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Creating PatternsReaction-Diffusion Theory
Heather A Harrington Kody John Hoffman Law
May 18, 2006
Math 534 Spring 2006
Creating Patterns: REACTION-DIFFUSION THEORY 1
Outline
• Goals
• Background
• Introduction
• Analytical and Stability Analysis
• Numerical Methods and Simulations
• Further Work
• Acknowledgments
Math 534 Spring 2006 May 18, 2006
Creating Patterns: REACTION-DIFFUSION THEORY 2
Goals for the Project
• Study nonlinear PDEs and the formation of patterns
• Use analytical techniques from Math 534 to find the stability of thereaction-diffusion system
• Use numerical techniques from Math 652 to verify the analyticalresults
Math 534 Spring 2006 May 18, 2006
Creating Patterns: REACTION-DIFFUSION THEORY 3
Background of Pattern Formation
History
• In 1952, Alan Turing created a mathematical model describing thegrowing embryo’s sequential changes that occur from fertilizationto birth.
• This discovery started a trend of modeling using a mathematicalapproach to describe the effects of different chemicals ormorphogens, and how they can react and diffuse throughout atissue.
Math 534 Spring 2006 May 18, 2006
Creating Patterns: REACTION-DIFFUSION THEORY 4
Background of Pattern Formation
Applications
• Animal Coat Patterns
• Ecology
• Embryonic cell growth
• Chemical reactions
Math 534 Spring 2006 May 18, 2006
Creating Patterns: REACTION-DIFFUSION THEORY 5
Introduction
Reaction-Diffusion Theory
• Creating an equation using the vector chemical concentration c
• Diffusivities in a diagonal matrix D
• Reaction kinetics R(c)
• Reaction-diffusion system
∂c∂t
= R(c) + D∇2c
Math 534 Spring 2006 May 18, 2006
Creating Patterns: REACTION-DIFFUSION THEORY 6
Introduction
Two-Component Model
• Two chemical species: c = (u, v)
• Let diffusion coefficients be D1 and D2
• Define reaction kinetics:
R(c) =(
f(u, v)g(u, v)
)• Two-component system(
ut
vt
)=
(f(u, v)g(u, v)
)+∇2
(D1uD2v
).
Math 534 Spring 2006 May 18, 2006
Creating Patterns: REACTION-DIFFUSION THEORY 7
Introduction
Nondimensionalization
• We nondimensionalize the variables since u and v which aredependent on space (x) and time (t).
• y = xL where the domain is x ∈ [0, L] which implies y ∈ [0, 1].
• Let t∗ = D1tL2 , d = D2
D1, and γ = L2
D1.
• This gives the nondimensionalized system(ut∗vt∗
)= γ
(f(u, v)g(u, v)
)+∇2
(u
d · v
)
Math 534 Spring 2006 May 18, 2006
Creating Patterns: REACTION-DIFFUSION THEORY 8
Introduction
Boundary and Initial Conditions
• Chemical interactions for the formation of patterns are onlysignificant in the interior of the domain Ω.
• Therefore, the Neumann B.C on the boundary of domain, ∂Ωwhere the outward normal gradient vector for the species c(x,t)must also vanish for the reaction diffusion system
(n · ∇) · c(x, t) = 0 for x ∈ ∂Ω ,
• The initial condition of the solution is given for the fixed boundaryproblem
c(x, 0) = c0(x) .
Math 534 Spring 2006 May 18, 2006
Creating Patterns: REACTION-DIFFUSION THEORY 9
Introduction
Linearization
• Since c0 is a spatially uniform homogeneous steady state solution,then R(c0) = 0.
• We approximate the solution in a region close to c0 where w ∈ Rby a slight perturbation
c = c0 + ε w ,
• Then the linearized system is
∂w∂t
−D∇2w = R′(c0)w
Math 534 Spring 2006 May 18, 2006
Creating Patterns: REACTION-DIFFUSION THEORY 10
Introduction
Reaction Kinetics
• One type of nonlinear reaction kinetics is the Schnakenbergkinetics where a and b are positive rate constants and defined as:
f(u, v) = a− u + u2v
g(u, v) = b− u2v .
Math 534 Spring 2006 May 18, 2006
Creating Patterns: REACTION-DIFFUSION THEORY 11
Analytical Analysis
Homogeneous Uniform Steady State
• In the absence of diffusion, we find
ut = γf(u, v) = 0
vt = γg(u, v) = 0 .
• Then our equation becomes
dwdt
= γ
(fu fv
gu gv
)(u0,v0)
·w
Math 534 Spring 2006 May 18, 2006
Creating Patterns: REACTION-DIFFUSION THEORY 12
• From ordinary differential equations, this yields a solution in theform
w(t) ∝ veλt ,
• The eigenvalue problem (γA− λI)w = 0 is stable for thehomogeneous steady state solution w = 0 when Re(λ) < 0.
• This yields the conditions for stability:
fu + gv < 0 ,
fugv − fvgu > 0
Math 534 Spring 2006 May 18, 2006
Creating Patterns: REACTION-DIFFUSION THEORY 13
Analytical Analysis
Diffusion-Driven Instability (DDI)
• We extend our methods to the full equations to include diffusion:
wt = γAw + D∇2w where D =(
1 00 d
)
• By separation of variables, we solve the Laplacian for any x ∈ Ωand let λ be dependent on the eigenvalue k2, where thewavenumber k is determined by the following equation:
k2X(x) + X ′′(x) = 0 , such that (n · ∇)w(x, t) = 0 on ∂B .
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Creating Patterns: REACTION-DIFFUSION THEORY 14
• The solution after applying the B.C. is
X(x) ∝ cos(nπx) ⇒ Xk(x) ∝ cos(kx)
• We find the full solution is
w(x, t) =∑
k
Fk cos(kx)eλt
• Substitute w into the full system and we have the have thefollowing eigenvalue problem
(−γA + k2D + λ(k2)I)w = 0.
