Math 5110/6830
Homework 3.1
1. Consider a discrete-time model of the concentration of medication in the
bloodstream, Mt is the concentration on day t,
Mt+1 = Mt − g(Mt )Mt + S
where g(MT ) is the fraction absorbed and S is the daily dosage. Suppose
that
Mp
g(M ) = p
.
K + Mp
Set K = 2 and S = 1.
a) Show analytically that M ∗ = 2 is the equilibrium point for any value
of p.
b) Study analytically the stability of the equilibrium for different values
of p: when is it stable? unstable? when does the solution oscillate toward
the equilibrium?
c) Sketch a cobweb diagram of a case when M ∗ = 2 is unstable.
2. Let an represent the whale population after n years. Consider the model
an+1 = an + k(M − an )(an − m),
where k > 0. Assume that M = 5000, m = 100, and k = 0.0001.
a) Find the fixed points of the model, and determine their stability via
linearization.
b Perform cobwebbing for a0 = 50 and a0 = 200. What will happen to
the population of whales in each of these cases? Describe in words and
also sketch the graphs of an versus n.
c) (MATLAB) Compute numerically and plot on the same graph the
dynamics of the whale population over 20 years starting with a0 equal 20,
60, 100, 140, 180, 220, 260, . . . up to 1060. What do you observe?
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