Lecture Series on Pattern Formation

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Fifty Years of Turing's Idea: a Lecture Series on Pattern Formation

last update: February 19, 2003

Spring Semester: Lecture 0, Lecture 1, Lecture 2, Lecture 3, Lecture 4

Fall Semester: Lecture 5, Lecture 6, Lecture 7, Appendix

In the celebrated paper published in 1952, A. M. Turing proposed a striking idea called "diffusion-driven instability" which has given a framework to understand the mechanism of pattern formation observed in many branches of science including developmental biology. There seems to have been no substantial development for the first decade or so after the publication of the paper, but by the late sixties people realized the significance of Turing's idea and proposed many mathematical models based on the idea to describe pattern formation. During the past three decades the field of reaction-diffusion systems has attracted many researchers in various disciplines such as mathematicians, physists, chemists, biologists and so on. Notable contributions have been made to this field by Japanese researchers.

To celebrate the fiftieth anniversary of the publication of the paper "The chemical basis of morphogenesis", we invite a researcher in the theory of pattern formation to address a ninety minute lecture in each month.

Extra Lecture: Pattern Trek: for the Next Generation

lecturer: Izumi Takagi (Mathematical Institute, Tohoku University)

date and time: March 6, 2003, 10:30 am -- 12:00 pm

place: Room 24, Kawai Hall

abstract: We have enjoyed seven lectures given in this special program. Each lecture presents us an aspect of the cutting-edge in the theory of reaction-diffusion systems. In this extra lecture we take a glance at the history of the research on pattern formation during these five decades and discuss the future directions. As a matter of fact, we are beginning to understand some part of the whole picture, but do not know even how much or how little of it.

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Lecture 7: Formation and Stability of Oscillatory Structures in Reactive Mixtures

lecturer: Professor Takao Ohta (Graduate School of Science, Hiroshima University)

date and time: January 14, 2003, 3:30 pm -- 5:00 pm

place: Room 802, Godoto Building

A model of phase separation of chemically reactive ternary mixtures is constructed to investigate the interaction and the dynamics of propagating domains. In this model, a Hopf-type bifurcation at a finite wave number appears. Numerical simulations show that, above the threshold, stripes or spots undergo coherent propagation in two dimensions. The traveling spots constitute a hexagonal structure. Amplitude equations for the traveling lamellae are derived and the stability of the traveling and standing waves are discussed [1], [2]. Stability and Synchronization of the traveling structures under external temporal modulations are also investigated [3].

[1] T. Okuzono and T. Ohta, "Self-Propulsion of Cellular Structures in Chemically Reacting Mixtures" Phys. Rev. E64 045201 (2001).

[2] T. Okuzono and T. Ohta, "Traveling waves in phase-separating reactive mixtures" Phys. Rev. E (submitted).

[3] S. Sugiura, T. Okuzono and T. Ohta, Phys. Rev. E (in press).

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Lecture 6: On the propagation speed of travelling wave solutions of reaction-diffusion systems in mathematical ecology

lecturer: Professor Yuzo Hosono (Kyoto Sangyo University)

date and time: December 10, 2002, 3:30 pm -- 5:00 pm

place: Room 802, Godoto Building

Reaction-diffusion systems have been playing an important role in the theoretical understanding of various phenomena arising in mathematical ecology. In this lecture we focus on reaction-diffusion systems in mathematical ecology including the Lotka-Volterra system and, by considering their travelling wave solutions, we discuss the invasion and propagation of a biological species. First, we describe how to analyse the speed of travelling waves, which expresses the prosperity and decay of a species, in the case of simple monostable single equations. Then, in the case of two species models, we show the existence of travelling wave solutions connecting an unstable steady state and a stable steady state, and study their properties. Based on these observations, we discuss the invasion of a foreign species into a region where one species has already occupied, or the simultaneous invasion of two species into an open space. Also we refer to a chemical reaction system which is closely related to models in mathematical ecology.

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Lecture 5: Modelling pattern formation on growing domains

lecturer: Professor Philip Maini (Oxford University)

date and time: October 22, 2002, 3:30 pm -- 5:00 pm

place: Room 802, Godoto Building

Recently, it has been shown that domain growth can play an important role in qualitative changes in patterning. We examine the classical Turing model from this point of view and explore how its behaviour with regard to pattern selection and transition is determined on a growing domain by the form and rate of growth, and the reaction kinetics. We apply the model to a number of biological examples, including ligament patterning in bivalues, and pigmentation patterning in butterfly wings and on fish.

