Creating Patterns
Reaction-Diffusion Theory
Heather A. Harrington and Kody John Hoffman Law
May 18, 2006
Abstract
The focus of our project is to model the dynamics of Turing pattern formation using activator/inhibitor reaction diffusion (R-D) equations. Cell
motility and chemical pattern models are the two types of mathematical
models for biological pattern formation. We focus on chemical patterns since
these are thought to have applications to tumor growth. We discuss multiple types of nonlinear kinetics common to reaction-diffusion theory such as
Gierer and Meinhardt, Thomas and then study Schnakenberg systems in detail. First, we will linearize the system, perform stability analysis and, based
on Neumann boundary conditions, we solve for the wavenumbers and corresponding eigenfunctions which yield spatially unstable solutions. In addition,
we describe the numerical methods used in our simulations. We define our
Turing parameter space for Schnakenberg necessary for diffusion driven instability, numerically implement some corresponding equations and observe
effects of domain change given particular parameters for one and two spatial
dimensions. Hence, we observe robust regimes of patterns with respect to
each parameter through variation of parameters. The solution to the R-D
bifurcates from spatially stable to unstable and we observe the convergence
to a heterogeneous steady state. Next, we numerically perform bifurcation
analysis to determine robustness of the diffusion driven instability. Finally,
we present our results and discuss further work in our last section.
Contents
1 Introduction
1.1 Reaction-Diffusion Theory . . . .
1.1.1 Nondimensionalization and
1.1.2 Linearization . . . . . . .
1.2 Reaction Kinetics . . . . . . . . .
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Boundary
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Conditions
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2 Stability Analysis
2.1 Homogeneous Uniform Steady State . . . .
2.1.1 Homogeneous Stability Conditions
for Schnakenberg Kinetics . . . . .
2.2 Diffusion-Driven Instability . . . . . . . .
2.2.1 Diffusion-Driven Instability
for Schnakenberg Kinetics . . . . .
2.3 Two-Dimensional Stability Analysis . . . .
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3 Numerical Methods
3.1 One-Dimensional Methods
3.1.1 Finite Difference .
3.1.2 Newton’s Method .
3.2 Two-Dimensional Methods
3.3 Error Analysis . . . . . . .
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4 Results and Discussion
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4.1 Bonus Section: Radial . . . . . . . . . . . . . . . . . . . . . . 27
Chapter 1
Introduction
In 1952, Alan Turing created a mathematical model describing the growing embryo’s sequential changes that occur from fertilization to birth [1].
This discovery started a trend of modeling using a mathematical approach
to describe the effects of different chemicals or morphogens, and how they
can react and diffuse throughout a tissue. Morphogenesis, the process of
the diffusion and reaction of morphogens, can cause patterns to form and
develop from a homogeneously steady state to a heterogeneous steady state
when a slight perturbation is created in this chemical system. Over the past
sixty years, this concept has been prevalent throughout biology in models
including ecology, animal coat patterns, embryonic development and cellular
chemical reactions in tumor growths [2, 3]. We are interested in equations
which are temporally stable, but unstable to spatial perturbation and their
heterogeneous equilibrium. The methods needed to understand and develop
patterns are explained in the following chapters. Drastically different biological processes all have one common factor: mathematical models based on
reaction-diffusion theory describe the given behavior.
1.1
Reaction-Diffusion Theory
When two chemicals diffuse within the same domain, a temporally stable, but
spatially unstable situation may occur resulting in pattern formation. The
reaction of the system is based on a basic diffusion equation of the chemical
concentration along with an additional nonlinear reaction term. Creating
an equation using the vector chemical concentration, c, diffusivities in a
diagonal matrix D, and reaction kinetics R(c) describe the reaction-diffusion
mechanics of the system.
∂c
= R(c) + D∇2 c
∂t
1
(1.1)
If two chemical species c = (u, v) are undergoing morphogenesis, it can be
generalized that each will have a specific diffusion coefficient D1 and D2 .
Moreover, to specify the two-component model system, we define
f (u, v)
R(c) =
.
g(u, v)
By (1.1), the system can be described as follows:
D1 u
ut
f (u, v)
2
.
=
+∇
D2 v
vt
g(u, v)
1.1.1
(1.2)
Nondimensionalization and Boundary Conditions
In (1.2), the nonlinear reaction functions, u and v which are dependent on
space (x) and time (t) have different diffusion coefficients. Depending on
the reaction and diffusion of the system, the reaction kinetics can vary. To
consider the kinetics of the system, we must first nondimensionalize the variables. Suppose y = Lx where the domain is x ∈ [0, L] which implies y ∈ [0, 1].
Then (1.2) can be rewritten as
1 2 D1 u(y(x), t)
f (u, v)
ut
=
+ 2∇
.
(1.3)
vt
g(u, v)
D2 v(y(x), t)
L
Next, divide both sides by D1 and multiply by L2
L2 u t
L2 f (u, v)
u(y(x), t)
2
.
