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Turing Systems Helsinki University of Technology Laboratory of Computational Engineering Publications Teknillisen korkeakoulun Laskennallisen tekniikan laboratorion julkaisuja Espoo 2004 REPORT B40 COMPUTATIONAL STUDIES OF PATTERN FORMATION IN TURING SYSTEMS Teemu Leppänen AB TEKNILLINEN KORKEAKOULU TEKNISKA HÖGSKOLAN HELSINKI UNIVERSITY OF TECHNOLOGY TECHNISCHE UNIVERSITÄT HELSINKI UNIVERSITE DE TECHNOLOGIE D’HELSINKI Helsinki University of Technology Laboratory of Computational Engineering Publications Teknillisen korkeakoulun Laskennallisen tekniikan laboratorion julkaisuja Espoo 2004 REPORT B40 COMPUTATIONAL STUDIES OF PATTERN FORMATION IN TURING SYSTEMS Teemu Leppänen Dissertation for the degree of Doctor of Science in Technology to be presented with due permission of the Department of Electrical and Communications Engineering, Helsinki University of Technology, for public examination and debate in Auditorium S4 at Helsinki University of Technology (Espoo, Finland) on the 27th of November, 2004, at 12 noon. Helsinki University of Technology Department of Electrical and Communications Engineering Laboratory of Computational Engineering Teknillinen korkeakoulu Sähkö- ja tietoliikennetekniikan osasto Laskennallisen tekniikan laboratorio Distribution: Helsinki University of Technology Laboratory of Computational Engineering P. O. Box 9203 FIN-02015 HUT FINLAND Tel. +358-9-451 4826 Fax. +358-9-451 4830 http://www.lce.hut.fi Online in PDF format: http://lib.hut.fi/Diss/2004/isbn9512273969 E-mail: Teemu.Leppanen@hut.fi c Teemu Leppänen ISBN 951-22-7395-0 (printed) ISBN 951-22-7396-9 (PDF) ISSN 1455-0474 PicaSet Oy Espoo 2004 Abstract This thesis is an analytical and computational treatment of Turing models, which are coupled partial differential equations describing the reaction and diffusion be- havior of chemicals. Under particular conditions, such systems are capable of generating stationary chemical patterns of finite characteristic wave lengths even if the system starts from an arbitrary initial configuration. The characteristics of the resulting dissipative patterns are determined intrinsically by the reaction and diffusion rates of the chemicals, not by external constraints. Turing patterns have been shown to have counterparts in natural systems and thus Turing systems could provide a plausible way to model the mechanisms of biological growth. Turing patterns grow due to diffusion-driven instability as a result of infinites- imal perturbations around the stationary state of the model and exist only under non-equilibrium conditions. Turing systems have been studied using chemical experiments, mathematical tools and numerical simulations. In this thesis a Turing model called the Barrio-Varea-Aragon-Maini (BVAM) model is studied by employing both analytical and numerical methods. In addi- tion to the pattern formation in two-dimensional domains, also the formation of three-dimensional structures is studied extensively. The scaled form of the BVAM model is derived from first principles. The model is then studied using the stand- ard linear stability analysis, which reveals the parameter sets corresponding to a Turing instability and the resulting unstable wave modes. Then nonlinear bifurc- ation analysis is carried out to find out the stability of morphologies induced by two-dimensional hexagonal symmetry and various three-dimensional symmetries (SC, BCC, FCC). This is realized by employing the center manifold reduction technique to obtain the amplitude equations describing the reduced chemical dy- namics on the center manifold. The main numerical results presented in this thesis include the study of the Turing pattern selection in the presence of bistability, and the study of the structure selection in three-dimensional Turing systems depending on the initial configuration. Also, the work on the effect of numerous constraints, such as random noise, changes in the system parameters, thickening domain and multistability on Turing pattern formation brings new insight concerning the state selection problem of non-equilibrium physics. Preface The research presented in this dissertation has been carried out at the Laboratory of Computational Engineering (LCE), Helsinki University of Technology (HUT), mostly during the years 2002-2004, while working in the Biophysics and Statist- ical Mechanics, and the Complex Systems groups. The work was initiated during my apprenticeship at LCE as a second-year student in the summer of 2000. My postgraduate studies were funded by the Graduate School in Computational Meth- ods of Information Technology (ComMIT) and the Academy of Finland under the Finnish Centre of Excellence Programme. I also gratefully acknowledge the fin- ancial support from the Finnish Academy of Science and Letters (Väisälä