Preprints

Theodore kolokolnikov and David Iron
Law of mass action and saturation in SIR model with application
to Coronavirus modelling

Shuangquan Xie, Theodore kolokolnikov, Yasumasa Nishiura,
Complex oscillatory motion of multiple spikes in a threecomponent Schnakenberg system

Chunyi Gai, David Iron, Theodore Kolokolnikov, and John Rumsey,
Spike dynamics in the presence of noise

T. Kolokolnikov, Christopher Ticknor, and P. G. Kevrekidis,
Toroidal Vortex Filament Knots and Links: Existence, Stability and Dynamics

N. Khalil D. Iron T. Kolokolnokov,
Stability and Dynamics of Spiketype Solutions to Delayed
GiererMeinhart Equations
Papers

Theodore Kolokolnikov, Frederic PaquinLefebvre and Michael Ward,
Stable asymmetric spike equilibria for the GiererMeinhardt model with a
precursor field,
The IMA Journal of Applied Mathematics, 2020.

Chunyi Gai, David Iron Theodore Kolokolnokov,
Localized outbreaks in SIR model with diffusion,
JMB, 2020

Theodore Kolokolnikov and Juncheng Wei,
Hexagonal spike clusters for some PDE's in 2D
, DCDSB

Theodore Kolokolnikov and Shuangquan Xie,
Spike density distribution for the GiererMeinhardt model with precursor
, Physica D, 2019.

Theodore Kolokolnikov, Michael Ward,
Justin Tzou and Juncheng Wei,
Stabilizing a homoclinic stripe,
Proc. Royal. Soc. London A., 2018, 376(2135): 20180110.

Kevin P. O'Keeffe, Joep H.M. Evers, Theodore Kolokolnikov,
Ring states in swarmalator systems
, Physical Review E 98, 022203 (2018)

Shuangquan Xie, Panayotis G. Kevrekidis and Theodore Kolokolnikov,
Multivortex crystal lattices in BoseEinstein Condensates with a
rotating trap , Proc. Royal. Soc. London A., 474.2213 (2018):20170553.

Julien SmithRoberge, David Iron and Theodore Kolokolnikov,
Pattern Formation in Bacterial Colonies with
DensityDependent
Diffusion, European Journal of Applied Mathematics (2018): 123.

T.Kolokolnikov and J. Wei,
Pattern formation in a reactiondiffusion system with spacedepenent feed
rate
, SIAM Review, 2018

Joep H.M. Evers, David Iron, Theodore Kolokolnikov and John Rumsey,
Agentbased model of the effect of globalization on
inequality and class mobility, Physica D (2017)

J.H.M. Evers, R.C. Fetecau and T. Kolokolnikov,
Equilibria for an aggregation model with two
species, SIADS, 2017.

ChiunChuan Chen, ChihChiang Huang and Theodore Kolokolnikov,
Critical Exponent of a Simple Model of Spot
Replication,
JDE, 2017

T.Kolokolnikov,
Mathematics on the chopping block The
Mathematical Intelligencer, 2017.

J.C. Tzou, S. Xie, T. Kolokolnikov, and M.J. Ward,
The stability and slow dynamics of localized spot patterns for the 3D
Schnakenberg reactiondiffusion model, SIAM J. Appl. Dyn. Sys. (2016): 35.

S.Xie and T.Kolokolnikov,
Moving and jumping spot in a two dimensional reactiondiffusion model,
Nonlinearity (2017).

A.E. Lindsay, J. C. Tzou, and T. Kolokolnikov,
Optimization of first passage times by multiple cooperating mobile traps,
SIAM Multiscale Modeling and Simulation (2017)

J.Evers and T.Kolokolnikov,
Metastable states for an aggregation model with noise,
SIAM Journal on Applied Dynamical Systems, 15(4): pp. 2213.2226 (2016).