=
+ ∇ D2
v(y(x), t)
D1 v t
D1 g(u, v)
D1
(1.4)
2
L
2
Finally, let t∗ = DL12t . Rename d = D
and γ = D
. By redefining γ, the
D1
1
size of the domain can easily be determined. The dimensionless system of
coupled nonlinear partial differential equations which model the development
and formation of patterns is seen in (1.5).
ut ∗
f (u, v)
u
2
=γ
+∇
(1.5)
vt ∗
g(u, v)
d·v
Suppose the diffusion coefficients, D1 and D2 equal zero, or that diffusion
is absent, then there exists suitable parameters for which the linearized system is stable. In the presence of diffusion, the homogeneous solution may be
2
unstable to spatial perturbation. Turing produced this novel idea of diffusiondriven instability, which is counterintuitive to the theories that diffusion always creates a stable solution. To take a more in depth look at how diffusiondriven instability can occur, next we explore the boundary conditions of the
reaction diffusion equation.
The boundary and initial conditions when solving a partial differential
equations, as learned in Math 534, can drastically change the solution of a
problem. Since the relevance of the chemical interactions and the formation
of patterns are only significant in the interior of the domain Ω, we define the
system to have zero flux boundary conditions and specified initial conditions.
According to [2], the Neumann B.C on the boundary of domain, ∂Ω where
the outward normal gradient vector for the species c(x,t) must also vanish
(n · ∇) · c(x, t) = 0 for x ∈ ∂Ω ,
for the reaction diffusion system. The initial condition of the solution is given
for the fixed boundary problem
c(x, 0) = c0 (x) .
1.1.2
Linearization
For this nonlinear partial differential equation, we set the initial conditions
as the constant equilibrium solution c0 . If we define a linear operator
L(c) =
∂c
− D∇2 c = R(c)
∂t
(1.6)
Since c0 is a spatially uniform homogeneous steady state solution, then
R(c0 ) = 0, similarly L(c0 ) = 0. We approximate the solution in a region
close to c0 where c1 ∈ R by a slight perturbation
c = c0 + c1 ,
and by substitution in (1.7) and Taylor expansion (1.8)
L(c0 + c1 ) = R(c0 + c1 )
L(c0 ) + L( c1 ) = R(c0 ) + R0 (c0 )c1 + 2
L( c1 ) = R0 (c0 ) c1 + O(2 )
3
(1.7)
00
R (c0 ) 2
c1
2
(1.8)
(1.9)
By linearity, we rewrite
L( c1 ) = L(c1 ) ⇒ L(c1 ) ≈ R0 (c0 )c1
(1.10)
Then we started with c0 as a solution, and found this approximation which we
redefine in later sections to be w = (w1 (t)w2 (x)) where c(x, t) = c0 + w(x, t).
In later sections, we apply this linearization method to the two species problem c(x, t) = (u(x, t), v(x, t)) using techniques from [2, 4]. Finally, this gives
the full linearized system
∂w
− D∇2 w = R0 (c0 )w
∂t
(1.11)
and understanding the linearization of the reaction diffusion equation is absolutely required for conducting stability analysis of the model.
1.2
Reaction Kinetics
We consider many different types of reaction kinetics that produce reaction
diffusion systems such as activator-inhibitor mechanisms. The simplest, the
Schnackenberg reaction (1.12) is used to describe the activator and inhibitor
interactions
f (u, v) = k1 − k2 u + k3 u2 v
g(u, v) = k4 − k3 u2 v .
(1.12)
Each ki is a positive rate constant and the stability analysis of this system
considered in the next section is based on these reaction kinetics. For the
scope of this project, we reduce via Murray’s change of variables to the
following equations
f (u, v) = a − u + u2 v
g(u, v) = b − u2 v .
(1.13)
Other types of kinetics, such as the Gierer and Meinhardt kinetics (1.14)
k3 u 2
v
2
g(u, v) = k4 u − k5 v ,
f (u, v) = k1 − k2 u +
4
(1.14)
and Thomas kinetics (1.15) are studied extensively in literature [2, 4] ; however, for the scope of this project, we will focus primarily on Schnackenberg
reaction kinetics.
f (u, v) = k1 − k2 u − h(u, v)
g(u, v) = k3 − k4 v − h(u, v)
k5 uv
h(u, v) =
k6 + k 7 u + k8 u2
5
(1.15)
Chapter 2
Stability Analysis
In this section we first consider the one-dimensional case conditions of the homogeneous steady state necessary for stability. Next, we explore our stability
conditions for Schnakenberg kinetics by finding the behavior the null clines
and Jacobian given different parameter values. We also generalize the conditions for diffusion-driven instability of the homogeneous steady state which
we verify numerically. Finally, we extend our findings to two-dimensions.
2.1
Homogeneous Uniform Steady State
To create a Turing-type spatial pattern, the system is highly dependent on
the reaction kinetics, R(c) as well as the parameters. In the presence of a
slight perturbation, we examine the homogeneous steady state
c0 = (u0 , v0 ) = 0 i.e. f = g = 0 .
With no diffusion the homogeneous steady state must be linearly stable.
