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. Growth inhibitor model. Find (by making c(r) plots for different R
values) R∗ (the maximum growth radius) of the spherical piece of tissue with
sensitivity level c1 = 0.2. How high the sensitivity level needs to be for the
tissue to continue to grow endlessly? Use β = 1, λ = 0.8, µ = 1.6, D = 1.
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