Math 4600, Homework 12
1. The following model describes signal transduction in the axon of a neuron:
ut = uxx + u(1 − u)(u − 0.5),
where u represents the membrane potential. We study the model on 0 ≤ x ≤ l
with boundary conditions
ux (0, t) = 0, ux (l, t) = 0.
a) Find equation for the steady state solution of this system. Write it in the
form of a system of first order ODEs (similar to what we did for the traveling
wave solution)
b) Find fixed points of the ODE system and study their stability
c) Sketch a phase portrait (representative trajectories in the phase plane)
d) Mark with a different color trajectories that satisfy boundary conditions and
sketch them as a function of x. What do these state solutions mean biologically?
2. In the cancer-immune system interaction model plot f (X) for different
values of E0 . Make observations on the shape and location of zeros of f , note the
cases when you think the traveling wave solution can exist. Do your observations
match what we found in class? (you can choose r = 1, K = 1, k1 = .1, k2 = 0.1
3. Consider the diffusion equation with u(0, t) = 0, u(1, t) = 0.
Using the separation of variables approach, we need to consider solutions of the
form
ũ(x, t) = eλt sin(ωx).
a) Find appropriate values of ω using the boundary conditions
b) Find corresponding values of λ by substituting the prospect solutions into
the equation
c) What is the qualitative behavior of ũ(x, t) as t → ∞?
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