Therefore the steady state must satisfy
ut = γf (u, v) = 0
vt = γg(u, v) = 0 .
(2.1)
Now we linearize around this steady state (u0 , v0 ). In order to find the
behavior of the kinetics in the absence of diffusion, we let
w = c − c0 = (u − u0 , v − v0 ) .
(2.2)
Where the Jacobian of our kinetics,
0
R (c0 ) is redefined as A =
6
fu fv
gu gv
.
(2.3)
dw
=γ
dt
fu fv
gu gv
·w
(2.4)
(u0 ,v0 )
From ordinary differential equations, this yields a solution in the form
w(t) ∝ veλt ,
where λ is an eigenvalue and v is the corresponding eigenvector. By substitution,
d (veλt )
= λ(veλt ) = γ A(veλt )
dt
wt = λw = γAw .
(2.5)
The eigenvalue problem (γA−λI)w = 0 is stable for the homogeneous steady
state solution w = 0 when Re(λ) < 0.
p
τA ± τA2 − 4∆A
λ1,2
=
,
(γA − λI)w = 0 ⇒
γ
2
where
τA = tr(A) = fu + gv
and ∆A = det(A) = fu gv − fv gu .
The conditions for stability for this time-dependent eigenvalue problem are
fu + gv < 0 ,
fu gv − fv gu > 0 .
2.1.1
(2.6)
(2.7)
Homogeneous Stability Conditions
for Schnakenberg Kinetics
The steady state (u0 , v0 ) for the Schnakenberg kinetics (1.13), exists when
f (u, v) = a − u + u2 v = 0 and g(u, v) = b − u2 v = 0
and v = ub2 intersect). By setting f = g = 0 , the behavior of
(i.e. v = (u−a)
u2
the kinetics in the absence of diffusion is found (see Figure 2.1). The solution
of this intersection is
b
(u0 , v0 ) = b + a,
.
(2.8)
(b + a)2
7
Figure 2.1: Null clines of the Schnakenberg kinetics along the domain [0.1, 1.5]. The
intersection of the steady state, (u0 , v0 ) occurs at (1.0, 0.9).
For the stability of Schnakenberg kinetics, our homogeneous steady state
conditions (2.6) and (2.7) must always satisfy:
b − a < (b + a)3
(a + b)2 > 0
(2.9)
(2.10)
Given the kinetics, the Jacobian is calculated
∂
f (u, v) = −1 + 2uv
∂u
∂
g(u, v) = −2uv
∂u
∂
f (u, v) = u2
∂v
∂
g(u, v) = −u2 .
∂v
At the steady state the linearized system is defined:
d u
−1 + 2u0 v0 u20
u
=γ
.
2
−2u0 v0
−u0
v
dt v
In this case, we solve the eigenvalue problem
wt = γAw = λw ,
8
(2.11)
A =
a11 a12
a21 a22
=
−1 +
−2b
a+b
2b
a+b
(b + a)2
−(b + a)2
(2.12)
The characteristic polynomial is
λ2 − λ(a11 + a22 ) + (a11 a22 − a12 a21 ) .
To guarantee stability in the absence of diffusion, we require the Re(λ) < 0,
a11 + a22 < 0 and (a11 a22 − a12 a21 ) > 0 .
(2.13)
We note
fv = a12 > 0 and gu = a21 < 0.
2.2
Diffusion-Driven Instability
In the presence of diffusion, there may be certain conditions that will yield
an instability. In our model, patterns may form when the system is unstable.
The necessary conditions are derived and explained here. From (2.2), we let
w ∈ B (w0 ) and then linearize around the full system
1 0
2
wt = γAw + D∇ w where D =
.
(2.14)
0 d
Then
w(x, t) = X(x)T (t)
with T (t) ∝ veλt .
(2.15)
(2.16)
By separation of variables, we next solve the Laplacian for any x ∈ Ω and let
λ be dependent on the eigenvalue k 2 , where the wavenumber k is determined
by the following equation:
k 2 X(x) + X 00 (x) = 0 ,
such that (n · ∇)w(x, t) = 0 on ∂B .
The general solution to the Laplacian eigenvalue problem for the nondimensionalized system on (0,1) is
X 00 (x) = −k 2 X(x)
⇒ X(x) = B cos(kx) + Csin(kx) .
9
(2.17)
(2.18)
To satisfy the zero flux conditions, we found
k = nπ ,
for mode numbers n = 1, 2, 3 . . .
and the eigenfunction is of the form
X(x) ∝ cos(nπx) ⇒ Xk (x) ∝ cos(kx) .
(2.19)
Through techniques learned in Math 534 [5], we determine there exists
a zero eigenvalue k = 0, which is precisely our stationary homogeneous
solution,(u0 , v0 ). Any solution X(x) is a linear combination of elements in the
set of Xk (x)’s. By substituting (2.16) and (2.19) in (2.15), the full solution
X
Xk (x)Tλ (t)
w(x, t) =
k
=
X
Fk cos(kx)eλt ,
(2.20)
k
where we let w(x, 0) = g(x). This implies
X
Fk cos(kx) = g(x)
k
where Fk are fourier cosine coefficients and g(x) is some perturbation of w0 .
