Math 5120, Homework 8, due March 9
1. Our general m-dimensional reaction-diffusion model, when written in polar
coordinates (R, φ), can be used as a model for the formation of tentacles in hydra. Hydra is a simple freshwater animal, it has tubular body, so it is annular is
cross-section (http://en.wikipedia.org/wiki/Hydra_(genus) ). If a spatial
mode with n peaks grows with time we will interpret it as n tentacles beginning
to grow.
a) For the general reaction-diffusion equation
∂~u
= αf~(~u) + D∇2 ~u
∂t
rewrite it in plane polar coordinates, using the fact that
∇2 ~u =
∂~u
1 ∂ 2 ~u
1 ∂
(R
)+ 2 2
R ∂R ∂R
R ∂φ
b) Show that growth of n tentacles is possible if the equation
n2
det σn I − αJ + 2 D = 0
a
has a solution with positive real part. Assume for simplicity that δ is so small
there is no variation in R direction.
2. Let us see in a simple example that diffusion can indeed destabilize a
steady state. Consider the equation
J=
~ut = J~u + D~uxx ,
3
13
1
, D=
−1 −3
0
0
9
.
a) Show that in equation without diffusion ~ut = J~u the steady state is stable
b) Now add diffusion to the equation and consider it on the interval (0, π) with
zero Neumann boundary condition. Are there any unstable spatial modes for
this problem? If so, which?
1