Math 5120, Homework 3
1. In the age-structured model we found in class the solution
Ra
−
(λ+µ(b))db
u(a, t) = ceλt e 0
We still need to determine constants c and λ using the initial and boundary
conditions:
u(a, 0) = u0 (a);
Z ∞
u(0, t) =
β(a)u(a, t)da.
0
a) Using the boundary condition show that λ satisfies
Z ∞
Ra
−
(λ+µ(b))db
β(a)e 0
da = 1.
0
b) If β(a) ≥ β0 > 0 then show for the left hand-side of equation in a):
- it is a strictly decreasing function of λ
- find limits for λ → ±∞
Then argue that the equation in a) has a unique solution.
c) Using the initial condition show that the equation for finding c is:
Ra
−
(λ+µ(b))db
ce 0
= u0 (a).
Solve for c. Note that it cannot be satisfied for an arbitrary choice of µ(a) and
u0 (a) with a constant c. This means that our method (separation of variables)
will only work for this system when this condition can be satisfied. Suppose
taht such c can be found.
d) Notice that the complete solution that we just finished finding involves time
only in eλt . As a result the following can be shown: with time the age distribution takes shape of w(a) and this shape grows to infinity (if λ > 0) or decays to
zero (if λ < 0) exponentially. Assume µ(a) = 0.01, u(a, 0) = 1 million for age
from 0 to 100, β(a) = 0.07 for a from 20 to 35. Find a plot the age-distribution
solution for several values of t to see this effect.
2. Certain ant species use pheromones as a signal for danger. In experiments
these ants were released in a long tube and stimulated one ant until it eleased a
pheromone. Use one-dimensional diffusion equation as a model for the spread
of the pheromones in the tube. Assume that at time t = 0 a signal of strength
α is released. The diffusion consant is D = 1. Other ants react to the stimulus
if the concentration they perceive is 10% of α and higher.
a) For each t > 0 find the region in the tube 0 ≤ x ≤ x(t) where the ants would
react to the stimulus (region of influence).
b) Sketch the time evolution of the boundary x(t)
c) Find the time t∗ such that the region of influence is empty for all t > t∗
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