Finally, we substitute the solution w for each k of (2.20) into (2.14) and
linearize,
wt = Fk cos(kx)λeλt
Fk cos(kx)T (t)λ(k 2 )
Xk (x)T (t)λ(k 2 )
wλ(k 2 )
⇒0
2.2.1
=
=
=
=
=
γAFk cos(kx)eλt + D∇2 Fk cos(kx)eλt
γAFk cos(kx)T (t) + DFk k 2 (− cos(kx))T (t)
γAXk (x)T (t) − Dk 2 Xk (x)T (t)
γAw − Dk 2 w
(−γA + k 2 D + λ(k 2 )I)w
(2.21)
Diffusion-Driven Instability
for Schnakenberg Kinetics
To solve for the eigenvalues, λ = λ(k 2 ) , we evaluate the eigenvalue problem
and solve the characteristic polynomial for Schnakenberg kinetics. Let
G = γA − k 2 D
10
and then evaluate the equation for these particular kinetics
0 = (G − λI)w
2
a11 a12
k
0
λ 0
= γ
−
−
· w.
a21 a22
0 dk 2
0 λ
(2.22)
(2.23)
Then,
det(G − λI) = γ 2 a11 a22 − γa11 k 2 d − γa11 λ − γa22 k 2 − γa22 λ + k 2 λ
+k 2 dλ + k 4 d + λ2 − γ 2 a12 a21
= λ2 + λ(k 2 + k 2 d − γ(a11 + a12 )) + k 4 d − γa11 k 2 d
+γ 2 (a11 a22 + a12 a21 )
= λ2 − λ −k 2 (1 + d) + γ(a11 + a22 ) + ϕ(k 2 )
where
ϕ(k 2 ) = dk 4 − k 2 (γ(a22 + a11 d)) + γ 2 (a11 a22 − a12 a21 ) .
(2.24)
Then, it follows
2
λ − λτG + ∆G
and we find λ1,2 =
τG ±
p
τG2 − 4∆G
2
(2.25)
given
τG = γ(a11 + a22 ) − k 2 (1 + d) and ∆G = ϕ(k 2 ) .
In the absence of diffusion we have stability if Re((λ(0)) < 0, so we have the
following
a11 + a22 < 0 .
(2.26)
Since k 2 (1 + d) > 0 ,
then τG < 0.
(2.27)
To satisfy
Re(λ(k 2 )) > 0, i.e.
q
q
2
2
τG2 − 4ϕ(k 2 ) ≥ |τ | ,
τG ± τG − 4ϕ(k ) > 0 ⇒
we need ϕ(k 2 ) < 0 . To determine what value of ϕ will yield this outcome,
we calculate when ϕmin < 0.
∂ϕ
= 2dk 2 − (γ(a22 + a11 d))
∂k 2
11
Thus, we find ϕ is minimum when
k2 =
γ(a22 + a11 d)
.
2d
By substituting k 2 into (2.24)
(da11 + a22 )2
2
ϕmin = γ (a11 a22 − a12 a21 −
4d
(da11 + a22 )2
⇒ a11 a22 − a12 a21 <
4d
(2.28)
(2.29)
Therefore, ϕmin = 0 is a bifurcation value of diffusion driven instability. We
can also find a critical value for the diffusion coefficient, dc at the bifurcation
point ϕ.
4dc (a11 a22 − a12 a21 ) = (dc a11 + a22 )2
d2c a211 + 2(2a12 a21 − a11 a22 )dc + a222 = 0
Note: By requiring ϕ(k 2 ) < 0, a necessary condition for instability is
da11 + a22 > 0 .
(2.30)
By (2.13) ⇒ d > 1. The following conditions are necessary, but not sufficient
for diffusion driven instability in conjunction with (2.13)
(a + b)3
a22
=
d>−
a11
b−a
2
(a22 + a11 d) > 4d(a11 a22 − a12 a21 )
⇒ [d(b − a) − (a + b)3 ]2 > 4d(a + b)4 .
(2.31)
(2.32)
An additional requirement for diffusion-driven instability is the existence of
a wavenumber, k 2 within the given domain in order for a spatially unstable
pattern to form. For
dc > 1 ,
we establish that d > dc
will yield unstable modes corresponding to k 2 which is given by
γL < k 2 < γM
12
(2.33)
p
(a22 + a11 d)2 − 4d(a11 a22 − a12 a21 )
KL = L = γ
2d
p
(a22 + a11 d) + (a22 + a11 d)2 − 4d(a11 a22 − a12 a21 )
KM = M = γ
.
2d
(2.34)
(a22 + a11 d) −
Therefore, our final condition is sufficient for large γ.
