
perturbed reactiondiffusion (RD) system with Brusselator kinetics.
Asymptotic analysis is used to derive a nonlocal eigenvalue problem
(NLEP) whose spectrum determines the linear stability of a multispike
steadystate solution. Similar to previous NLEP stability analyses
of spike patterns for other RD systems, a multispike steadystate
solution can become unstable to either a competition or an oscillatory
instability. In the parameter regime where a Hopf bifurcation
occurs, it is shown from a numerical study of the NLEP that an
asynchronous, rather than synchronous, oscillatory instability
of the spike amplitudes can be the dominant instability. The existence
of robust asynchronous temporal oscillations of the spike amplitudes
has not been observed in NLEP stability studies of other RD systems.
A similar NLEP stability analysis of a quasisteady state two
spike pattern reveals the possibility of dynamic bifurcations
leading to either a competition or an oscillatory instability
of the spike amplitudes. It is shown that the novel asynchronous
oscillatory instability mode can again be the dominant instability.
Results from NLEP theory are confirmed by numerical computations
of the full PDE system.
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David
Uminsky, Dept. of Mathematics, UCLA, Los Angeles, U.S.
Predicting patterns and designing interactions for nonlocal
particle interactions
Pairwise
particle interactions arise in diverse physical systems ranging
from insect swarms and bacterial distributions, to selfassembly
of nanoparticles. In the presence longrange attraction and shortrange
repulsion, such systems may exhibit rich patterns in there bound
states. In this talk we present a theory to classify the morphology
of various patterns in N dimensions from a given confining potential.
We also present a method to solve the inverse statistical mechanics
problem: Given an observed pattern, can we construct a confining
interaction potential which exhibits that pattern.
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Michael
Ward Dept. of Mathematics, University of British Columbia,
The Stability of HotSpot Patterns of Urban Crime
Over
the past five years, agentbased stochastic models have been developed
by various researchers to predict spatiotemporal concentrations
of criminal activity in urban settings. The continuum
limit of these models leads to reactiondiffusion systems with
chemotactic terms. In this context, and in a particular singularly
perturbed limit, we analytically construct localized equilibrium
and quasiequilibrium solutions characterizing hotspots of criminal
activity for the reactiondiffusion system of Short et al. (MMAS,
Vol. 18, Suppl. (2008), pp.~12491267). Explicit thresholds for
the diffusivity of the criminal density determining the stability
of these localized patterns are obtained for both 1D and 2D
domains by first deriving and then analyzing a novel class of
nonlocal eigenvalue problem (NLEP). The implication of these results
are discussed,together with some open problems related to the
dynamics of hotspot patterns.
Joint work with Theodore Kolokolnikov (Dalhousie U.) and Juncheng
Wei (Chinese U. of Hong Kong).
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CONTRIBUTED
TALKS
Tarini
Kumar Dutta
Professor
of Mathematics, Gauhati University, Guwahati 781014, INDIA
**Determination of Lyapunov exponents and various Fractal Dimensions
in Population Chaotic Models.
This
paper is primarily concerned with the determination of Lyapunov
exponents and some fractal dimensions in one dimensional Ricker
population model : f(x)=x e^(r(1x/k)) ,where r is the control
parameter and k is the carrying capacity. Suitable formulae have
been developed in order to determine Lyapunov exponents, box
counting dimension, information dimension, operational dimension
, and correlation dimension, and some significant results are
obtained. Moreover, Hausdorff dimension is calculated with suitable
bounds.
Key words: population model / fractal dimension / carrying capacity
2010 AMS subject classification : 37G25 , 37G15

*Tripti
Dutta
York university
Coauthors: Jane Heffernan, Centre for Disease Modelling, Mathematics
& Statistics, York University
Variability in HIV infection inhost: A Monte Carlo Markov Chain
model.
HIV
targets cells with CD4 receptors, including CD4 Tcells, the main
driver of immune response.Through infection it kills CD4 Tcells
making a patient susceptible to opportunistic infections. Changes
in CD4 lymphocyte counts and viral load are used to monitor the
disease status of HIV patients and inform decisions regarding
the initiation or continuance of antiretroviral therapy.The natural
variability in these measurements thus needs to be known so that
informed decisions regarding patient health can be made. We have
developed and employed a Markov Chain Monte Carlo (MCMC) simulation
to measure the variability in important immunological measurements
such as the infected equilibrium, basic reproductive ratio, and
initial growth rate.The simulation is also used to determine the
probability of extinction of an initial viral load, variation
in the time to peak viremia and variation in peak magnitude.

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Waseem
Asghar Khan
CIIT, Islamabad
He’s frequency formulation for higherorder nonlinear oscillators
and nonlinear oscillator with discontinuous
Based
on an ancient Chinese algorithm, J H He suggested a simple but
effective method to find the frequency of a higherorder nonlinear
oscillator and nonlinear oscillator with discontinuous. In this
paper, we use this method on higherorder nonlinear oscillators
and nonlinear oscillator with discontinuous to improve the accuracy
of the frequency; these two higherorder examples are given, revealing
that the obtained solutions are of remarkable accuracy and are
valid for the whole solution domain.

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Matt
Kloosterman
University of Waterloo
Coauthors: Sue Ann Campbell Francis Poulin
A Closed NPZ Model with Delayed Nutrient Recycling
We
consider a closed NutrientPhytoplanktonZooplankton (NPZ) model
that allows for a delay in the nutrient recycling. A delaydependent
conservation law allows us to quantify the total biomass in the
system. With this, we can investigate how a planktonic ecosystem
is affected by the quantity of biomass it contains and by the
properties of the delay distribution. The quantity of biomass
and the length of the delay play an significant role in determining
the existence of equilibrium solutions, since a sufficiently small
amount of biomass or a long enough delay can lead to the extinction
of a species. Furthermore, the quantity of biomass and length
of delay are important since a small change in either can change
the stability of an equilibrium solution. We explore these effects
for a variety of delay distributions using both analytical and
numerical techniques, and verify results with simulations.
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Nemanja
Kosovalic
York University
Coauthors: Dr. Felicia Magpantay, Dr. Jianhong Wu
An Age Structured Population Model with State Dependent Time Delay
In
this talk we consider an age structured population model in which
the age to maturity at a given time depends on whether or not
the food consumed by the immature population within that time
span reaches a prescribed threshold value. This introduces a state
dependent delay into the model. In contrast with other works on
this problem, we consider it from the point of view of a hyperbolic
partial differential equation with a state dependent time delay.
Wilten
Nicola, University of Waterloo
Coauthors: Sue Ann Campbell
Bifurcations of Large Networks of PulseCoupled Oscillators
Many
functional subunits of the brain contain a large number of neurons.
These regions are often modeled as networks of pulsecoupled oscillators.
The models can be conductance based or of the integrateandfire
type. When fit properly, these large network models replicate
the bifurcations of the original data. Since these models are
nonsmooth systems, determining the bifurcation types of these
networks is outside the realm of classical bifurcation theory.
Population density equations have been used to extend the classical
theory to these systems. The theory is applied to a model consisting
of a network of Izhikevich neurons fit to hippocampal region CA3.

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