[ preprints | papers | talks | other] see also Google scholar author profile

Preprints

  1. Theodore kolokolnikov and David Iron Law of mass action and saturation in SIR model with application to Coronavirus modelling
  2. Shuangquan Xie, Theodore kolokolnikov, Yasumasa Nishiura, Complex oscillatory motion of multiple spikes in a three-component Schnakenberg system
  3. Chunyi Gai, David Iron, Theodore Kolokolnikov, and John Rumsey, Spike dynamics in the presence of noise
  4. T. Kolokolnikov, Christopher Ticknor, and P. G. Kevrekidis, Toroidal Vortex Filament Knots and Links: Existence, Stability and Dynamics
  5. N. Khalil D. Iron T. Kolokolnokov, Stability and Dynamics of Spike-type Solutions to Delayed Gierer-Meinhart Equations

Papers

  1. Theodore Kolokolnikov, Frederic Paquin-Lefebvre and Michael Ward, Stable asymmetric spike equilibria for the Gierer-Meinhardt model with a precursor field, The IMA Journal of Applied Mathematics, 2020.
  2. Chunyi Gai, David Iron Theodore Kolokolnokov, Localized outbreaks in S-I-R model with diffusion, JMB, 2020
  3. Theodore Kolokolnikov and Juncheng Wei, Hexagonal spike clusters for some PDE's in 2D , DCDS-B
  4. Theodore Kolokolnikov and Shuangquan Xie, Spike density distribution for the Gierer-Meinhardt model with precursor , Physica D, 2019.
  5. Theodore Kolokolnikov, Michael Ward, Justin Tzou and Juncheng Wei, Stabilizing a homoclinic stripe, Proc. Royal. Soc. London A., 2018, 376(2135): 20180110.
  6. Kevin P. O'Keeffe, Joep H.M. Evers, Theodore Kolokolnikov, Ring states in swarmalator systems , Physical Review E 98, 022203 (2018)
  7. Shuangquan Xie, Panayotis G. Kevrekidis and Theodore Kolokolnikov, Multi-vortex crystal lattices in Bose-Einstein Condensates with a rotating trap , Proc. Royal. Soc. London A., 474.2213 (2018):20170553.
  8. Julien Smith-Roberge, David Iron and Theodore Kolokolnikov, Pattern Formation in Bacterial Colonies with Density-Dependent Diffusion, European Journal of Applied Mathematics (2018): 1-23.
  9. T.Kolokolnikov and J. Wei, Pattern formation in a reaction-diffusion system with space-depenent feed rate , SIAM Review, 2018
  10. Joep H.M. Evers, David Iron, Theodore Kolokolnikov and John Rumsey, Agent-based model of the effect of globalization on inequality and class mobility, Physica D (2017)
  11. J.H.M. Evers, R.C. Fetecau and T. Kolokolnikov, Equilibria for an aggregation model with two species, SIADS, 2017.
  12. Chiun-Chuan Chen, Chih-Chiang Huang and Theodore Kolokolnikov, Critical Exponent of a Simple Model of Spot Replication, JDE, 2017
  13. T.Kolokolnikov, Mathematics on the chopping block The Mathematical Intelligencer, 2017.
  14. J.C. Tzou, S. Xie, T. Kolokolnikov, and M.J. Ward, The stability and slow dynamics of localized spot patterns for the 3-D Schnakenberg reaction-diffusion model, SIAM J. Appl. Dyn. Sys. (2016): 35.
  15. S.Xie and T.Kolokolnikov, Moving and jumping spot in a two dimensional reaction-diffusion model, Nonlinearity (2017).
  16. 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)
  17. 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).
  18. P-C Chavy-Waddy and T. Kolokolnikov, A local PDE model of aggregation formation in bacterial colonies, Nonlinearity 29.10 (2016): 3174
  19. J.C. Tzou, P.G. Kevrekidis, T. Kolokolnikov, and R. Carretero-Gonzales, Weakly nonlinear analysis of vortex formation in a dissipative variant of the Gross-Pitaevskii equation, SIADS, 2016
  20. R. Carretero-González, P.G. Kevrekidis and T. Kolokolnikov, Vortex Nucleation in a Dissipative Variant of the Nonlinear Schrödinger Equation under Rotation, Physica D (2016):1-14.
  21. H. Tompkins and T. Kolokolnikov, Swarm shape and its dynamics in a predator-swarm model, SIAM Undergraduate Research Online, 2014
  22. Y. Chen, T. Kolokolnikov, J. Tzou, and C. Gai, Patterned vegetation, tipping points, and the rate of climate change, EJAM, 2015
  23. Shuangquan Xie and T. Kolokolnikov, Oscillations of many interfaces in the near-shadow regime of two-component reaction-diffusion systems, DCDS-B, 2016
  24. 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.
  25. 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
