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 Introduction to Ordinary Differential Equations Stephen Sapesrtone x: Prey or Activator y: Predator or Inhibitor

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. 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. 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. 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 How to solve the equation

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

20. First group

21. Development - X x0=14

22. Development - Y x0=14

23. XY 9 holes

24. XY x0=15 9 holes

25. Second group

26. Development - X

27. XY 8 holes

28. Third group

29. A 3 holes

30. B 4 holes

31. C

32. Double the length of the axes

33. A x0=44a 1/10

34. A x0=44a 2/10

35. x0=44a A 3/10

36. x0=44a A 4/10

37. x0=44a A 5/10

38. x0=44a A 6/10

39. x0=44a A 7/10

40. x0=44a A 8/10

41. x0=44a A 9/10

42. x0=44a A 10/10

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

52. Flat-top cone 1/4 x0=44ac50

53. Flat-top cone 2/4 x0=44ac50

54. Flat-top cone 3/4

55. Flat-top cone 4/4

56. Pyramid initial conditions Similar to the cone

57. Pyramid 1/2

58. Pyramid 2/2

59. Flat-top pyramid 100px1/2

60. Flat-top pyramid 2/2

61. Same holes as before, but four of them Flat-top pyramid

62. x0=44ac701/7

63. Flat-top pyramid x0=44ac702/7

64. Flat-top pyramid x0=44ac703/7

65. Flat-top pyramid x0=44ac704/7

66. Flat-top pyramid x0=44ac705/7

67. Flat-top pyramid x0=44ac706/7

68. Flat-top pyramid x0=44ac707/7

69. Holes ‘repel’ each other Flat-top pyramid

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