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
• Non-homogenous solutions to DE
Turing Patterns
Phys Rev Lett 64 (1990) 2953
Castets, Dulos, Boissonade, De Kepper
Turing Patterns
http://chaos.utexas.edu/research/spots/spots.html
Lotka-Volterra Model
x  x  y
y  y  xy
x: Prey or Activator
y: Predator or Inhibitor
Introduction to Ordinary Differential Equations
Stephen Sapesrtone
Lotka-Volterra Model
http://mathworld.wolfram.com/Lotka-VolterraEquations.html
Modified Lotka-Volterra Model
• Change from a single value to one
dimension of space
• Add diffusion
• Add intraspecies interaction term
Modified Lotka-Volterra Model
 
dX
X  kY 
2
 
 X r0 1 
 X
d
X
1

X
0




dY
 kX

2
Y
 1  k f Y   d Y
d
1  X

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
Modified Lotka-Volterra Model
X
Modified Lotka-Volterra Model
Y
1D results reproduced, now
expand to two dimensions
How to solve the equation
• To reduce the runtime, use an implicit
Euler method for time
• Space is in a 321x321 grid
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
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
How to solve the equation
Initial conditions
Initial conditions
First group
Development - X
x0=14
Development - Y
x0=14
X
9 holes
Y
x0=14
X
9 holes
Y
x0=15
Second group
Development - X
X
8 holes
Y
Third group
A
3 holes
B
4 holes
C
• Double the length of the axes
A
1/10
x0=44a
A
2/10
x0=44a
A
3/10
x0=44a
A
4/10
x0=44a
A
5/10
x0=44a
A
6/10
x0=44a
A
7/10
x0=44a
A
8/10
x0=44a
A
9/10
x0=44a
A
10/10
x0=44a
B
x0=44b
C
x0=44c
A
x0=45a
B
x0=45b
C
x0=45c
Varied initial values
Conic initial conditions
r=
real(i) - rnx2  + real(j) - rnx2 
2
2
 xbar2 - xbar1 
xbar = 
r - rxb1 + xbar1
 rnx2 - rxb1 
Cone
Cone
Flat-top cone
1/4
x0=44ac50
Flat-top cone
2/4
x0=44ac50
Flat-top cone
3/4
Flat-top cone
4/4
Pyramid initial conditions
• Similar to the cone
Pyramid
1/2
Pyramid
2/2
Flat-top pyramid
1/2
100px
Flat-top pyramid
2/2
Flat-top pyramid
• Same holes as before, but four of them
Flat-top pyramid
1/7
x0=44ac70
Flat-top pyramid
2/7
x0=44ac70
Flat-top pyramid
3/7
x0=44ac70
Flat-top pyramid
4/7
x0=44ac70
Flat-top pyramid
5/7
x0=44ac70
Flat-top pyramid
6/7
x0=44ac70
Flat-top pyramid
7/7
x0=44ac70
Flat-top pyramid
• Holes ‘repel’ each other
Pattern Formation in a
Reaction-diffusion System
Noel R. Schutt, Desiderio A. Vasquez
Department of Physics, IPFW,
Fort Wayne IN