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 sub-themes include pattern formation, blow-up 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
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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 square-root 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 non-negativity preserving cross-diffusion models

Starting from a discrete lattice master equation or from continuum mechanical principles we derive a class of cross-diffusion equations. We show that the members of this class automatically satisfy a necessary and sufficient condition to preserve non-negativity. We demonstrate that this class of models includes certain ad-hoc models that have been used in spatial population dynamics, such as the well-known Shigesada-Kawasaki-Teramoto 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 self-organized biological aggregations

Pattern formation is one of most studied aspects of animal communities. Here, we discuss a class of non-local 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 high-speed 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 attractive-repulsive 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 fixed-speed self-propelled individuals that align.
In this talk, I will present two simple models extending standard agent-based swarming algorithms to include variable speed and strong attraction-repulsion 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 elasticity-driven 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 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.
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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

Group-living 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 well-known learning rule (based on the Rescorla-Wagner 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 Lotka-Volterra competition equations. We will then compare the results of this model to spatio-temporal data on inter-gang 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 Edelstein-Keshet.
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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 majority-rule 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 one-dimensional domain, the stability of localized steady-state and quasi-steady-state spike patterns is analyzed for a singularly perturbed reaction-diffusion (RD) system with Brusselator kinetics. Asymptotic analysis is used to derive a nonlocal eigenvalue problem (NLEP) whose spectrum determines the linear stability of a multi-spike steady-state solution. Similar to previous NLEP stability analyses of spike patterns for other RD systems, a multispike steady-state 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 quasi-steady 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 non-local particle interactions

Pairwise particle interactions arise in diverse physical systems ranging from insect swarms and bacterial distributions, to self-assembly of nanoparticles. In the presence long-range attraction and short-range 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 Hot-Spot Patterns of Urban Crime

Over the past five years, agent-based stochastic models have been developed by various researchers to predict spatio-temporal concentrations of criminal activity in urban settings. The continuum
limit of these models leads to reaction-diffusion systems with chemotactic terms. In this context, and in a particular singularly perturbed limit, we analytically construct localized equilibrium and quasi-equilibrium solutions characterizing hot-spots of criminal activity for the reaction-diffusion system of Short et al. (MMAS, Vol. 18, Suppl. (2008), pp.~1249--1267). Explicit thresholds for the diffusivity of the criminal density determining the stability of these localized patterns are obtained for both 1-D and 2-D 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 hot-spot 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(1-x/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
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*Tripti Dutta
York university
Coauthors: Jane Heffernan, Centre for Disease Modelling, Mathematics & Statistics, York University
Variability in HIV infection in-host: A Monte Carlo Markov Chain model.

HIV targets cells with CD4 receptors, including CD4 T-cells, the main driver of immune response.Through infection it kills CD4 T-cells 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 higher-order 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 higher-order nonlinear oscillator and nonlinear oscillator with discontinuous. In this paper, we use this method on higher-order nonlinear oscillators and nonlinear oscillator with discontinuous to improve the accuracy of the frequency; these two higher-order 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 Nutrient-Phytoplankton-Zooplankton (NPZ) model that allows for a delay in the nutrient recycling. A delay-dependent 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 Pulse-Coupled Oscillators

Many functional subunits of the brain contain a large number of neurons. These regions are often modeled as networks of pulse-coupled oscillators. The models can be conductance based or of the integrate-and-fire type. When fit properly, these large network models replicate the bifurcations of the original data. Since these models are non-smooth 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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