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

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Turing patterns in a modified Lotka-Volterra model

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

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

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

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

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

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

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Modified Lotka-Volterra Model

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Now patterns can develop In 2005 patterns were found in this model in one dimension Use finite difference equation to Reproduce results

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Modified Lotka-Volterra Model X

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Y

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1D results reproduced, now expand to two dimensions

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

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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

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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

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How to solve the equation

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Initial conditions

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First group

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Development - X x0=14

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Development - Y x0=14

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XY 9 holes

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XY x0=15 9 holes

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Second group

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Development - X

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XY 8 holes

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Third group

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A 3 holes

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B 4 holes

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C

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Double the length of the axes

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A x0=44a 1/10

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A x0=44a 2/10

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x0=44a A 3/10

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x0=44a A 4/10

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x0=44a A 5/10

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x0=44a A 6/10

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x0=44a A 7/10

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x0=44a A 8/10

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x0=44a A 9/10

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x0=44a A 10/10

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B x0=44b

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C x0=44c

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A x0=45a

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B x0=45b

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C x0=45c

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Varied initial values

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Conic initial conditions

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Cone

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Flat-top cone 1/4 x0=44ac50

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Flat-top cone 2/4 x0=44ac50

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Flat-top cone 3/4

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Flat-top cone 4/4

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Pyramid initial conditions Similar to the cone

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Pyramid 1/2

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Pyramid 2/2

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Flat-top pyramid 100px1/2

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Flat-top pyramid 2/2

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

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x0=44ac701/7

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Flat-top pyramid x0=44ac702/7

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Flat-top pyramid x0=44ac703/7

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Flat-top pyramid x0=44ac704/7

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Flat-top pyramid x0=44ac705/7

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Flat-top pyramid x0=44ac706/7

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Flat-top pyramid x0=44ac707/7

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Holes ‘repel’ each other Flat-top pyramid

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

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