Math 4600, Homework 11
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. One of the well-known phenomenological (capturing the phenomena, but
not necessarily the mechanisms) models of cancer is represnted by Gompertz
equation
dN
= −bN ln(N/K).
dt
a) Solve this equation with N (0) = N0 using the substitution u = ln(N/K), to
obtain the solution N (t) = K exp(−Ae−bt ), where A = − ln(N0 /K)
b) (computing) Plot the solution as a function of time for K = 1, N0 = 0.1,
b = 1. Illustrate with a plot and describe in words what happens to the tumor
dynamics if the growth rate b is varied. Compare with the logistic dynamics
Ṅ = −bN (N − K)
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