Math 534 Spring 2006 May 18, 2006
Creating Patterns: REACTION-DIFFUSION THEORY 15
• The remaining conditions for DDI (i.e. k2 such that λ(k2) > 0)
dfu + gv > 0 ,
(dfu + gv)2 − 4d(fugv − fugu) > 0 .
• This implies the condition on critical d:
For dc > 1 , ⇒ d > dc
will yield unstable modes corresponding to k2 which is given by
γL < k2 < γM
Math 534 Spring 2006 May 18, 2006
Creating Patterns: REACTION-DIFFUSION THEORY 16
KL = L = γ(gv + fud)−
√(gv + fud)2 − 4d(fugv − fvgu)
2d
KM = M = γ(gv + fud) +
√(gv + fud)2 − 4d(fugv − fvgu)
2d.
• In the two-dimensional case, we find k2 = (mπ)2 + (nπ)2 wheren, m ∈ Z This gives the full two-dimensional solution:
w(x, y, t) ≈∑m,n
Fm,n cos(nπx) cos(mπy)eλ(k2)t.
Math 534 Spring 2006 May 18, 2006
Creating Patterns: REACTION-DIFFUSION THEORY 17
Numerical Methods
Finite Difference Method
• Since the two-component system has already beennondimensionalized where x ∈ [0, 1], we let
∆x =1M
with x = ∆x ·m for m = 0, . . . ,M ,
then we have the second order finite difference approximation
uxx =u(∆x(m + 1)) + u(∆x(m− 1))− 2u(∆xm)
(∆x)2+O((∆x)2)
uxx ≈ u(∆x(m + 1)) + u(∆x(m− 1))− 2u(∆xm)(∆x)2
= δ2xu
Math 534 Spring 2006 May 18, 2006
Creating Patterns: REACTION-DIFFUSION THEORY 18
• We denote uxx ≈ δ2xu. Then we find the first order finite difference
method for the temporal equation to be:
ut =u(∆t(n + 1))− u(∆tn)
∆t+O(∆t)
ut ≈ u(∆t(n + 1))− u(∆tn)∆t
• Then Unm ≈ u(∆xm,∆tn) . Hence, with given initial conditions, we
iterate this system solving for each subsequent time level n+1. Themethod we use to iterate the changes in two species with respectto time is
Un+1m = Un
m + ∆t[γ(a− Un
m + V nm(Un
m)2) + δ2xUn
m
]V n+1
m = V nm + ∆t
[γ(b− V n
m(Unm)2) + δ2
xV nm
]• There exists a point ∆x · (M + 1), and the Neumann boundary
Math 534 Spring 2006 May 18, 2006
Creating Patterns: REACTION-DIFFUSION THEORY 19
condition is hence:
∂UnM
∂x=
UnM+1 − Un
M−1
2∆x= 0
⇒ UnM+1 = Un
M−1.
This yields
δ2xUn
M =2(Un
M−1 − UnM)
(∆x)2
Math 534 Spring 2006 May 18, 2006
Creating Patterns: REACTION-DIFFUSION THEORY 20
Numerical Methods
Newton’s Method
• The generalized iterative Newton method is
xn+1 = xn −f(xn)f ′(xn)
• For the reaction-diffusion system, we found(uv
)n+1
=(
uv
)n
−(
FG
)n
· J−1n
Math 534 Spring 2006 May 18, 2006
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Numerical Simulations
• Cartesian Coordinates
? Evolution from homogeneous to heterogeneous- 1D? Evolution from homogeneous to heterogeneous- 2D? One-dimensional changing domain? Two-dimensional growing domain
• Polar Coordinates
? No angular dependence homogeneous to heterogeneous(traveling front)
? With angular dependence (limit cycle)
Math 534 Spring 2006 May 18, 2006
Creating Patterns: REACTION-DIFFUSION THEORY 22
(a) (b)
(c) (d)
Figure 1: 2(a) and 1(b): Solution of the heterogeneous steady state of the activator u with one-dimensional Schnakenbergkinetics (a = 0.3, b = 1, d = 40) over changing domains where L ranges from 10 to 50. 2(c) and 1(d): Solution of thedominant mode of heterogeneous steady state of the activator u with one-dimensional Schnakenberg kinetics (a = 0.3, b= 1, d = 40) over changing domains where L ranges from 10 to 50.
Math 534 Spring 2006 May 18, 2006
Creating Patterns: REACTION-DIFFUSION THEORY 23
(a) (b)
(c) (d)
Math 534 Spring 2006 May 18, 2006
Creating Patterns: REACTION-DIFFUSION THEORY 24
Numerical Simulations
• Cartesian Coordinates
? Evolution from homogeneous to heterogeneous- 1D? Evolution from homogeneous to heterogeneous- 2D? One-dimensional changing domain? Two-dimensional growing domain
• Polar Coordinates
? No angular dependence homogeneous to heterogeneous(traveling front)
? With angular dependence (limit cycle)
Math 534 Spring 2006 May 18, 2006
Creating Patterns: REACTION-DIFFUSION THEORY 25
Future Work
• Consider a model in a third dimension using the Schnakenbergkinetics
• Study the growing domain and radial analytical analysis
• Consider the other types of kinetics
• Conduct a thorough error analysis
Math 534 Spring 2006 May 18, 2006
Creating Patterns: REACTION-DIFFUSION THEORY 26
Acknowledgments
• Nathan Fidalgo
• Panayotis Kevrekidis, PhD
• Mike Satz
• Nathaniel Whitaker, PhD
• Robin Young, PhD
Math 534 Spring 2006 May 18, 2006