BIO 5910: Homework Assignment 2
Due on September 12, 2013
1. Our model of the concentration of medication in the bloodstream,
Mt+1 = 0.5Mt + 1.0
can be generalized to
Mt+1 = ρMt + S.
a. What do ρ and S mean?
b. Find the equilibrium.
c. Graph the equilibrium as a function of S, and explain the result.
d. Graph the equilibrium as a function of ρ, and explain the result.
e. Cobweb a representative case. Can the equilibrium be unstable?
2. The model considered in the previous problem becomes nonlinear if the fraction of medication used by the body is a function of the concentration. Suppose that the fraction used is a
decreasing function of the concentration. Set S = 1.0.
a. Write the basic updating function as
new concentration = old concentration − fraction used × old concentration + supplement
and figure out what each term is.
b. Consider the expression
fraction used =
0.5
.
1.0 + 0.1Mt
Is this fraction larger or smaller than the basic value of 0.5? What happens to the fraction
used as Mt becomes large? What does this mean in biological terms?
c. Substitute this expression to write the updating function.
d. Solve for the equilibrium. Can you explain why it is larger?
e. Cobweb starting from M0 = 1.0.
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