2.3
Two-Dimensional Stability Analysis
As with the one-dimensional case, we analytically solve for the spatial Laplacian in two dimensions for two spatial components, W = X(x)Y (y) with
(n · ∇)W = 0 for (x, y) ∈ ∂B. Once again we solve:
∆W = −k 2 W
Wxx + Wyy = −k 2 W
X 00 (x)Y (y) + X(x)Y 00 (y) = X(x)Y (y)(−k 2 )
(2.35)
(2.36)
(2.37)
By separating variables, we get
X 00 (x) Y 00 (y)
+
= −k 2 = −(k12 + k22 ) .
X(x)
Y (y)
where
X 00 (x)
= −k12
X(x)
and
Y 00 (y)
= −k22 .
Y (y)
Now, we solve
X 00 (x)
X(x)
⇒ X(x)
⇒ X 0 (x)
X 0 (0)
X 0 (1)
= −k12
(2.38)
=
=
=
=
(2.39)
(2.40)
(2.41)
(2.42)
A cos(k1 x) + B cos(k1 x)
Ak1 sin(k1 x) + Bk1 sin(k1 x)
Bk1 = 0 ⇒ B = 0
Ak1 sin(k1 x) = 0 ⇒ k1 = nπ .
Similarly, since solving for Y (y) is analogous to (2.42) for the given boundary
conditions. We find k2 = mπ.
k 2 = (mπ)2 + (nπ)2
where modes n, m ∈ Z
13
This gives the full two-dimensional solution similar to (2.20) to be
X
2
w(x, y, t) ≈
Fm,n cos(nπx) cos(mπy)eλ(k )t .
(2.43)
m,n
The unstable wavenumbers must contain at least one possible mode, so k 2
must satisfy the sufficient condition (2.33) for diffusion driven instability.
14
Chapter 3
Numerical Methods
In order to solve the nonlinear problem, we use two numerical methods: finite
difference and Newton’s method and analyze the error using methods from
Math 652.
3.1
3.1.1
One-Dimensional Methods
Finite Difference
The finite difference scheme represents derivatives discretized by a mesh
which represents reality to the specified accuracy. This accuracy is to be
determined by preference according to what is required as well as the relation between the sizes of the mesh dimensions as dictated by error analysis.
Since the two-component system has already been nondimensionalized where
x ∈ [0, 1], we let
∆x =
1
M
with x = ∆x · m for m = 0, . . . , M ,
then we have the second order finite difference approximation
u(∆x(m + 1)) + u(∆x(m − 1)) − 2u(∆xm)
+ O((∆x)2 ) (3.1)
(∆x)2
u(∆x(m + 1)) + u(∆x(m − 1)) − 2u(∆xm)
≈
= δx2 u
(3.2)
(∆x)2
uxx =
uxx
We denote uxx ≈ δx2 u which is accurate on the order of (∆x)2 , and is certainly
sufficient if M = 100. The first order finite difference method for the temporal
derivative we have
ut =
u(∆t(n + 1)) − u(∆tn)
+ O(∆t)
∆t
15
(3.3)
Similar to (3.2), by dropping the O(∆t), we find
ut ≈
u(∆t(n + 1)) − u(∆tn)
.
∆t
(3.4)
Let
n
Um
≈ u(∆xm, ∆tn) ,
then we have
n
n+1
− Um
Um
n
n 2
n
= γ(a − Um
+ Vmn · (Um
) ) + δx2 Um
,
∆t
Vmn+1 − Vmn
n 2
n
= γ(b − Vmn · (Um
) ) + δx2 Um
,
∆t
(3.5)
with error bounded by O(∆t) + O(∆x)2 . Hence, with given initial conditions, we iterate this system solving for each subsequent time level n+1. The
method we use to iterate the changes in two species with respect to time is
n+1
n
n
n 2
n
Um
= Um
+ ∆t γ(a − Um
+ Vmn (Um
) ) + δx2 Um
(3.6)
n+1
n
n
n 2
2 n
Vm
= Vm + ∆t γ(b − Vm (Um ) ) + δx Vm
(3.7)
We handle the boundary conditions via the following scheme: Assume without loss of generality there exists a point ∆x · (M + 1), and the Neumann
boundary condition is hence:
n
n
n
UM
∂UM
+1 − UM −1
=
=0
∂x
2∆x
n
n
⇒ UM
+1 = UM −1 .
(3.8)
(3.9)
Therefore, this yields
n
=
δx2 UM
n
n
2(UM
−1 − UM )
(∆x)2
(3.10)
We iterate this method until the solution is sufficiently removed from the
homogeneous steady state. Experimentation proved that n = 10e+5 was
adequate. Subsequently, we iterate until we reach a suitable threshold (10e-8
in the L2 norm) for convergence to the heterogeneous steady state. Finally,
we polish our solution with Newton’s method, exacting our solution to 10e15.
16
3.1.2
Newton’s Method
Newton’s method is a root finding algorithm which is an iterative process.
Given a point (xn , f (xn )), f 0 (xn ) and a point near the solution f(x) = 0, the
method converges to a fixed point giving better approximation of the solution
at each step. We use the first few terms of the Taylor series to get
f (x) − f (xn ) = f 0 (xn )(x − xn )
(3.11)
By letting f (x) = 0, we substitute into (3.11) to derive the generalized
iterative Newton method
xn+1 = xn −
f (xn )
f 0 (xn )
(3.12)
In our case, we apply Newton’s method to the already vectorized space
u = (u0 , u1 , · · · um ) and v = (v0 , v1 , · · · vm ) .
