New Book

Radek Erban and Jonathan Chapman, "Stochastic Modelling of Reaction-Diffusion Processes", Cambridge Texts in Applied Mathematics, Cambridge University Press, 308 pages (2020)

Our new book provides an introduction to stochastic methods for modelling biological systems, covering a number of applications, ranging in size from molecular dynamics simulations of small biomolecules to stochastic modelling of groups of animals. The focus is on the underlying mathematics, i.e. it is not assumed that the reader took any advanced courses in biology or chemistry. The book can be used both for self-study and as a supporting text for advanced undergraduate or beginning graduate-level courses in applied mathematics. It discusses the essence of mathematical methods which appear (under different names) in a number of interdisciplinary scientific fields bridging mathematics and computations with biology and chemistry. New mathematical approaches and their analysis are explained using simple examples of biological models. The book starts with stochastic modelling of chemical reactions, introducing stochastic simulation algorithms and mathematical methods for analysis of stochastic models. Different stochastic spatio-temporal models are then studied, including models of diffusion and stochastic reaction-diffusion modelling. The methods covered include molecular dynamics, Brownian dynamics, velocity jump processes and compartment-based models. [download the table of contents] [online course based on our book]

Papers/preprints available to download (pdf files)

86. Gheorghe Craciun and Radek Erban, "Planar chemical reaction systems with algebraic and non-algebraic limit cycles", submitted to Journal of Mathematical Biology (2024)
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85. Radek Erban and Stefanie Winkelmann, "Multi-grid reaction-diffusion master equation: applications to morphogen gradient modelling", submitted to Bulletin of Mathematical Biology (2024)
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84. Radek Erban and Yuichi Togashi, "Asymmetric periodic boundary conditions for molecular dynamics and coarse-grained simulations of nucleic acids", Journal of Physical Chemistry B, Volume 127, Number 38, pp. 8257-8267 (2023)
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83. Radek Erban and Hye-Won Kang, "Chemical Systems with Limit Cycles", Bulletin of Mathematical Biology, Volume 85, Article 76 (2023)
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82. Jonathan Utterson and Radek Erban, "Symmetries of many-body systems imply distance-dependent potentials", Physical Review E, Volume 108, Number 1, Article 014122 (2023)
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81. Ravinda Gunaratne, Carlos Floyd, Haoran Ni, Garegin Papoian and Radek Erban, "On Stretching, Bending, Shearing and Twisting of Actin Filaments II: Multi-Resolution Modelling", submitted to Journal of Chemical Theory and Computation (2022)
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80. Carlos Floyd, Haoran Ni, Ravinda Gunaratne, Radek Erban and Garegin Papoian, "On Stretching, Bending, Shearing and Twisting of Actin Filaments I: Variational Models", Journal of Chemical Theory and Computation, Volume 18, Number 8, pp. 4865-4878 (2022)
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79. Jonathan Utterson and Radek Erban, "On standardised moments of force distribution in simple liquids", Physical Chemistry Chemical Physics, Volume 24, pp. 5646-5657 (2022)
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78. Radek Erban and Robert van Gorder, "On the sum of positive divisors functions", Research in Number Theory, Volume 7, article number 25 (2021)
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77. German Enciso, Radek Erban and Jinsu Kim, "Identifiability of stochastically modelled reaction networks", European Journal of Applied Mathematics, Volume 32, Number 5, pp. 865-887 (2021)
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76. Dragos Crisan and Radek Erban, "On the counting function of semiprimes", Integers: Journal of Combinatorial Number Theory, article number A122 (2021)
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75. Radek Erban, "Coarse-graining molecular dynamics: stochastic models with non-Gaussian force distributions", Journal of Mathematical Biology, Volume 80, pp. 457-479 (2020)
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74. Hye Won Kang and Radek Erban, "Multiscale stochastic reaction-diffusion algorithms combining Markov chain models with stochastic partial differential equations", Bulletin of Mathematical Biology, Volume 81, Number 8, pp. 3185-3213 (2019)
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73. Ravinda Gunaratne, Daniel Wilson, Mark Flegg and Radek Erban, "On short-range and long-range interactions in multi-resolution dimer models", Interface Focus, Volume 9, Number 3, rsfs.2018.0070 (2019)