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Lecture 4: Biological invasions into periodically fragmented environments: Reaction-diffusion models

lecturer: Professor Nanako Shigesada (Department of Information and Computer Sciences, Nara Women's University)

date and time: July 5, 2002, 10:30 am -- 12:00 pm

place: Room 24, Kawai Hall

Range expansions of invading species in homogeneous environments have been extensively studied, since the pioneer works by Fisher and Skellam. Here, we focus on range expansion in heterogeneous environments, which is generated by segmenting an original favorable habitat into a regularly striped pattern. When the invading species enters an unfavorable habitat, it will be able to expand its range only if it successfully survives in that habitat and reaches a favorable one lying ahead. If unfavorable habitats dominate, the population may become extinct without expanding the range. To deal with range expansion in such fragmented environments, we modify Fisher's model by assuming that the intrinsic growth rate and diffusion coefficient vary depending on habitat properties.

The model is analyzed for two initial conditions: the initial population density is concentrated on a straight line or at the origin. For each case, we derive a mathematical formula, which characterizes the spatio-temporal pattern of range expansion. When initial distribution starts from a straight line, it evolves to a traveling periodic wave, whose frontal speed is analytically derived. When the range starts from the origin, it tends to expand radially at a constant average speed in each direction (ray speed) keeping its frontal envelope in a similar shape. The envelopes show varieties of patterns, nearly circular, oval-like, spindle-like or vanishing in the extreme case, depending on parameter values. By deriving the formula for the ray speed, we discuss how the pattern and speed of the range expansion are affected by the size of fragmentation, and the qualities of favorable and unfavorable habitats.

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Lecture 3: Spatio-temporal patterns in far from equilibrium systems
-- Turing's lateral inhibition instability --

lecturer: Professor Masayasu Mimura (Hiroshima University)

date and time: June 21, 2002, 10:30 am -- 12:00 pm

place: Room 24, Kawai Hall

In 1952, A. Turing stated by using a simple reaction-diffusion equations that "diffusion" possibly enhances spatial inhomogeneity. His contribution is not to propose concrete models to understand morphogenesis in differential biology but to show theoretically a paradoxical fact of "diffusion" by using mathematical equations. In my talk, I would like to show some extension of Turing' idea to understand patterns arising in far from equilibrium systems.

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Lecture 2: On a double heteroclinic loop pulse of saddle-focus type arising in a 3-component reaction diffusion system

lecturer: Professor Yasumasa Nishiura (Hokkaido University)

date and time: June 7, 2002, 10:30 am -- 12:00 pm

place: Room 24, Kawai Hall

abstract: Double heteroclinic loop pulse of "saddle-focus" type has been regarded as an unstable object, in fact it was shown rigorously that such a pulse is unstable for 2x2 system under natural conditions, however, for 3x3 systems, we found that stable one exists and it is closely related to what is called transient instability. Such a pulse is believed to play an important role to detect the onset of spatial-temporal complex patterns.

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Lecture 1: Periodic patterns and Eckhaus instability

lecturer: Eiji Yanagida (Mathematical Institute, Tohoku University)

date and time: May 17, 2002, 10:30 am -- 12:00 pm

place: Room 24, Kawai Hall

In nonlinear systems such as reaction-diffusion systems, it is commonly observed that when a steady state with a spatial periodic structure is destabilized, there appears another spatially periodic steady state with a different period. Such a phenomenon is called the Eckhaus instability. This lecture is meant for the introduction to mathematical techniques used in the study of this kind of phenomenon. Also recent progress in the stability analysis of periodic patterns in (skew) gradient systems are described.

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Lecture 0: Turing's pioneering paper

lecturer: Izumi Takagi (Mathematical Institute, Tohoku University)

date and time: April 26, 2002, 10:30 am -- 12:00 pm

place: Room 24, Kawai Hall

abstract:

Brief introduction to the following paper:

A. M. Turing, "The chemical basis of morphogenesis", Phil. Trans. Roy. Soc. London Ser. B 237 (1952), 37-72.

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organized by
Izumi Takagi and Eiji Yanagida
Mathematical Institute, Tohoku University
Sendai, Japan 980-8578