PC ChavyWaddy and T. Kolokolnikov,
A local PDE model of aggregation formation in bacterial colonies,
Nonlinearity 29.10 (2016): 3174

J.C. Tzou, P.G. Kevrekidis, T. Kolokolnikov, and R. CarreteroGonzales,
Weakly nonlinear analysis of vortex formation in a dissipative variant of
the GrossPitaevskii equation,
SIADS, 2016

R. CarreteroGonzález,
P.G. Kevrekidis and T. Kolokolnikov,
Vortex Nucleation in a Dissipative Variant of the Nonlinear
Schrödinger Equation under Rotation, Physica D (2016):114.

H. Tompkins and T. Kolokolnikov,
Swarm shape and its dynamics in a predatorswarm model,
SIAM Undergraduate Research Online, 2014

Y. Chen, T. Kolokolnikov, J. Tzou, and C. Gai,
Patterned vegetation, tipping points, and the rate of climate change,
EJAM, 2015

Shuangquan Xie and T. Kolokolnikov,
Oscillations of many interfaces in the nearshadow regime of
twocomponent reactiondiffusion systems,
DCDSB, 2016

A.E. Lindsay, J.C. Tzou, T. Kolokolnikov,
Narrow escape problem with a mixed trap and
the effect of orientation, Phys. Rev. E 91, 032111.

A.E. Lindsay, M.J. Ward and T. Kolokolnikov,
The Transition to a Point Constraint in a Mixed
Biharmonic Eigenvalue Problem
SIAM Journal on Applied Mathematics, 2015

Theodore Kolokolonikov, Maximizing algebraic
connectivity for certain
families of graphs, Linear Algebra and its Applications 471 (2015):122140.

Justin Tzou, Shuangquan Xie and T. Kolokolnikov,
Firstpassage times, mobile traps, and
Hopf bifurcations, Physical Review E 90 (6), 062138

J. Tzou and T. Kolokolnikov,
Mean first passage time for a small rotating trap inside a reflective
disk,
SIAM Multiscale Modeling & Simulation 13.1(2015)

J. Tzou, M.J. Ward and T. Kolokolnikov,
Slowly varying control parameters, delayed bifurcations,
and the stability of spikes in reactiondiffusion systems,
Physica D: Nonlinear Phenomena 290, 24 (2015).

T. Kolokolnikov, P.G. Kevrekidis, and R. CarreteroGonzales,
A Tale of Two Distributions: From Few To Many Vortices In
QuasiTwoDimensional BoseEinstein Condensates.
Proceedings of the Royal Society A: Mathematical, Physical and Engineering Science 470, 20140048.

Alan Mackey, Theodore Kolokolnikov and Andrea L. Bertozzi,
Twospecies
particle aggregation and stability of codimension one solutions.
, Discrete and Continuous Dynamical Systems B, 19(5), pp. 14111436, 2014
 Yuxin Chen and Theodore Kolokolnikov,
A minimal model of
predatorswarm dynamics,
Journal of the Royal Society Interface 11:20131208 (2014)

T. Kolokolnikov, J. Wei, and A. Alcolado,
Basic mechanisms driving complex spike dynamics in
a chemotaxis model with logistic growth,
SIAM J. Appl. Math., 74(5), pp.13751396 (2014)

T. Kolokolnikov and Alan Lindsay,
Recovering multiple small inclusions inside an a
three dimensional domain using a single measurement
, Published online first,
Inverse Problems in Science & Engineering, 14 Apr
2014.

Theodore Kolokolonikov,
Jose Carrillo,
Andrea Bertozzi,
Razvan Fetecau
and Mark Lewis,
Introduction to Physica D special issue on
Emergent behaviour in multiparticle systems with nonlocal interactions
, Physica D Vol.260, pp.14 (2013)

Andrea L. Bertozzi, James von Brecht and Hui Sun, T. Kolokolnikov and
David Uminsky,
Ring Patterns and Their Bifurcations in a Nonlocal
Model of Biological Swarms, Communications in Mathematical
Sciences 13.4 (2015): 955985

Sorathan Chaturapruek, Jonah Breslau, Daniel Yazdi,
Theodore Kolokolnikov and Scott G. McCalla,
Crime Modeling with Levy Flights
, SIAM J. Appl. Math. (2013), 73(4), 1703.1720.