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72. Tomislav Plesa, Hans Othmer and Radek Erban, "Noise-induced mixing and multimodality in reaction networks", European Journal of Applied Mathematics, Volume 30, pp. 887-911 (2019)
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71. Edward Rolls and Radek Erban, "Multi-resolution polymer Brownian dynamics with hydrodynamic interactions", Journal of Chemical Physics, Volume 148, Number 19, 194111 (2018)
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70. Radek Erban, "Coupling all-atom molecular dynamics simulations of ions in water with Brownian dynamics", Proceedings of the Royal Society A, Volume 472, Number 2186, 20150556 (2016)
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69. Julia Stadlmann and Radek Erban, "Limiting stochastic processes of shift-periodic dynamical systems", Royal Society Open Science, Volume 6, Number 11, rsos.191423 (2019)
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68. Edward Rolls, Yuichi Togashi and Radek Erban, "Varying the resolution of the Rouse model on temporal and spatial scales: application to multiscale modelling of DNA dynamics", Multiscale Modeling and Simulation, Volume 15, Number 4, pp. 1672-1693 (2017)
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67. Tomislav Plesa, Konstantinos Zygalakis, David Anderson and Radek Erban, "Noise control for molecular computing", Journal of the Royal Society Interface, Volume 15, Number 144, 20180199 (2018)
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66. Ulrich Dobramysl, Garegin Papoian and Radek Erban, "Steric effects induce geometric remodeling of actin bundles in filopodia", Biophysical Journal, Volume 110, pp. 2066-2075 (2016)
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65. Martin Robinson, Chong Luo, Patrick Farrell, Radek Erban and Apala Majumdar, "From molecular to continuum modelling of bistable liquid crystal devices", Liquid Crystals, DOI: 10.1080/02678292.2017.1290284 (2017)
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64. Tomislav Plesa, Tomas Vejchodsky and Radek Erban, "Test models for statistical inference: two-dimensional reaction systems displaying limit cycle bifurcations and bistability", in Stochastic Dynamical Systems, Multiscale Modeling, Asymptotics and Numerical Methods for Computational Cellular Biology (2017)
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63. Andrew Duncan, Radek Erban, Konstantinos Zygalakis, "Hybrid framework for the simulation of stochastic chemical kinetics", Journal of Computational Physics, Volume 326, pp. 398-419 (2016)
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62. Tomislav Plesa, Tomas Vejchodsky and Radek Erban, "Chemical reaction systems with a homoclinic bifurcation: an inverse problem", Journal of Mathematical Chemistry, Volume 54, Number 10, pp. 1884-1915 (2016)
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61. Jonathan Chapman, Radek Erban and Samuel Isaacson, "Reactive boundary conditions as limits of interaction potentials for Brownian and Langevin dynamics", SIAM Journal on Applied Mathematics, Volume 76, Number 1, pp. 368-390 (2016)
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60. Ulrich Dobramysl, Sten Rudiger and Radek Erban, "Particle-based multiscale modeling of calcium puff dynamics", Multiscale Modelling and Simulation, Volume 14, Number 3, pp. 997-1016 (2016)
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59. Radek Erban, "From molecular dynamics to Brownian dynamics", Proceedings of the Royal Society A, Volume 470, Number 2167, 20140036 (2014)
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58. Radek Erban, Jan Haskovec and Yongzheng Sun, "On Cucker-Smale model with noise and delay", SIAM Journal on Applied Mathematics, Volume 76, Number 4, pp. 1535-1557 (2016)
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57. Martin Robinson, Steven Andrews and Radek Erban, "Multiscale reaction-diffusion simulations with Smoldyn", Bioinformatics, Volume 31, Number 14, pp. 2406-2408 (2015)
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56. Shuohao Liao, Tomas Vejchodsky and Radek Erban, "Tensor methods for parameter estimation and bifurcation analysis of stochastic reaction networks", Journal of the Royal Society Interface, Volume 12, Number 108, 20150233 (2015)
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55. Benjamin Franz, Jake Taylor-King, Christian Yates and Radek Erban, "Hard-sphere interactions in velocity jump models", Physical Review E, Volume 94, Number 1, 012129 (2016)
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54. Mihai Cucuringu and Radek Erban, "ADM-CLE approach for detecting slow variables in continuous time Markov chains and dynamic data", SIAM Journal on Scientific Computing, Volume 39, Number 1, pp. B76-B101 (2017)
