Download presentation

Presentation is loading. Please wait.

Published byShyanne Wakeham Modified over 2 years ago

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

Similar presentations

© 2017 SlidePlayer.com Inc.

All rights reserved.

Ads by Google