Ground patterns

Project description: Derive and analyze a partial differential equation modeling the formation of sorted patterned ground in polar and high alpine regions.
Methods:
1. Read the paper by M.A. Kessler & B.T. Werner, in which a particulate model is proposed and numerically implemented.
2. Understand relevant background literature on this topic.
3. Derive a continuous analog of this model.
4. Perform numerical simulations of the two models and compare their behaviors.
5. Analyze pattern formation as described by the continuous model.

Vegetation patterns

Project description: Understand and analyze, in terms of envelope equations, reaction-diffusion models for vegetation pattern formation.
Methods:
1. Read and understand the papers by the following authors, in which models for vegetation pattern formation are proposed:
  - C.A. Klausmeier
  - R. Lefever, O. Lejeune & P. Couteron [Generic modelling of vegetation patterns. A case study of Tiger bush in sub-saharian Sahel, in Mathematical Models for Biological Pattern Formation, edited by P.K. Maini & H.G. Othmer, pp. 83-112, Springer, New York, 2001]
  - J. von Hardenberg, E. Meron, M. Shachak & Y. Zarmi
2. Perform a numerical simulation of C. Klausmeier's model.
3. Derive an envelope equation describing pattern formation in C. Klausmeier's model.
4. Perform a numerical simulation of the envelope equation you have derived and compare the results with the simulation of the original model.

Project description: Understand and analyze, in terms of envelope equations, reaction-diffusion models for fish skin patterns.
Methods:
1. Read and understand the papers by the following authors, in which models for fish skin pattern formation are proposed:
  - S. Kondo & R. Asai [A reaction-diffusion wave on the skin of the marine angelfish Pomacanthus, Nature 376, 765-768 (1995)]
  - See also the comment by T. Höfer & P. Maini [Turing patterns in fish?, Nature 380, 678 (1996)]
  - K.J. Painter, H.G. Othmer & P.K. Maini
  - K.J. Painter [Models for pigment pattern formation in the skin of fishes, in Mathematical Models for Biological Pattern Formation, edited by P.K. Maini & H.G. Othmer, pp. 59-81, Springer, New York, 2001]
2. Collect information on the different types of fish skin patterns seen in nature (there are many pictures of fish available on the web. See for instance the FishBase the Digital Library Project at Berkeley, the Reef Base, or the image collection of the Great Barrier Reef Marine Park Authority).
3. Perform a numerical simulation of the models by Kondo & Asai and by Painter et al.
4. Derive an envelope equation describing pattern formation in one of these models.
5. Perform a numerical simulation of the envelope equation you have derived and compare the results with the simulation of the original model.

Project description: Understand and analyze simple chemical reaction models.
Methods:
1. In the Applied Math Laboratory, experiment with the Belousov-Zhabotinsky reaction (for more pictures, see the picture gallery at the University of Leeds).
2. Compare the two following models: the Oregonator and the Brusselator models with spatial diffusion terms.
3. Analyze the bifurcation properties of the two models as well as pattern formation in the one of them, numerically and by means of an envelope equation.