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Alan Turing's morphogenesis: on the wonders of Nature

• Background:
  
  Is Nature predictable? Can we know with certainty the evolution of physical, or even, engineering systems? Does unexpected complexity arise from simplicity?
  
  In 1952, Alan Turing proposed a mathematical model than can explain the formation of structures and patterns in living organisms (for instance, the patterns of spots and stripes on animals' coats). This model was originally proposed for explaining the phenomenon of morphogenesis - from the Greek, formation of shapes - in cells, and is well-known as Turing's morphogenesis. It models the self-organization phenomena present in a wide variety of biological systems. The transitions and behaviour patterns associated with this type of phenomena are exhibited by nonlinear dynamical systems. They are typically related to the loss of stability of an equilibrium point. This loss of stability usually has its origin in what in mathematical terms is known as a bifurcation. That is, the point which marks a structural change. Turing's ideas of the mathematical formalization of the transition between different structures in biological systems have inspired many innovations in engineering and mathematics: from the computational morphogenesis in optimization problems to evolvable software systems; from computer graphics to building design.
  
  
• Description:
  
  This project has two main goals. First, to simulate Turing's reaction-diffusion model of morphogenesis. For this purpose, simplifications and different variations of the original model will be considered. Second, to explore applications of the results achieved. Depending on the student's interests and background, we could explore the possibility of proposing other computational modelling frameworks (hybrid systems or complex networks' models) for describing pattern formation and change.
  
  This project celebrates the 100th anniversary of Alan Turing's birth.


• Deliverables:
  
  
  • Analysis and identification of the main dynamical patterns present in the basic Turing's reaction-diffusion model of morphogenesis.
  • Simulation of the model.
  • Playing with the parameters of the model to obtain self-organising patterns.
  • Optional: proposing new types of computational models to expand Turing's reaction-diffusion model for different connections of cells.


• Supervisor: Dr. Eva Navarro López.


• Suitability:
  
  This project is suitable for students with enthusiasm for discovering the answers to many questions that Nature poses in our day-to-day life. It is essential to have some knowledge on dynamical systems theory and simulation, programming and differential equations.


• Some references:
  
  [1] A.M. Turing. 'The Chemical Basis of Morphogenesis'. Philosophical Transactions of the Royal Society of London. Series B, Biological Sciences, vol. 237, no. 641. (Aug. 14, 1952), pp. 37-72..
  
  [2] D'Arcy W. Thompson. 'On Growth and Form'. Cambridge University Press; Abridged edition, 1992. Originally published in 1917. Available at this link .