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53. Tomas Vejchodsky, Radek Erban and Philip Maini, "Reduction of chemical systems by delayed quasi-steady state assumptions", submitted to Physica D (2014)
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52. Yongzheng Sun, Wei Lin and Radek Erban, "Time delay can facilitate coherence in self-driven interacting particle systems", Physical Review E, Volume 90, Number 6, 062708 (2014)
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51. Simon Cotter and Radek Erban, "Error analysis of diffusion approximation methods for multiscale systems in reaction kinetics", SIAM Journal on Scientific Computing, Volume 38, Number 1, pp. B144-B163 (2016)
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50. Andrew Duncan, Shuohao Liao, Tomas Vejchodsky, Radek Erban and Ramon Grima, "Noise-induced multistability in chemical systems: discrete vs continuum modelling", Physical Review E, Volume 91, 042111 (2015)
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49. Martin Robinson, Mark Flegg and Radek Erban, "Adaptive two-regime method: application to front propagation", Journal of Chemical Physics, Volume 140, Number 12, 124109 (2014)
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48. Jake Taylor-King, Benjamin Franz, Christian Yates and Radek Erban, "Mathematical modelling of turning delays in swarm robotics", IMA Journal of Applied Mathematics, Volume 80, Number 5, pp. 1454-1474 (2015)
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47. Yang Cao and Radek Erban, "Stochastic Turing patterns: analysis of compartment-based approaches", Bulletin of Mathematical Biology, Volume 76, Number 12, pp. 3051-3069 (2014)
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46. Radek Erban, Mark Flegg and Garegin Papoian, "Multiscale stochastic reaction-diffusion modelling: application to actin dynamics in filopodia", Bulletin of Mathematical Biology, Volume 76, Number 4, pp. 799-818 (2014)
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45. Benjamin Franz, Chuan Xue, Kevin Painter and Radek Erban, "Travelling waves in hybrid chemotaxis models", Bulletin of Mathematical Biology, Volume 76, Number 2, pp. 377-400 (2014)
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44. Mark Flegg, Stefan Hellander and Radek Erban, "Convergence of methods for coupling of microscopic and mesoscopic reaction-diffusion simulations", Journal of Computational Physics, Volume 289, pp. 1-17(2015)
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43. Mark Flegg, Jonathan Chapman, Likun Zheng and Radek Erban, "Analysis of the two-regime method on square meshes", SIAM Journal on Scientific Computing, Volume 36, Number 3, pp. B561-B588 (2014)
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42. Margarita Schlackow, Samuel Marguerat, Nicholas Proudfoot, Jurg Bahler, Radek Erban and Monika Gullerova, "Genome-wide analysis of poly(A) site selection in Schizosaccharomyces pombe", RNA, Volume 19, Number 12, pp. 1617-1631 (2013)
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41. Benjamin Franz, Mark Flegg, Jonathan Chapman and Radek Erban, "Multiscale reaction-diffusion algorithms: PDE-assisted Brownian dynamics", SIAM Journal on Applied Mathematics, Volume 73, Number 3, pp. 1224-1247 (2013)
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40. Mark Flegg, Sten Ruediger and Radek Erban, "Diffusive spatio-temporal noise in a first-passage time model for intracellular calcium release", Journal of Chemical Physics, Volume 138, Number 15, 154103 (2013)
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39. Wan Chen, Radek Erban and Jonathan Chapman, "From Brownian dynamics to Markov chain: an ion channel example", SIAM Journal on Applied Mathematics, Volume 74, Number 1, pp. 208-235 (2014)
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38. Simon Cotter, Tomas Vejchodsky and Radek Erban, "Adaptive finite element method assisted by stochastic simulation of chemical systems", SIAM Journal on Scientific Computing, Volume 35, Number 1, pp. B107-B131 (2013)
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37. Mark Flegg, Jonathan Chapman and Radek Erban, "Two Regime Method for optimizing stochastic reaction-diffusion simulations", Journal of the Royal Society Interface, Volume 9, Number 70, pp. 859-868 (2012)
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36. Tamas Szekely, Kevin Burrage, Radek Erban and Konstantinos Zygalakis, "A higher-order numerical framework for stochastic simulation of chemical reaction systems", BMC Systems Biology, Volume 6, 85 (2012)
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35. Chong Luo, Apala Majumdar and Radek Erban, "Multistability in planar liquid crystal wells", Physical Review E, Volume 85, Number 6, 061702, 15 pages (2012)