Yuxin Chen, T. Kolokolnikov and Daniel Zhirov,
Collective behavior of large number of vortices in
the plane
, Proceedings of the Royal Society A (2013), 469:20130085

Theodore Kolokolnikov, Juncheng Wei and Wen Yang,
On Large ring solutions for GiererMeinhardt
system in R^{3}, Journal of Differential Equations (2013).

J. von Brecht, T. Kolokolnikov, A.L. Bertozzi and H. Sun,
Swarming on Random Graphs
, Journal of Statistical Physics,
pp.124 (2013).

ChiunChuan Chen and T. Kolokolnikov,
Simple pde model of spot replication in any dimension,
SIAM
Journal of Mathematical Analysis, 44(5), pp.35643593 (2012)

T. Kolokolnikov, M.J. Ward and J. Wei,
The Stability of SteadyState HotSpot Patterns for a
ReactionDiffusion Model of Urban Crime,
DCDSB, 2014

T. Kolokolnikov, Y. Huang and M. Pavlovski,
Singular patterns for an aggregation model with a confining potential,
Physica D Vol.260, pp.6576 (2013)

Rebecca McKay
Theodore Kolokolnikov and Paul Muir,
Interface oscillations in reactiondiffusion systems above the Hopf
bifurcation
, DCDSB, 17(7), pp. 25232543, 2012.

Wangyi Liu, Andrea L. Bertozzi, and Theodore Kolokolnikov,
Diffuse interface surface tension models in an
expanding flow
, Comm. Math. Sci., 10(1), pp. 387418, 2012.

James von Brecht, David Uminsky, Theodore Kolokolnikov and Andrea L.
Bertozzi,
Predicting pattern formation in particle interactions
, M3AS, vol. 22, Supp. 1, 1140002, 2012

T. Kolokolnikov,
Hui Sun, David Uminsky and Andrea L. Bertozzi,
Stability of ring patterns arising from twodimensional particle
interactions
Phys. Rev. E 84, 015203(R) (2011)

R. C. Fetecau,
Y. Huang and
T. Kolokolnikov,
Swarm dynamics and equilibria for a nonlocal aggregation model
Nonlinearity, Vol. 24, No. 10, pp. 26812716 (2011).

T. Kolokolnikov and Juncheng Wei,
Stability of spiky solutions in a competition model with
crossdiffusion
, SIAM J. Appl. Math. 71, pp. 14281457 (2011).

Rebecca McKay and Theodore Kolokolnikov,
Stability transitions and dynamics of localized patterns near the
shadow limit of reactiondiffusion systems
, DCDSB, 17(1), pp. 191220 (2012)

Sina Adl, David Iron and T. Kolokolnikov,
A threshold area ratio of organic to conventional agriculture causes
recurrent pathogen outbreaks in organic agriculture
, Science of The Total Environment 409(11):21922197, 2011.

T. Kolokolnikov and Xiaofeng Ren
Smokering solutions of GiererMeinhardt System in R^{3}
, SIAM Journal on Applied Dynamical Systems 10(1):251277, 2011.

Adam Alcolado, T. Kolokolnikov and David Iron,
Instability thresholds in the microwave heating model with
exponential nonlinearity, European Journal of Applied Mathematics,
22(3):187216, 2011.

M. J. Ward,
Samara Pillay, Anthony Peirce and T. Kolokolnikov,
An Asymptotic Analysis of the Mean First Passage Time for Narrow Escape
Problems: Part I: TwoDimensional Domains
SIAM J. on Multiscale Modelling and Simulation, 8(3):803835, 2010.