(3.13)
We define our f (xn ) as
n
n 2
n
Fn = γ(a − Um
+ Vmn (Um
) ) + δx2 Um
n 2
Gn = γ(b − Vmn (Um
) ) + δx2 Vmn
(3.14)
(3.15)
and then we calculate the f 0 (xn ) in (3.12), which corresponds to the Jacobian
of (3.14), (3.15)
∂F ∂F ∂u
∂v
Jn = ∂G
which is a 2M X 2M quindiagonal matrix.
∂G
∂u
∂v
n
Hence, substitute (3.13), (3.14), (3.15) into (3.12)
u
u
F
=
−
· J −1
v n+1
v n
G n n
(3.16)
And we are able to substantially refine our answer within a few iterations
(we use ten).
3.2
Two-Dimensional Methods
As with one-dimensional stability analysis of the two-component system, we
also extend the aforementioned numerical method to an additional spatial
dimension. Without loss of generality, we assume that the finite difference
17
first order approximation for ut is the same discrete approximation (3.4). The
same logic generalizes the finite difference method, except for two dimensions
we use five points rather three to calculate the solution at each time step.
We let
1
with j∆x for j = 0, · · · , M
M
1
with p∆y for p = 0, · · · , P
∆y =
P
∆x =
(3.17)
(3.18)
Thus we extend our finite difference method where
n
Ujp
≈ u(∆xj, ∆yp, ∆tn) then:
n
n
∇2 Ujp
≈ (δx2 + δy2 )Ujp
n
n
n
n
n
n
Uj+1,p
+ Uj−1,p
− 2Uj,p
Uj,p+1
+ Uj,p−1
− 2Uj,p
≈
+
(∆x)2
(∆y)2
n
= ∇2 Ujp
+ O(∆x2 ) + O(∆y 2 )
The full finite difference using (3.5), the two-component model is
n+1
n
n
n 2
n
Ujp
= Ujp
+ ∆t γ(a − Ujp
+ Vjpn (Ujp
) ) + ∇2 Ujp
n 2
Vjpn+1 = Vjpn + ∆t γ(b − Vjpn (Ujp
) ) + ∇2 Vjpn
(3.19)
(3.20)
(3.21)
(3.22)
(3.23)
Due to the complexity of the two-dimensional system of equations, we refrain from the use of Newton’s Method and we allow ourselves convergence
to 10e-6 in L2 norm, which reflects pointwise convergence of 10e-8. For
additional simplification of the two-dimensional problem, our domain Ω is
a square where ∆x = ∆y. Moreover, this way we are able to utilize our
non-dimensionalization technique.
3.3
Error Analysis
For numerical simulations of standard diffusion equations, it has been shown
that the appropriate choice for ∆t to ensure convergence and stability is
∆t ≤
(∆x)2
.
2
Now, the calculation for ∆t becomes quite complicated and also somewhat
ambiguous for the case of a nonlinear reaction diffusion difference approximation. From Morton and Mayers [6], we know there exists such a ∆t based
18
on a function similar to this including the maximum of the functions u and
v and their derivatives provided they exist (i.e. provided the functions are
adequately smooth). We also know that due to our nondimensionalization,
we will find γ in the denominator of the equation for this ∆t. Therefore, we
empirically test for ∆t, starting at only a fraction of the necessary value for
a standard diffusion, until we observe convergence to a steady state. For our
one-dimensional case with ∆t = 0.1(∆x2 ), our solution seems well behaved.
However, from a little experience with nonlinear problems, we use a smaller
∆t = 0.01∆x2 to be safe (especially considering our imprecise method). In
the two-dimensional regime, we find the system to be less well-behaved, as
expected, and reduce to 0.001(∆x2 ).
19
Chapter 4
Results and Discussion
We numerically find the Turing parameter space (a, b, d) for Schnakenberg kinetics which allows us to choose the parameters values necessary for
diffusion-driven instability. Notice that figure 4.1(a) appears to have disjoint
points, but there are merely the discrete points that numerical implementation creates. If we increase the number of points, this would appear to be
a continuous space. The Schnakenberg kinetics have an interesting property
since the sign of the matrix Jacobian A is
+ +
.
(4.1)
− −
The sign of the Jacobian influences the behavior of the aforementioned null
clines in figure 2.1 and also show a figure eight behavior between the two
species, u and v at the heterogeneous steady state solution (see figure 4.1).