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34. Simon Cotter, Konstantinos Zygalakis, Ioannis Kevrekidis and Radek Erban, "A Constrained Approach to Multiscale Stochastic Simulation of Chemically Reacting Systems", Journal of Chemical Physics, Volume 135, Number 9, 094102, 12 pages (2011)
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33. Benjamin Franz and Radek Erban, "Hybrid modelling of individual movement and collective behaviour", in Dispersal, individual movement and spatial ecology: A mathematical perspective (edited by M. Lewis, P. Maini and S. Petrovskii), Springer, pp. 129-157 (2013)
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32. Radek Erban and Jan Haskovec, "From individual to collective behaviour of coupled velocity jump processes: a locust example", Kinetic and Related Models, Volume 5, Number 4, pp. 817-842 (2012)
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31. Jana Lipkova, Konstantinos Zygalakis, Jonathan Chapman and Radek Erban, "Analysis of Brownian Dynamics Simulations of Reversible Bimolecular Reactions", SIAM Journal on Applied Mathematics, Volume 71, Number 3, pp. 714-730 (2011)
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30. Guido Klingbeil, Radek Erban, Mike Giles and Philip Maini, "STOCHSIMGPU: Parallel stochastic simulation for the Systems Biology Toolbox 2 for MATLAB", Bioinformatics, Volume 27, Number 8, pp. 1170-1171 (2011)
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29. Ornella Cominetti, Anastasios Matzavinos, Sandhya Samarasinghe, Don Kulasiri, Sijia Liu, Philip Maini and Radek Erban, "DifFUZZY: A fuzzy spectral clustering algorithm for complex data sets", International Journal of Computational Intelligence in Bioinformatics and Systems Biology, Volume 1, Number 4, pp. 402-417 (2010)
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28. Chuan Xue, Hyung Ju Hwang, Kevin Painter and Radek Erban, "Travelling waves in hyperbolic chemotaxis equations", Bulletin of Mathematical Biology Volume 73, Number 8, pp. 1695-1733 (2011)
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27. Guido Klingbeil, Radek Erban, Mike Giles and Philip Maini, "Fat vs. thin threading approach on GPUs: application to stochastic simulation of chemical reactions", IEEE Transactions on Parallel & Distributed Systems, Volume 23, Number 2, pp. 280-287 (2012)
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26. Carlos Escudero, Christian Yates, Jerome Buhl, Iain Couzin, Radek Erban, Ioannis Kevrekidis and Philip Maini, "Ergodic directional switching in mobile insect groups", Physical Review E, Volume 82, Number 1, 011926 (2010)
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25. Christian Yates, Ruth Baker, Radek Erban and Philip Maini, "Refining self-propelled particle models for collective behaviour", Canadian Applied Mathematics Quarterly, Volume 18, Number 3, pp. 299-350 (2010)
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24. Ruth Baker, Christian Yates and Radek Erban, "From microscopic to macroscopic descriptions of cell migration on growing domains", Bulletin of Mathematical Biology, Volume 72, Number 3, pp. 719-762 (2010)
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23. Radek Erban and Jonathan Chapman, "Stochastic modelling of reaction-diffusion processes: algorithms for bimolecular reactions", Physical Biology, Volume 6, Number 4, 046001, 18 pages (2009)
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22. Radek Erban, Jonathan Chapman, Ioannis Kevrekidis and Tomas Vejchodsky, "Analysis of a stochastic chemical system close to a SNIPER bifurcation of its mean-field model", SIAM Journal on Applied Mathematics, Volume 70, Number 3, pp. 984-1016 (2009)
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21. Amit Singer, Radek Erban, Ioannis Kevrekidis and Ronald Coifman, "Detecting the slow manifold by anisotropic diffusion maps", Proceedings of the National Academy of Sciences (PNAS), Volume 106, Number 38, pp. 16090-16095 (2009)
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20. Christian Yates, Radek Erban, Carlos Escudero, Iain Couzin, Jerome Buhl, Ioannis Kevrekidis, Philip Maini and David Sumpter, "Inherent noise can facilitate coherence in collective swarm motion", Proceedings of the National Academy of Sciences (PNAS), Volume 106, Number 14, pp. 5464-5469 (2009)
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19. Chuan Xue, Hans Othmer and Radek Erban, "From individual to collective behavior of unicellular organisms: Recent results and open problems", Multiscale Phenomena in Biology: Proceedings of the 2nd Okinawa Conference on Mathematics and Biology, AIP Conference Proceedings, Volume 1167, Issue 1, pp. 3-14 (2009)
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18. Radek Erban and Jonathan Chapman, "Reactive boundary conditions for stochastic simulations of reaction-diffusion processes", Physical Biology, Volume 4, Number 1, pp. 16-28 (2007)