fund), and the Jenny and Antti Wihuri foundation. First and foremost, I wish to express my gratitude to Prof. Kimmo Kaski for directing me to the amazingly interesting field of Turing systems, giving me the opportunity to combine the research flexibly with my interdisciplinary studies, and encouraging me all the way through. My sincere thanks go to Prof. Rafael Barrio for his support and inspiring guidance. I also wish to thank Dr. Mikko Karttunen, who has instructed me numerous times during this project. In addi- tion, I am grateful to Dr. Murat Ünalan for sharing his biological expertise, and Prof. Philip Maini for his hospitality during my one-month visit to the Centre for Mathematical Biology at the University of Oxford. Thanks are also due to LCE personnel for making the laboratory an enjoyable environment to work. I would especially like to thank my office mate Sebastian von Alfthan for advice and making the work days fruitful, and Jukka-Pekka On- nela for the intensive and motivating discussions. I would also like to thank my colleagues Ville Mäkinen, Markus Miettinen, Petri Nikunen, Toni Tamminen and Ilkka Kalliomäki for keeping up the cheerful and encouraging atmosphere. I would not have been able to complete my studies at HUT as fast as I did without the unfailing support from my family. I am obliged to my parents Anita and Olli for teaching me the meaning of hard work through their own example. I also wish to thank my delightful sisters Helmi and Marjaana for tolerating me. Lastly, thank you Satu for your unconditional love and optimism. Teemu I. Leppänen List of publications Although this thesis has been written as a monograph, it is based on the results that have been published in the following scientific articles: 1. T. Leppänen, M. Karttunen, K. Kaski, R. A. Barrio, and L. Zhang, A new dimension to Turing patterns, Physica D 168-169C, 35-44 (2002). 2. T. Leppänen, M. Karttunen, K. Kaski, and R. A. Barrio, The effect of noise on Turing patterns, Prog. Theor. Phys. (Suppl.) 150, 367-370 (2003). 3. T. Leppänen, M. Karttunen, K. Kaski, and R. Barrio, Dimensionality effects in Turing pattern formation, Int. J. Mod. Phys. B 17, 5541-5553 (2003). 4. T. Leppänen, M. Karttunen, K. Kaski, and R. A. Barrio, Turing systems as models of complex pattern formation, Braz. J. Phys. 34, 368-372 (2004). 5. T. Leppänen, M. Karttunen, R. A. Barrio, and K. Kaski, Morphological transitions and bistability in Turing systems, Phys. Rev. E 70, (2004). 6. T. Leppänen, The theory of Turing pattern formation, in Current Topics in Physics (Imperial College Press), a book chapter in press (2004). 7. T. Leppänen, M. Karttunen, R. A. Barrio, and K. Kaski, Spatio-temporal dynamics in a Turing model, InterJournal, submitted (2004). The work has also been presented in the following popular science articles: • T. Leppänen and K. Kaski, Matemaatikko selätti seepran raidat (in Finnish), Tiede 1, 48-49 (2004). • T. Leppänen and K. Kaski, Matemaatikko luonnonilmiöiden jäljillä (in Finnish), Arkhimedes 2, 13-16 (2004). Contents Abstract i Preface iii List of publications v Contents vii 1 Introduction 1 2 Some general aspects on pattern formation 7 2.1 Basic concepts . ....................... 9 2.1.1 Symmetry-breaking . ................... 9 2.1.2 Stability . ....................... 11 2.1.3 Bifurcations . ....................... 13 2.2 Hydrodynamical systems . .................... 15 2.2.1 Rayleigh-Bénard convection . ........... 15 2.2.2 Taylor-Couette flow . ................. 18 2.3 Chemical reaction-diffusion systems . ............. 20 2.3.1 Belousov-Zhabotinsky reaction . ............. 21 2.3.2 Complex Ginzburg-Landau equation ............ 23 2.4 Patterns in natural systems ..................... 24 2.5 Summary . .......................... 29 3 Pattern formation in Turing systems 31 3.1 Turing instability .......................... 32 3.1.1 Linear stability . ...................... 32 3.1.2 Pattern selection problem . ............. 35 3.1.3 Degeneracies ........................ 36 3.2 Examples of Turing models . .................. 37 3.2.1 The Brusselator model ................... 38 3.2.2 The Gray-Scott model ................... 39 viii Contents 3.2.3 The Lengyel-Epstein model . ............. 42 3.3 Experimental Turing structures . .................. 43 3.3.1 CIMA reaction ....................... 43 3.3.2 Recent experimental work . ............. 45 3.4 Summary . .......................... 45 4 Mathematical analysis of Barrio-Varea-Aragon-Maini model 47 4.1 Derivation of BVAM model . ................... 47 4.2 Linear stability analysis . ..................... 49 4.3 Nonlinear bifurcation analysis . ................. 53 4.3.1 Derivation of amplitude equations ............. 54 4.3.2 Center manifold reduction . ............. 58 4.3.3 Stability analysis
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