Seminar on Spatio-Temporal Patterns

The Seminar is conducted at the Mathematical Institute of Utrecht University under the auspices of the NWO cluster NDNS+ and will take place on Wednesdays 13.15-15.00 in Room BBL 071, Utrecht University, starting on February 8, 2012. It is intended for Master students and (beginning) PhD students and is supervised by the organizers. It has not yet been decided for how long the seminar will continue, but presumably the last session will be on June 13.


Odo Diekmann
Arjen Doelman
Stephan van Gils
Yuri Kuznetsov
Heinz Hanßmann
Vivi Rottschäfer
Paul Zegeling

Right now we think of the following topics : travelling waves, asymptotic speed of propagation, Turing patterns, reaction-diffusion equations, integro-differential equations, epidemic models, neurodynamic models, ...

More precisely, a list of possible topics, with between ( ) the main math technique involved, includes:

Scalar equations :
-- existence of travelling waves (homoclinic and heteroclinic orbits and their bifurcations)
-- asymptotic speed of propagation aka spreading speed (maximum principle, comparison functions i.e. upper and lower solutions)
-- on convex domains no stable patterns (linearized stability + spectral analysis)

special topics :
-- Weinberger's iterated maps (order, so kind of maximum principle)
-- advection (motivation : rivers, moving climate)

Systems of equations :
-- Turing patterns (bifurcation theory, spectral theory)
-- spiral (rotating) patterns in bounded and unbounded domains (singular perturbations, normal forms)
-- stability of travelling waves (Evans function, small parameter asymptotics)

special topics :
-- cross diffusion

Here follow some links to interesting mathematical studies of patterns. They can be used to prepare presentations.

Further topics as well as specific papers/book chapters can be suggested by the participants.

Study points:
An active participant can earn 8 ECTS. Passive participants are welcome, but will not receive any credits. It is assumed that each participant will give a presentation. Each presentation is 45 minutes long, so every week there are two presentations. It is possible that two students work together and fill a complete session, but then each of them should lecture for 45 minutes. In general, we expect that a lecture pays attention to
-- motivation
-- mathematical model (formulation/derivation of equation)
-- results
-- proofs (always main ideas; details when these matter/help)
-- interpretation (what do the math results tell us about the motivating question ?)

This text (in Dutch) explains 'how to prepare a good seminar talk'.


Specific reference material
8 Feb 2012
Stephan van Gils: A birds eye view of neurodynamics.    Lecture notes
15 Feb 2012
Odo Diekmann: A birds eye view of spatial ecology. Lecture notes
The first two sessions intend to provide motivated problem formulation, to explain concepts and to formulate striking results, but neither will present proofs. In these sessions we should also schedule the subsequent sessions and, in particular, let participants choose topics and material for presentation.

Specific reference material
22 Feb 2012
Arjen Doelman: The dynamics of reaction-diffusion patterns.    Lecture notes
29 Feb 2012
Angela Stevens:
I. Turing patterns and other pattern forming mechanisms in developmental systems.
II. Orientational selection and aggregation in stage structured population models.

Lecture notes
07 Mar 2012
Stefanie Postma: Pattern formation in gradient systems.
P.C. Fife "Models for phase separation and their mathematics"
14 Mar 2012

Corine Meerman:
I. Complex patterns in a simple system (John E. Pearson: Science, Vol. 261, 9 juli 1993, 189 - 192)
II. Experimental observation of self-replicating spots in a reaction-diffusion system
(Kyoung-Jin Lee, William D. McCormick, John E. Pearson & Harry L. Swinney: Nature, Vol 369, 19 mei 1994, 215 - 218)

21 Mar 2012
Sid Visser: Analysis of a lumped model of neocortex to study epileptiform activity.
KaYin Leung:  Spatial deterministic epidemics and the asymptotic speed of propagation.
Lecture notes
28 Mar 2012
Eric Siero:  Vegetation patterns in arid ecosystems - computing spectra of differential operators on the real line using continuation. J. Rademacher, B. Sandstede, and A. Scheel
"Computing absolute and essential spectra using continuation"
04 Apr 2012
Gosse Overal: Excitatory and inhibitory interactions in localized populations of model neurons.
Odo Diekmann: Linear determinacy of spreading speeds.
11 Apr 2012

18 Apr 2012
Víctor F. Breña-Medina: Cracking a non-homogeneous reaction-diffusion system: a root hair plant initiation model.
Dirk van Kekem : On the top of a function. Maximum principle and sub-/supersolutions.
Lecture notes
25 Apr 2012 NO SEMINAR
02 May 2012
Arthur Vromans: Stability of travelling waves: dichotomies, spectra, and Fredholm properties (B. Sandstede: Handbook of Dynamical Systems, vol. 2, Elsiveir Science, Amsterdam, 2002, pp. 983-1055)
Jurgen Hebbink: Dynamics of pattern formation in lateral-inhibition type neural fields (S. Amari: Biol. Cybernetics 27, 77- 87 (1977))

09 May 2012
Philip Klop: Turing instability in a 1D neural wave equation
Corine Meerman: Pattern formation in the one-dimensional Gray-Scott model.
16 May 2012
Stefanie Postma: Nonlinear parabolic equations.
Martino Pitruzella: Application of semigroup theory to reaction-diffusion equations.

Presentation (edited by S. Janssens)
23 May 2012
Gosse Overal:  A mathematical theory of the functional dynamics of cortical and thalamic nervous tissue.
Dirk van Kekem: Convergence of solutions of bistable nonlinear diffusion equations to travelling front solutions.
30 May 2012
Jurgen Hebbink:
Philip Klop: Tumor-stromal interactions in acid-mediated invasion.

06 Jun 2012
Arthur Vromans: Stability of travelling waves: spectrum and Evans function.
Yuri A. Kuznetsov: Bifurcations of standing and rotating waves from homogeneous states in reaction-diffision systems.

Last updated: Wed Jun 6, 2012