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17. Radek Erban, Jonathan Chapman and Philip Maini, "A practical guide to stochastic simulations of reaction-diffusion processes", Lecture Notes, available as http://arxiv.org/abs/0704.1908, 35 pages (2007)
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16. Radek Erban and Jonathan Chapman, "Time scale of random sequential adsorption", Physical Review E, Volume 75, Number 4, 041116, 5 pages (2007)
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15. Radek Erban, Thomas Frewen, Xiao Wang, Timothy Elston, Ronald Coifman, Boaz Nadler and Ioannis Kevrekidis, "Variable-free exploration of stochastic models: a gene regulatory network example", Journal of Chemical Physics, Volume 126, Issue 15, 155103, 12 pages (2007)
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14. Radek Erban and Jonathan Chapman, "On chemisorption of polymers to solid surfaces", Journal of Statistical Physics, Volume 127, Number 6, pp. 1255-1277 (2007)
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13. Michail Kavousanakis, Radek Erban, Andreas Boudouvis, William Gear and Ioannis Kevrekidis, "Projective and coarse projective integration for problems with continuous symmetries", Journal of Computational Physics, Volume 225, Issue 1, pp. 382-407 (2007)
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12. Radek Erban and Hans Othmer, "Taxis equations for amoeboid cells", Journal of Mathematical Biology, Volume 54, Number 6, pp. 847-885 (2007)
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11. Radek Erban, Jonathan Chapman, Kerry Fisher, Ioannis Kevrekidis and Leonard Seymour, "Dynamics of polydisperse irreversible adsorption: a pharmacological example", Mathematical Models and Methods in Applied Sciences (M3AS), Volume 17, Number 5, pp. 759-781 (2007)
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10. Liang Qiao, Radek Erban, Carl T. Kelley and Ioannis Kevrekidis, "Spatially distributed stochastic systems: equation-free and equation-assisted preconditioned computation", Journal of Chemical Physics, Volume 125, Issue 20, 204108, 8 pages (2006)
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9. Radek Erban and Hyung Ju Hwang, "Global existence results for complex hyperbolic models of bacterial chemotaxis", Discrete and Continuous Dynamical Systems Series B, Volume 6, Number 6, pp. 1239-1260 (2006)
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8. Radek Erban and Hans Othmer, "Macroscopic taxis equations from cell-based models", Oberwolfach Reports, Volume 3, Issue 2, pp. 1409-1412 (2006)
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7. Radek Erban, Ioannis Kevrekidis, David Adalsteinsson and Timothy Elston, "Gene regulatory networks: a coarse-grained, equation-free approach to multiscale computation", Journal of Chemical Physics, Volume 124, Issue 8, 084106, 17 pages (2006)
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6. Radek Erban, Ioannis Kevrekidis and Hans Othmer, "An equation-free computational approach for extracting population-level behavior from individual-based models of biological dispersal", Physica D, Volume 215, Issue 1, pp. 1-24 (2006)
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5. Radek Erban and Hans Othmer, "From signal transduction to spatial pattern formation in E.coli : A paradigm for multi-scale modeling in biology", Multiscale Modeling and Simulation, Volume 3, Number 2, pp. 362-394 (2005)
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4. Radek Erban & Hans Othmer, "From individual to collective behavior in bacterial chemotaxis", SIAM Journal on Applied Mathematics, Volume 65, Number 2, pp. 361-391 (2004)
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3. Radek Erban, "On the existence of solutions to the Navier-Stokes equations of a two dimensional compressible flow", Mathematical Methods in the Applied Sciences, Volume 26, Issue 6, pp. 489-517 (2003)
   
   The original 60 page version of the manuscript is available here. It contains detailed proofs of useful theorems about Orlicz spaces and other results which had to be skipped in the published version. A shorter version of the paper (30 pages) is available here. This short version is similar to the published version. The published version of the manuscript is available online on the journal webpage www3.interscience.wiley.com/cgi-bin/jhome/2197.
2. Radek Erban "Navier-Stokes equations of compressible flow in exterior domains - existence of weak solutions", unpublished notes (2002)
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1. Radek Erban, "On the static limit solutions to the Navier-Stokes equations of compressible flow," Journal of Mathematical Fluid Mechanics, Volume 3, Number 4, pp. 393-408 (2001)
   [download the paper] The original publication is available at springerlink.com.

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