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Pattern Formation in a Reaction-diffusion System

Pattern Formation in a Reaction-diffusion System. Noel R. Schutt, Desiderio A. Vasquez Department of Physics, IPFW, Fort Wayne IN. Turing patterns in a modified Lotka-Volterra model. Turing Patterns. Predicted by Alan Turing in 1952 Patterns in chemical/biological systems

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Pattern Formation in a Reaction-diffusion System

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  1. Pattern Formation in a Reaction-diffusion System Noel R. Schutt, Desiderio A. Vasquez Department of Physics, IPFW, Fort Wayne IN

  2. Turing patterns in a modified Lotka-Volterra model

  3. Turing Patterns • Predicted by Alan Turing in 1952 • Patterns in chemical/biological systems • Non-homogenous solutions to DE

  4. Turing Patterns Phys Rev Lett 64 (1990) 2953 Castets, Dulos, Boissonade, De Kepper

  5. Turing Patterns http://chaos.utexas.edu/research/spots/spots.html

  6. Lotka-Volterra Model x: Prey or Activator y: Predator or Inhibitor Introduction to Ordinary Differential Equations Stephen Sapesrtone

  7. Lotka-Volterra Model http://mathworld.wolfram.com/Lotka-VolterraEquations.html

  8. Modified Lotka-Volterra Model • Change from a single value to one dimension of space • Add diffusion • Add intraspecies interaction term

  9. Modified Lotka-Volterra Model

  10. Modified Lotka-Volterra Model • Now patterns can develop • In 2005 patterns were found in this model in one dimension • Use finite difference equation to Reproduce results

  11. Modified Lotka-Volterra Model X

  12. Modified Lotka-Volterra Model Y

  13. 1D results reproduced, now expand to two dimensions

  14. How to solve the equation • To reduce the runtime, use an implicit Euler method for time • Space is in a 321x321 grid

  15. How to solve the equation • Original math code in FORTRAN • Math code is fairly simple • Perl wrapper code to simplify working with math code • php code to organize results • Results take 20MB to 2.8GB per run

  16. Initial conditions • Solve equation for steady states • Each set of values gives three steady states e.g. 7.99 (unstable), 11.48 (unstable), 22.22 (stable) • Filled the grid with this value ± small disturbance

  17. How to solve the equation

  18. Initial conditions

  19. Initial conditions

  20. First group

  21. Development - X x0=14

  22. Development - Y x0=14

  23. X Y 9 holes x0=14

  24. X Y 9 holes x0=15

  25. Second group

  26. Development - X

  27. X Y 8 holes

  28. Third group

  29. A 3 holes

  30. B 4 holes

  31. C

  32. Double the length of the axes

  33. A 1/10 x0=44a

  34. A 2/10 x0=44a

  35. A 3/10 x0=44a

  36. A 4/10 x0=44a

  37. A 5/10 x0=44a

  38. A 6/10 x0=44a

  39. A 7/10 x0=44a

  40. A 8/10 x0=44a

  41. A 9/10 x0=44a

  42. A 10/10 x0=44a

  43. B x0=44b

  44. C x0=44c

  45. A x0=45a

  46. B x0=45b

  47. C x0=45c

  48. Varied initial values

  49. Conic initial conditions

  50. Cone

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