  26. Theodore Kolokolonikov, Maximizing algebraic connectivity for certain families of graphs, Linear Algebra and its Applications 471 (2015):122-140.
  27. Justin Tzou, Shuangquan Xie and T. Kolokolnikov, First-passage times, mobile traps, and Hopf bifurcations, Physical Review E 90 (6), 062138
  28. 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)
  29. J. Tzou, M.J. Ward and T. Kolokolnikov, Slowly varying control parameters, delayed bifurcations, and the stability of spikes in reaction-diffusion systems, Physica D: Nonlinear Phenomena 290, 24 (2015).
  30. T. Kolokolnikov, P.G. Kevrekidis, and R. Carretero-Gonzales, A Tale of Two Distributions: From Few To Many Vortices In Quasi-Two-Dimensional Bose-Einstein Condensates. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Science 470, 20140048.
  31. Alan Mackey, Theodore Kolokolnikov and Andrea L. Bertozzi, Two-species particle aggregation and stability of co-dimension one solutions. , Discrete and Continuous Dynamical Systems B, 19(5), pp. 1411-1436, 2014
  32. Yuxin Chen and Theodore Kolokolnikov, A minimal model of predator-swarm dynamics, Journal of the Royal Society Interface 11:20131208 (2014)
  33. 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.1375-1396 (2014)
  34. 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.
  35. Theodore Kolokolonikov, Jose Carrillo, Andrea Bertozzi, Razvan Fetecau and Mark Lewis, Introduction to Physica D special issue on Emergent behaviour in multi-particle systems with non-local interactions , Physica D Vol.260, pp.1-4 (2013)
  36. 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): 955-985
  37. 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.
  38. 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
  39. Theodore Kolokolnikov, Juncheng Wei and Wen Yang, On Large ring solutions for Gierer-Meinhardt system in R3, Journal of Differential Equations (2013).
  40. J. von Brecht, T. Kolokolnikov, A.L. Bertozzi and H. Sun, Swarming on Random Graphs , Journal of Statistical Physics, pp.1-24 (2013).
  41. Chiun-Chuan Chen and T. Kolokolnikov, Simple pde model of spot replication in any dimension, SIAM Journal of Mathematical Analysis, 44(5), pp.3564-3593 (2012)
  42. T. Kolokolnikov, M.J. Ward and J. Wei, The Stability of Steady-State Hot-Spot Patterns for a Reaction-Diffusion Model of Urban Crime, DCDS-B, 2014
  43. T. Kolokolnikov, Y. Huang and M. Pavlovski, Singular patterns for an aggregation model with a confining potential, Physica D Vol.260, pp.65-76 (2013)
  44. Rebecca McKay Theodore Kolokolnikov and Paul Muir, Interface oscillations in reaction-diffusion systems above the Hopf bifurcation , DCDS-B, 17(7), pp. 2523-2543, 2012.
  45. Wangyi Liu, Andrea L. Bertozzi, and Theodore Kolokolnikov, Diffuse interface surface tension models in an expanding flow , Comm. Math. Sci., 10(1), pp. 387-418, 2012.
  46. James von Brecht, David Uminsky, Theodore Kolokolnikov and Andrea L. Bertozzi, Predicting pattern formation in particle interactions , M3AS, vol. 22, Supp. 1, 1140002, 2012
  47. T. Kolokolnikov, Hui Sun, David Uminsky and Andrea L. Bertozzi, Stability of ring patterns arising from two-dimensional particle interactions Phys. Rev. E 84, 015203(R) (2011)
  48. R. C. Fetecau, Y. Huang and T. Kolokolnikov, Swarm dynamics and equilibria for a nonlocal aggregation model Nonlinearity, Vol. 24, No. 10, pp. 2681-2716 (2011).
  49. T. Kolokolnikov and Juncheng Wei, Stability of spiky solutions in a competition model with cross-diffusion , SIAM J. Appl. Math. 71, pp. 1428-1457 (2011).
  50. Rebecca McKay and Theodore Kolokolnikov, Stability transitions and dynamics of localized patterns near the shadow limit of reaction-diffusion systems , DCDS-B, 17(1), pp. 191-220 (2012)
  51. 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):2192-2197, 2011.
  52. T. Kolokolnikov and Xiaofeng Ren Smoke-ring solutions of Gierer-Meinhardt System in R3 , SIAM Journal on Applied Dynamical Systems 10(1):251-277, 2011.
  53. Adam Alcolado, T. Kolokolnikov and David Iron, Instability thresholds in the microwave heating model with exponential nonlinearity, European Journal of Applied Mathematics, 22(3):187-216, 2011.
  54. 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: Two-Dimensional Domains SIAM J. on Multiscale Modelling and Simulation, 8(3):803-835, 2010.