T. Kolokolnikov, Book review: Mathematics and Technology by
Christiane Rousseau and Yvan SaintAubin
, CMS notes, vol. 41 no. 2 March 2009

T. Kolokolnikov, Juncheng Wei and Matthias Winter,
Existence and stability analysis of spiky solutions for the
gierermeinhardt system with large reaction rates,
Physica D: Nonlinear Phenomena
Volume 238(16) 2009, Pages 16951710

T. Kolokolnikov, Chunhua Ou and Yuan Yuan,
Profiles of selfshading, sinking phytoplankton with finite depth
J. Math. Bio, 2009 Jul;59(1):10522

T. Kolokolnikov, M.J. Ward and J. Wei,
Spot selfreplication and dynamics for Schnakenburg model in two dimensional domain
J. Nonlinear Science, 2008, 14321467

David Iron, T. Kolokolnikov, John Rumsey and Juncheng Wei,
Stability of curved interfaces in the perturbed twodimensional
AllenCahn system
SIAM J. Appl. Math. Volume 69, Issue 5, pp. 12281243 (2009)

T. Kolokolnikov and J. Wei,
Positive clustered layered solutions for the
GiererMeinhardt system,
JDE 2008, 245(15), 964993

Mustapha Tlidi, Majid Taki and T. Kolokolnikov,
Introduction: Dissipative localized structures in extended systems,
CHAOS 17, 037101 ,2007

T. Kolokolnikov and Mustapha Tlidi,
Spot deformation and replication in the twodimensional
BelousovZhabotinski reactiondiffusion system,
Physical Review Letters, Vol.98
2007(18), article 188303.

T. Kolokolnikov, M. Ward and J. Wei,
Selfreplication of mesa patterns in reactiondiffusion
models
Physica D, Vol.236(2), 2007, Pages 104122

T. Kolokolnikov, M. Ward, W. Sun and J. Wei,
The Stability of a Stripe for the
GiererMeinhardt Model and the Effect of Saturation
SIAM J. Appl. Dyn. Systems, Vol. 5, No. 2, (2006), pp.~313363.

K. Kang, T. Kolokolnikov and M.J. Ward,
The Stability and Dynamics of a Spike in the OneDimensional
KellerSegel model
,
IMA J Appl Math, April 2007; 72: 140  162.

T. Kolokolnikov, T. Erneux and J. Wei,
Mesatype patterns in the onedimensional Brusselator and their stability
Physica D 214(2006) 6377

T. Kolokolnikov, M.J. Ward and J. Wei,
Slow Translational
Instabilities of Spike Patterns in the OneDimensional GrayScott Model,
,
Interfaces and Free Boundaries, Vol. 8, No. 2, (2006), pp.~185222.

T. Kolokolnikov, M.J. Ward and J. Wei,
The Existence and
Stability of Spike Equilibria in the OneDimensional GrayScott Model:
The Low Feed Rate Regime,
Studies in Appl. Math., Vol. 115, No. 1, (2005), pp.~2171.

T. Kolokolnikov, M. J. Ward and Michele Titcombe,
Optimizing the Fundamental Neumann Eigenvalue for the Laplacian in a Domain with
Small Traps,
European Journal of Applied Math, 2005(16), 161200.

T. Kolokolnikov, M. J. Ward and Juncheng Wei,
The Existence and Stability of Spike Equilibria in the OneDimensional GrayScott
Model on a FiniteDomain,
Applied Math Letters, Vol. 18, No. 8, (2005), pp.~951956.

T. Kolokolnikov, M.
J. Ward and Juncheng Wei
Zigzag and Breakup Instabilities of Stripes and Rings in the
TwoDimensional GrayScott Model
Studies in Appl. Math., Vol. 16, No. 1, (2006), pp.~3595

T. Kolokolnikov, Thomas Erneux, Michele Nizette, Nicolas Joly and
Serge Bielawski,
The Qswitching instability in passively
modelocked lasers
Physica D 219 (2006) 1321

T. Kolokolnikov, M.
J. Ward and Juncheng Wei,
The Existence and Stability of Spike Equilibria in the OneDimensional GrayScott
Model: The PulseSplitting Regime
,
Physica D, Vol. 202, No. 34, (2005), pp. 258293.

T. Kolokolnikov and
Juncheng Wei,
On RingLike Solutions for the GrayScott Model:
Existence, Instability and SelfReplicating Rings
,
Eur. J. Appl. Math., 16(2005), no.2, 201237.

T. Kolokolnikov and M.
J. Ward,
Bifurcation of Spike Equilibria in the
NearShadow GiererMeinhardt Model ,
Discrete and Continuous Dynamical Systems B,
Vol. 4, No. 4, November 2003, 32 pages.

