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Creating Patterns Reaction-Diffusion Theory Heather A Harrington Kody John Hoffman Law May 18, 2006 Math 534 Spring 2006

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Page 1: Creating Patterns Reaction-Diffusion Theoryhharring/research/slideshow1.pdfCreating Patterns: REACTION-DIFFUSION THEORY 8 Introduction Boundary and Initial Conditions • Chemical

Creating PatternsReaction-Diffusion Theory

Heather A Harrington Kody John Hoffman Law

May 18, 2006

Math 534 Spring 2006

Page 2: Creating Patterns Reaction-Diffusion Theoryhharring/research/slideshow1.pdfCreating Patterns: REACTION-DIFFUSION THEORY 8 Introduction Boundary and Initial Conditions • Chemical

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

Page 3: Creating Patterns Reaction-Diffusion Theoryhharring/research/slideshow1.pdfCreating Patterns: REACTION-DIFFUSION THEORY 8 Introduction Boundary and Initial Conditions • Chemical

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

Page 4: Creating Patterns Reaction-Diffusion Theoryhharring/research/slideshow1.pdfCreating Patterns: REACTION-DIFFUSION THEORY 8 Introduction Boundary and Initial Conditions • Chemical

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

Page 5: Creating Patterns Reaction-Diffusion Theoryhharring/research/slideshow1.pdfCreating Patterns: REACTION-DIFFUSION THEORY 8 Introduction Boundary and Initial Conditions • Chemical

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

Page 6: Creating Patterns Reaction-Diffusion Theoryhharring/research/slideshow1.pdfCreating Patterns: REACTION-DIFFUSION THEORY 8 Introduction Boundary and Initial Conditions • Chemical

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

Page 7: Creating Patterns Reaction-Diffusion Theoryhharring/research/slideshow1.pdfCreating Patterns: REACTION-DIFFUSION THEORY 8 Introduction Boundary and Initial Conditions • Chemical

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

Page 8: Creating Patterns Reaction-Diffusion Theoryhharring/research/slideshow1.pdfCreating Patterns: REACTION-DIFFUSION THEORY 8 Introduction Boundary and Initial Conditions • Chemical

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

Page 9: Creating Patterns Reaction-Diffusion Theoryhharring/research/slideshow1.pdfCreating Patterns: REACTION-DIFFUSION THEORY 8 Introduction Boundary and Initial Conditions • Chemical

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

Page 10: Creating Patterns Reaction-Diffusion Theoryhharring/research/slideshow1.pdfCreating Patterns: REACTION-DIFFUSION THEORY 8 Introduction Boundary and Initial Conditions • Chemical

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

Page 11: Creating Patterns Reaction-Diffusion Theoryhharring/research/slideshow1.pdfCreating Patterns: REACTION-DIFFUSION THEORY 8 Introduction Boundary and Initial Conditions • Chemical

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

Page 12: Creating Patterns Reaction-Diffusion Theoryhharring/research/slideshow1.pdfCreating Patterns: REACTION-DIFFUSION THEORY 8 Introduction Boundary and Initial Conditions • Chemical

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

Page 13: Creating Patterns Reaction-Diffusion Theoryhharring/research/slideshow1.pdfCreating Patterns: REACTION-DIFFUSION THEORY 8 Introduction Boundary and Initial Conditions • Chemical

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

Page 14: Creating Patterns Reaction-Diffusion Theoryhharring/research/slideshow1.pdfCreating Patterns: REACTION-DIFFUSION THEORY 8 Introduction Boundary and Initial Conditions • Chemical

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 .

Math 534 Spring 2006 May 18, 2006

Page 15: Creating Patterns Reaction-Diffusion Theoryhharring/research/slideshow1.pdfCreating Patterns: REACTION-DIFFUSION THEORY 8 Introduction Boundary and Initial Conditions • Chemical

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

Page 16: Creating Patterns Reaction-Diffusion Theoryhharring/research/slideshow1.pdfCreating Patterns: REACTION-DIFFUSION THEORY 8 Introduction Boundary and Initial Conditions • Chemical

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

Page 17: Creating Patterns Reaction-Diffusion Theoryhharring/research/slideshow1.pdfCreating Patterns: REACTION-DIFFUSION THEORY 8 Introduction Boundary and Initial Conditions • Chemical

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

Page 18: Creating Patterns Reaction-Diffusion Theoryhharring/research/slideshow1.pdfCreating Patterns: REACTION-DIFFUSION THEORY 8 Introduction Boundary and Initial Conditions • Chemical

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

Page 19: Creating Patterns Reaction-Diffusion Theoryhharring/research/slideshow1.pdfCreating Patterns: REACTION-DIFFUSION THEORY 8 Introduction Boundary and Initial Conditions • Chemical

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

Page 20: Creating Patterns Reaction-Diffusion Theoryhharring/research/slideshow1.pdfCreating Patterns: REACTION-DIFFUSION THEORY 8 Introduction Boundary and Initial Conditions • Chemical

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

Page 21: Creating Patterns Reaction-Diffusion Theoryhharring/research/slideshow1.pdfCreating Patterns: REACTION-DIFFUSION THEORY 8 Introduction Boundary and Initial Conditions • Chemical

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

Page 22: Creating Patterns Reaction-Diffusion Theoryhharring/research/slideshow1.pdfCreating Patterns: REACTION-DIFFUSION THEORY 8 Introduction Boundary and Initial Conditions • Chemical

Creating Patterns: REACTION-DIFFUSION THEORY 21

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

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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

Page 24: Creating Patterns Reaction-Diffusion Theoryhharring/research/slideshow1.pdfCreating Patterns: REACTION-DIFFUSION THEORY 8 Introduction Boundary and Initial Conditions • Chemical

Creating Patterns: REACTION-DIFFUSION THEORY 23

(a) (b)

(c) (d)

Math 534 Spring 2006 May 18, 2006

Page 25: Creating Patterns Reaction-Diffusion Theoryhharring/research/slideshow1.pdfCreating Patterns: REACTION-DIFFUSION THEORY 8 Introduction Boundary and Initial Conditions • Chemical

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

Page 26: Creating Patterns Reaction-Diffusion Theoryhharring/research/slideshow1.pdfCreating Patterns: REACTION-DIFFUSION THEORY 8 Introduction Boundary and Initial Conditions • Chemical

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

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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