MA256 Spatial Patterns Tutorial
Problem 1:
Consider the one-dimensional di↵usion equation on the whole of R .
@u
@2u
=D 2
(1)
@t
@x
with initial conditions u(x, 0) = x
x is the Dirac delta function, which is zero except at x = 0 and whose
integral is unity
The solution of this initial value problem is given by
✓
◆
1
x2
u(x, t) = p
exp
(2)
4Dt
4⇡Dt
Show by di↵erentiation that the function u(x, t) (eq 2) satisfies (eq 1)
Hint: Some algebra can be avoided by using the fact that
log u =
1
log(4⇡Dt)
2
x2
4Dt
Problem 2:
Consider the dimensionless Bruesselator model which can give rise to spatial
patterns through a di↵usion-driven Turing instability.
@u
@t
@v
@t
=
=
@2u
@2u
2
=
a
(b
+
1)
u
+
u
v
+
D
u
@x2
@x2
2
2
@ v
@ v
g (u, v) + Dv 2 = bu u2 v + Dv 2
@x
@x
f (u, v) + Du
(3)
(4)
with parameters a, b, Du , and Dv all positive.
a)
Which term describes this model best: Activator-inhibitor model,
enhanced degradation model, or substrate depletion model?
b)
⇤ ⇤
Determine the steady
✓ @f state
◆(u , v ) of equations (3,4) and compute
@f
@v
the Jacobian J = @u
at (u⇤ , v ⇤ ). Considering its trace tr (J)
@g
@g
@u
@v
and determinant det (J), what condition must b meet to obtain a
stable equilibrium?
1
c)
The existence of a stable fixed point in the absence of di↵usion as
shown in b) is one condition for a di↵usion driven Turing instability.
Furthermore we require
Du
p
@g
@f
+ Dv
> 2 Du Dv ⇤ det (J) > 0
@v
@u
(5)
u
Using equation (5) and = D
Dv , determine bcrit above which you
expect stable spatial patterns to occur.
d)
Setting a = 2, Du = 4, and Dv = 16, obtain a numerical
for
qvalue
@g
@f
Du @v
+Dv @u
bcrit and compute the corresponding wave number kcrit =
2Du Dv
and the corresponding wavelength crit .
e)
Assume parameters as in d) and that b is slightly above bcrit so that
an initial random pertubation grows into a stable spatial pattern.
No-flux boundary conditions shall apply. For a domain of size L =
3
2 crit sketch all possible solutions for the spatial profiles you can
expect for u (x) and v (x) . Label the x-axis in units of ⇡. On the
y-axis show the numerical value of the steady state values u⇤ and v ⇤
which can be taken as 2.
2