T. Kolokolnikov and M.
J. Ward,
Reduced Wave Green's Functions and Their
Effect on the Dynamics of a Spike for the GiererMeinhardt Model,
European J. Applied Math, Vol. 14, No. 5, (2003), pp. 513545.

T. Kolokolnikov and Edgardo S. ChebTerrab
Firstorder Ordinary Differential Equations,
Symmetries and Linear Transformations
,
European J. Appl. Math, Vol 14(2), 2003, p. 231246
Selected conference presentations and talks

Law of mass action and saturation in SIR model with applications to coronavirus.
, talk given at CAIMSPIMS corona conference (June 2020).
Abstract: It is common in SIR models to assume that the infection rate is proportional
to the product S*I of susceptible and infected individuals.
This form is motivated by the law of mass action from chemistry.
While this assumption works at the onset of the outbreak, it needs to
be modified at higher rates such as seen currently in much of the
world (as of June 2020). We propose a physicsbased model which leads
to a simple saturation formula based on first principles incorporating the spread radius and population density.
We then apply this modified SIR model to coronavirus and show that it fits
much better than the ``classical'' law of mass action.
 Vortex knots on a torus and their stability
, talk given at SIAM Snowbird conference, May 2019.
 Pattern density distribution in PDE's
, talk given in Shanghai, March 2019 Workshop on reactiondiffusion systems.
 Talk about a local PDE model of bacterial aggregation given at SIAM PDE 2015: pdf
pdf plus movies (movies can be played by
clicking on them from the pdf file)

Patterned vegetation, tipping points, and the rate
of climate change, a talk given at CAIMS 2015 (Waterloo)
 Lectures given at GSSI, L'Aquila, Italy, June 2015
 Talk 1 and 2: see stuff on Turing and Delayed bifurcations
here.

Talk 3 and 4. See
here
for detailed calculation
for spikes stability for GM model. Uses movies see link below.
 Talk 5 Uses movies see link below.
 Talk 6 Uses movies see link below.
 Movies for talks 3 to 6 (35 meg; place in the same directory as pdf
files and then the movies can be invoked from the links in pdf files).
 See also course on pattern
formation.
 Predatorprey interactions, SIAM Snowbird 2015

Applications of aggregation model
with Newtonian repulsion, Madrid 2014.

Vortex crystals, animal skin patterns and ice fishing
(Click here for full version including
all the movies for this talk);
click here for abbreviated PDF version
Abstract:
I discuss three very different topics which turn out to have a
related mathematical formulation.
The first topic is classical vortex dynamics in the plane. We look for
relative equilibria (lattice crystals) in the limit of large number of
vortices. Many results for the steady state and its local stability
can be obtained taking the meanfield limit.
Next, we consider hotspot solutions to certain reactiondiffusion PDE
system. Similar PDE's are often used to model spots on the animal
skins. We derive a reduced system of ODE's which describe the motion
and locations of these spots. In certain regimes, we show that these
ODE's are related to the relative equlibria of the vortex model.
Finally, we describe the problem of mean first passage time. Suppose
you live in Canada and you want to catch a fish in a lake covered by
ice. Where do you drill a hole to maximize your chances? This question
can be reformulated in terms of a random walk of brownian particle;
the answer is given in terms of a certain optimization problem. Its
solution is again related to vortex crystals.

Hot spots in crime model,
See the
video of the talk; or the pdf slides.
Presented during PIMS Hot Topics Workshop on Computational Criminology,
Sept 2012.

Localized structures, their stability and
dynamics in PDEs,
Click here for full version including all of the movies (20meg)
;
Click here for abbreviated PDF version;

Complex patterns in patricle aggregation
models of biological formation
CRM/McGill Applied math seminar, Montreal, March 2011

Instability thresholds for a crossdiffusion model and for a crime
model
, Croucher advanced study institute, Hong Kong, March 2011.

Exact solutions and dynamics for the aggregation model with singular
repulsion and longrange attraction
, joint AMSSIAM meeting, New Orelans, January 2011.

