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
Lotka-Volterra Model Introduction to Ordinary Differential Equations Stephen Sapesrtone x: Prey or Activator y: Predator or Inhibitor
Lotka-Volterra Model
Modified Lotka-Volterra Model Change from a single value to one dimension of space Add diffusion Add intraspecies interaction term
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
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
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
Initial conditions Solve equation for steady states –Each set of values gives three steady states e.g (unstable), (unstable), (stable) Filled the grid with this value ± small disturbance
How to solve the equation
Initial conditions
First group
Development - X x0=14
Development - Y x0=14
XY 9 holes
XY x0=15 9 holes
Second group
Development - X
XY 8 holes
Third group
A 3 holes
B 4 holes
C
Double the length of the axes
A x0=44a 1/10
A x0=44a 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
B x0=44b
C x0=44c
A x0=45a
B x0=45b
C x0=45c
Varied initial values
Conic initial conditions
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 100px1/2
Flat-top pyramid 2/2
Same holes as before, but four of them Flat-top pyramid
x0=44ac701/7
Flat-top pyramid x0=44ac702/7
Flat-top pyramid x0=44ac703/7
Flat-top pyramid x0=44ac704/7
Flat-top pyramid x0=44ac705/7
Flat-top pyramid x0=44ac706/7
Flat-top pyramid x0=44ac707/7
Holes ‘repel’ each other Flat-top pyramid
Pattern Formation in a Reaction-diffusion System Noel R. Schutt, Desiderio A. Vasquez Department of Physics, IPFW, Fort Wayne IN