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. The diffusion-limited growth model.
a)Find integration constants and plot steady state solution for the nutrient
concentration as a function of distance from the center of the tumor. Clearly
mark on your graph the boundary of the necrotic core. Also from you graph,
find the thickness of the proliferating cells shell. Use c2 = 2, c1 = 1, r2 = 2,
k = 1 and D = 0.5.
b) Now use the same parameters, but r2 = 10. Does the thickness of the
proliferating layer match the large size approximation we found in class?
c) Predict (without much computation) the size of necrotic core for r2 = 19.
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