Boundary value problems with very sharp structures: numerical challenges
, BIRS workshop on geometric and numeric tools for differential
equations, Aug 2010.

Ring patterns in patrticle aggregation models
, CAIMS meeting, July 2010, St.John's Newfoundland.

A simple PDE model of selfreplication in any dimension
, CMS Winter meeting, Dec 2009, Windsor, Canada.

Combinatorics and PDEs
Abstract: How many solutions does the equation
+/1 +/ 2 +/ 3 + . . .+/ n = 0 have?
In the limit of large n, we derive an
asymptotic formula by using the fundamental solution of the
heat equation.

Boundary value problems with extremely sharp interfaces
, Bluenose numerical analysis days, Dalhousie, June 2008

Ring and smokering solutions in GiererMeinhardt model
, Second CanadaFrance Congress, June 2008

Stability of curved interfaces in the perturbed twodimensional
AllenCahn system
, Applied math Seminar, invited speaker, George
Washigton University, 2007.

First Joint CMS/SMM Meeting, Guanajuato, Mexico, 2006,
Spot replication in BelouzovZhabotinskii reaction.

Invited speaker, Workshop on reactiondiffusion systems, Hong Kong 2006.

Selfreplication
in reactiondiffusion
systems, Limit Problems in Analysis, May 2006, Leiden.

Coarsening and selfreplication of
mesa patterns in reactiondiffusion
systems, Dalhouse colloquium, February 2006

The Qswitching instability in passively
modelocked lasers
CLEO 2005, Munich, Germany

Spots, stripes, and labyrinths
in reaction diffusion systems,
SIAM Dynamical systems 2005, Snowbird,
Utah

Mesatype structures and their stability in the Brusselator
,
Free boundary problems: theory and applications, June 2005,
University of Coimbra, Portugal
 Young mathematicians Conference in PDE and Dynamical Systems (Jan 2004).
Speaker, Stripe Instabilities in the TwoDimensional GrayScott Model
 SIAM Dynamical systems conference, Utah (May 2003).
Speaker, Nearshadow GiererMeinhardt system: spike bifurcations and spikes
near the boundary
 BIRS workshop on pattern formation, Alberta (August 2003).
Speaker, Coupled regime of the GrayScott model: selfreplication and oscillatory
instabilities

With E.S. ChebTerrab and A.D. Roche, 1999, The Search for and Classification
of Integrable Abel ODE Classes. Poster for ISSAC
1999
Other
PhD thesis:
Pattern formation in reactiondiffusion models far from the Turing
regime,
2004. Supervisor: M.J. Ward.
T. Kolokolnikov, Braxton Osting and James von Brecht,
Algebraic connectivity of ErdosRenyi graphs near
the connectivity threshold
, unpublished.

Toroidal knots and their stability, from the paper:
T. Kolokolnikov, C. Ticknor, and P. G. Kevrekidis,
Toroidal Vortex Filament Knots and Links: Existence, Stability and Dynamics
T.Kolokolnikov,
Mathematics on the chopping block
Mathematical Intelligencer, Dec 2017
Flock of sheep in Argentina avoiding the shepherd in the
middle.
Photograph by
Yann Arthus Bertrand.
To accompany the paper
Yuxin Chen and Theodore Kolokolnikov,
A minimal model of
predatorswarm dynamics,
Journal of the Royal Society Interface 11:20131208 (2014)
Cover image for Proceedings of the Royal Society A, to accompany the paper
Collective behavior of large number of vortices in the plane by Y.Chen, D.Zhirov and T.Kolokolnikov.
Physica D, Volume 260, (1 October 2013)
Special issue on Emergent Behaviour in Multiparticle Systems with Nonlocal Interactions,
edited by Theodore Kolokolnikov, Andrea Bertozzi, Razvan Fetecau and Mark Lewis
Chaos 17, 037101 (2007),
Focus issue on Dissipative localized structures in extended systems,
edited by
Mustapha Tlidi, Majid Taki and T. Kolokolnikov. Cover artwork by Yasumasa Nishiura.