  55. T. Kolokolnikov, Book review: Mathematics and Technology by Christiane Rousseau and Yvan Saint-Aubin , CMS notes, vol. 41 no. 2 March 2009
  56. T. Kolokolnikov, Juncheng Wei and Matthias Winter, Existence and stability analysis of spiky solutions for the gierer-meinhardt system with large reaction rates, Physica D: Nonlinear Phenomena Volume 238(16) 2009, Pages 1695-1710
  57. T. Kolokolnikov, Chunhua Ou and Yuan Yuan, Profiles of self-shading, sinking phytoplankton with finite depth J. Math. Bio, 2009 Jul;59(1):105-22
  58. T. Kolokolnikov, M.J. Ward and J. Wei, Spot self-replication and dynamics for Schnakenburg model in two dimensional domain J. Nonlinear Science, 2008, 1432-1467
  59. David Iron, T. Kolokolnikov, John Rumsey and Juncheng Wei, Stability of curved interfaces in the perturbed two-dimensional Allen-Cahn system SIAM J. Appl. Math. Volume 69, Issue 5, pp. 1228-1243 (2009)
  60. T. Kolokolnikov and J. Wei, Positive clustered layered solutions for the Gierer-Meinhardt system, JDE 2008, 245(15), 964-993
  61. Mustapha Tlidi, Majid Taki and T. Kolokolnikov, Introduction: Dissipative localized structures in extended systems, CHAOS 17, 037101 ,2007
  62. T. Kolokolnikov and Mustapha Tlidi, Spot deformation and replication in the two-dimensional Belousov-Zhabotinski reaction-diffusion system, Physical Review Letters, Vol.98 2007(18), article 188303.
  63. T. Kolokolnikov, M. Ward and J. Wei, Self-replication of mesa patterns in reaction-diffusion models Physica D, Vol.236(2), 2007, Pages 104-122
  64. T. Kolokolnikov, M. Ward, W. Sun and J. Wei, The Stability of a Stripe for the Gierer-Meinhardt Model and the Effect of Saturation SIAM J. Appl. Dyn. Systems, Vol. 5, No. 2, (2006), pp.~313-363.
  65. K. Kang, T. Kolokolnikov and M.J. Ward, The Stability and Dynamics of a Spike in the One-Dimensional Keller-Segel model , IMA J Appl Math, April 2007; 72: 140 - 162.
  66. T. Kolokolnikov, T. Erneux and J. Wei, Mesa-type patterns in the one-dimensional Brusselator and their stability Physica D 214(2006) 63-77
  67. T. Kolokolnikov, M.J. Ward and J. Wei, Slow Translational Instabilities of Spike Patterns in the One-Dimensional Gray-Scott Model, , Interfaces and Free Boundaries, Vol. 8, No. 2, (2006), pp.~185-222.
  68. T. Kolokolnikov, M.J. Ward and J. Wei, The Existence and Stability of Spike Equilibria in the One-Dimensional Gray-Scott Model: The Low Feed Rate Regime, Studies in Appl. Math., Vol. 115, No. 1, (2005), pp.~21-71.
  69. 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), 161-200.
  70. T. Kolokolnikov, M. J. Ward and Juncheng Wei, The Existence and Stability of Spike Equilibria in the One-Dimensional Gray-Scott Model on a Finite-Domain, Applied Math Letters, Vol. 18, No. 8, (2005), pp.~951-956.
  71. T. Kolokolnikov, M. J. Ward and Juncheng Wei Zigzag and Breakup Instabilities of Stripes and Rings in the Two-Dimensional Gray-Scott Model Studies in Appl. Math., Vol. 16, No. 1, (2006), pp.~35-95
  72. T. Kolokolnikov, Thomas Erneux, Michele Nizette, Nicolas Joly and Serge Bielawski, The Q-switching instability in passively mode-locked lasers Physica D 219 (2006) 13-21
  73. T. Kolokolnikov, M. J. Ward and Juncheng Wei, The Existence and Stability of Spike Equilibria in the One-Dimensional Gray-Scott Model: The Pulse-Splitting Regime , Physica D, Vol. 202, No. 3-4, (2005), pp. 258-293.
  74. T. Kolokolnikov and Juncheng Wei, On Ring-Like Solutions for the Gray-Scott Model: Existence, Instability and Self-Replicating Rings , Eur. J. Appl. Math., 16(2005), no.2, 201-237.
  75. T. Kolokolnikov and M. J. Ward, Bifurcation of Spike Equilibria in the Near-Shadow Gierer-Meinhardt Model , Discrete and Continuous Dynamical Systems B, Vol. 4, No. 4, November 2003, 32 pages.
  76. T. Kolokolnikov and M. J. Ward, Reduced Wave Green's Functions and Their Effect on the Dynamics of a Spike for the Gierer-Meinhardt Model, European J. Applied Math, Vol. 14, No. 5, (2003), pp. 513-545.
  77. T. Kolokolnikov and Edgardo S. Cheb-Terrab First-order Ordinary Differential Equations, Symmetries and Linear Transformations , European J. Appl. Math, Vol 14(2), 2003, p. 231-246