In figure 4.2, we can observe the solution for particular parameters bifurcating as a function of the size of the domain, L. The axes (x, L, u)
respectively correspond to the standard (x, y, z) axes, and the graph of the
steady state of u by x is presented over changing domain size. If we compare figures 4.2(c) and 4.2(e), we can qualitatively see that as the domain
increases, the number of modes for the solution increases. At first it is apparent that the largest wavenumber corresponds to the dominant growing
mode by comparison of figures 4.3(a) and 4.2(e). However, we observe that
as more unstable modes emerge in our linearized solution, it becomes clear
the distinction of dominance cannot be determined based on wavenumber
alone. Mode seven did not take over as the dominant mode until we not
only included mode nine, but lost mode five, (see figure 4.3(a))which occurs
at domain size L=48. Here, we can also observe the distinction between the
dominant unstable mode and the heterogeneous steady state solution. From
our simulation figure 4.2(c), we see the solution exhibits behavior similar to
20
mode seven (seven nodes) within domain sizes L=[45, 50] as contrasted with
its dominance from L = 48.
21
(a) Turing Parameter Space with Schnakenberg Kinetics
(b) One-dimensional steady state for Schnakenberg kinetics, activator
u and inhibitor v (a = 0.2, b = 0.5, d = 20, L = 50)
Figure 4.1: Turing parameter space and steady state behavior for Schnakenberg kinetics.
22
(a)
(c)
(e)
(b)
(d)
(f)
Figure 4.2: 4.2(a)-4.2(d): Solution of the 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. 4.2(e) and 4.2(f): Solution of the dominant mode of
heterogeneous steady state of the activator u with one-dimensional Schnakenberg kinetics
23
(a = 0.3, b = 1, d = 40) over changing domains where L ranges from 10 to 50.
The simulations do have approximately the same qualitative behavior as
the dominant unstable mode which suggests our analytical findings are correct, that is, the dominant mode does influence the solution of the reactiondiffusion system and this correlation decreases with increase in total number
of modes.
In addition to formulating the Turing space, the behavior of the Schnakenberg kinetics and bifurcations, we also experiment with the perturbation size
to the system. By satisfying the four conditions for diffusion driven instability and a sufficient domain size, we find that for any size perturbation,
instability emerges, which confirms our suspicions. While conducting numerical simulations, we realized the role of the mode is dependent on the
size of the domain. We see the number of modes given a domain in figure
4.3(b) and have similar results as [4] where the modes overlap, and we note
that no unstable mode exists in the interval [9,10]. In figure 4.3(a) we determine which mode numbers exist as the size of the domain increases. We
realize the specific nodes correspond to the number of nodes illustrated in
figure 4.3(b).
Finally, we extend our one-dimensional simulations to two dimensions
using the numerical methods previously discussed. Similar to the onedimensional case, our numerics also confirm our analytical analysis findings
that the mode numbers increase as the length increases. Here we allow domain grown contiguously from the previous domain’s heterogeneous steady
state rather than initializing from a perturbation of the homogeneous. As
we iterate, we notice the change in the heterogeneous steady state solution
as the domain grows from L = 10 to L = 50 is symmetric with respect to
the line y = x. The evolution of the pattern at approximately equal increase
in the number of nodes over the increase in domain is consistent with what
we expect from a generalization from the analysis as shown in figure 4.4.
However, it does not have the full completely symmetric as a steady state
from an initial perturbation of the homogeneous steady state has.
24
(a) Mode Numbers vs L
(b) Number of Modes vs L
(c) I.P=1e-2, a = 0.2, b = 0.5, d = 20, L =200. (d) I.P= 1e-6, a = 0.1, b = 0.9, d = 10, L = 4.
Figure 4.3: The dominant modes for the linearized system as the domain changes are
illustrated in 4.3(a)-4.3(b). In 4.3(c) and 4.3(d), show the emergence of instability or
unstable modes of the solution of u for the linearized system for Schnakenberg kinetics
given an initial perturbation(I.P) at x = 5.
25
(a)
(b)
(c)
(d)
(e)
(f)
Figure 4.4: Solution of the two-dimensional steady state with a growing square domain
from L = 10x10-50x50 over time and kinetics (a = 0.1, b = 0.9, d = 10). 4.4(a) is at t=1,
4.4(b) corresponds to t 10, 4.4(c) occurs at t = 20, 4.4(d) t = 25, 4.4(e) t =33 and 4.4(e)
is the final t = 40 and L = 50.
26
After thoroughly analyzing and numerically implementing activatorinhibitor systems of reaction-diffusion equations, we came to truly understand the complexity of nonlinear partial differential equations. For most of
the problems we studied in this project, we were able to fairly accurately predict the qualitative behavior of the solution of the full nonlinear PDE with
our analysis of the linearized system around the equilibrium point; however,
as we began to explore and investigate the same problems except in a radial
domain, only then did we fully appreciate the new world of nonlinearity.
In concluding our project, we came up with questions and possible problems to consider in the future. Two possible extensions to the project we
can explore are three-dimensional pattern formation, and more complicated
surface patterns, such as a surface of a three-dimensional object, or a radially oriented diffusion (which we begin to explore in the bonus section, with
the disk). These have applications to realistic biological problems such as a
tumor growth. Also, we can extend our study of two-dimensional systems to
other Cartesian shapes, such as the rectangle (near future) and other, more
complicated shapes (such as a triangle or hexagon). In the future, we plan
to employ additional numerical methods such as an implicit and combination (Θ) [6] scheme that will allow us to examine the tradeoffs in calculation
time since this yields a recursive system of equations and less restrictions
on our time step. More analysis can be done on our two-dimensional square
growing domain, as well as growth of the domain beyond those which we
have explored, and the growing domain analysis in these other geometries.
One more possible avenue to mention is the affect of initial perturbation size
and location, if and how this alters the heterogeneous solution (possibly even
opening up the door to perturbation-based bifurcation).
Finally, we note that our exploration of activator inhibitor systems of
reaction diffusion equations which are spatially unstable in the field of biology
has merely scratched the surface, and that possible applications are extensive
(if not infinite). Other applications include the variations of these systems
which can model instabilities in neural firings and conformal mapping from
the retina to the visual cortex (which the authors suspect may be related to
macular degeneration). Thank you.
4.1
Bonus Section: Radial
For the one-dimensional Laplacian, we solve the eigenvalue problem
1
R00 (r) + R0 (r) = βR(r)
r
27
(4.2)
So, we solve this radially dependent eigenvalue problem.
βR =
R0 =
x
X
n=2
∞
X
an−2 rn−2
(4.3)
(n − 1)an−1 rn−2
(4.4)
n(n − 2)an rn−2
(4.5)
n=2
R0 =
∞
X
n=2
∞
a1 X
R0
⇒
=
+
nan rn−2
r
r
n=2
(4.6)
∞
⇒
a1 X
[βan−2 + nan + n(n − 1)an ] f n−2 = 0 .
+
r
n=2
(4.7)
Then, we can simplify this to find
n2 an + βan−2 = 0 and an = −β
an−2
for n ≥ 2 and a1 = 0.
n2
This implies
β n (−1)n a0
a2n+1 = 0.
(2n n!)2
We observe this is exactly the Bessel function:
a2n =
R(r) = a0
∞
X
(−1)n β n r2n
22n n!2
n=1
R0 (1) = a0
∞
X
(−1)n β n 2n
n=1
22n n!2
+ a0
(4.8)
=0
(4.9)
With angular dependence, (4.2) becomes
R00 (r)
R0 (r)
Θ00 (θ)
+
+ 2
=β
(4.10)
R(r)
rR(r) r Θ(θ)
We expect similar analysis to reveal the Bessel function of order m for
each eigenvalue m of the (Θ) eigenfunction:
Θ(θ) = Am cos(m(θ)) + Bm sin(m(θ)),
28
m = 1, 2, 3, . . .
(4.11)
(a)
(b)
(c)
(d)
Figure 4.5: Traveling front for strictly radial dependence of the solution u on the disk
29
(a)
(b)
(c)
(d)
Figure 4.6: Limit cycle for radially and angularly dependent solution for the activator (u)
on the disk
30
p
∞
X
(−1)n ( (β)r)2n+m
R(r) = a0
+ a0
2n+m n!(n + m)!
2
n=1
p
∞
X
(−1)n ( (β))2n+m (2n + m)
0
R (1) = a0
=0
22n+m n!(n + m)!
n=1
(4.12)
(4.13)
When we numerically implement these functions with aforementioned
techniques, we find that the empirical error analysis required an even smaller
∆t, namely 0.001(∆x2 ) for the strictly radially dependent case. For the case
of angular dependence, the solution was unstable until 0.0001(∆x2 ).
We use a slightly modified finite difference method and also employ the
scheme from [6] for handling the boundary condition necessary at r=0. In
one dimension, the radial solution appears to converge to a traveling front,
while with azimuthal dependence we observe a limit cycle. Unfortunately,
time does not permit further investigation for this project. It should be
noted (or perhaps confessed) that for the case of angular dependence, we
also instated an artificial boundary condition uθθ (1, θ) = 0. As well we note
that this section could surely be elaborated another ten pages, but for the
reader’s sake, since the material is not ground breaking, we refrain from doing
so.
Acknowledgements to individuals who have helped us
Panayotis Kevrekidis
Nathaniel Whitaker
Robin Young
Nathan Fidalgo
31
Bibliography
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R. Soc. Lond. 237, 37-72.
[2] J.D. Murray (2002). Mathematical Biology Volume 17. Springer-Verlag,
Berlin Heidelberg, third edition.
[3] M.A.J Chaplain, M. Ganesh, I.G.Graham (2001). Spatio-temporal pattern formation on spherical surfaces: numerical and application to solid
tumour growth. J. Math. Biol., 42, 387-432.
[4] E.J.
Crampin
(2000).
Reaction-Diffusion
Patterns
on
Growing
Domains:
DPhil
Thesis.
Magdalen
College,
University
of
Oxford.
Introduction
and
Chapter
One
http://www.bioeng.auckland.ac.nz/people/crampin/thesis.html
[5] Strauss, W.A. (3 March 1992). Partial Differential Equations: An Introduction. New York: Wiley.
[6] K.W. Morton and D.F. Mayers,(1994). Numerical Solution of Partial
Differential Equations, Cambridge University Press, Cambridge.
32