Selected conference presentations and talks

  1. Law of mass action and saturation in SIR model with applications to coronavirus. , talk given at CAIMS-PIMS 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 physics-based 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.
  2. Vortex knots on a torus and their stability , talk given at SIAM Snowbird conference, May 2019.
  3. Pattern density distribution in PDE's , talk given in Shanghai, March 2019 Workshop on reaction-diffusion systems.
  4. 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)
  5. Patterned vegetation, tipping points, and the rate of climate change, a talk given at CAIMS 2015 (Waterloo)
  6. Lectures given at GSSI, L'Aquila, Italy, June 2015
    1. Talk 1 and 2: see stuff on Turing and Delayed bifurcations here.
    2. Talk 3 and 4. See here for detailed calculation for spikes stability for GM model. Uses movies see link below.
    3. Talk 5 Uses movies see link below.
    4. Talk 6 Uses movies see link below.
    5. 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).
    6. See also course on pattern formation.
  7. Predator-prey interactions, SIAM Snowbird 2015
  8. Applications of aggregation model with Newtonian repulsion, Madrid 2014.
  9. 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 mean-field limit. Next, we consider hot-spot solutions to certain reaction-diffusion 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.
  10. 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.
  11. 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;
  12. Complex patterns in patricle aggregation models of biological formation CRM/McGill Applied math seminar, Montreal, March 2011
  13. Instability thresholds for a cross-diffusion model and for a crime model , Croucher advanced study institute, Hong Kong, March 2011.
  14. Exact solutions and dynamics for the aggregation model with singular repulsion and long-range attraction , joint AMS-SIAM meeting, New Orelans, January 2011.
  15. Boundary value problems with very sharp structures: numerical challenges , BIRS workshop on geometric and numeric tools for differential equations, Aug 2010.
  16. Ring patterns in patrticle aggregation models , CAIMS meeting, July 2010, St.John's Newfoundland.
  17. A simple PDE model of self-replication in any dimension , CMS Winter meeting, Dec 2009, Windsor, Canada.
  18. 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.
  19. Boundary value problems with extremely sharp interfaces , Bluenose numerical analysis days, Dalhousie, June 2008
  20. Ring and smoke-ring solutions in Gierer-Meinhardt model , Second Canada-France Congress, June 2008
  21. Stability of curved interfaces in the perturbed two-dimensional Allen-Cahn system , Applied math Seminar, invited speaker, George Washigton University, 2007.
  22. First Joint CMS/SMM Meeting, Guanajuato, Mexico, 2006, Spot replication in Belouzov-Zhabotinskii reaction.
  23. Invited speaker, Workshop on reaction-diffusion systems, Hong Kong 2006.
  24. Self-replication in reaction-diffusion systems, Limit Problems in Analysis, May 2006, Leiden.
  25. Coarsening and self-replication of mesa patterns in reaction-diffusion systems, Dalhouse colloquium, February 2006
  26. The Q-switching instability in passively mode-locked lasers CLEO 2005, Munich, Germany
  27. Spots, stripes, and labyrinths in reaction diffusion systems, SIAM Dynamical systems 2005, Snowbird, Utah
  28. Mesa-type structures and their stability in the Brusselator , Free boundary problems: theory and applications, June 2005, University of Coimbra, Portugal
  29. Young mathematicians Conference in PDE and Dynamical Systems (Jan 2004). Speaker, Stripe Instabilities in the Two-Dimensional Gray-Scott Model
  30. SIAM Dynamical systems conference, Utah (May 2003). Speaker, Near-shadow Gierer-Meinhardt system: spike bifurcations and spikes near the boundary
  31. BIRS workshop on pattern formation, Alberta (August 2003). Speaker, Coupled regime of the Gray-Scott model: self-replication and oscillatory instabilities
  32. With E.S. Cheb-Terrab 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 reaction-diffusion models far from the Turing regime, 2004. Supervisor: M.J. Ward.
  • T. Kolokolnikov, Braxton Osting and James von Brecht, Algebraic connectivity of Erdos-Renyi 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 predator-swarm 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 Multi-particle Systems with Non-local 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.

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