Assignment 3
Math 345, Prof. Shi
Due: Wednesday , Sept 27 (11am)
1. Determining a clearance rate Consider a patient receiving a drug intravenously at a rate
of 10 mg/hour. An hour later the concentration of drug in the patients body is 1 mg/liter.
Assuming the patient has 5 liters of blood and the drug is lost at a rate proportional to
amount of drug in the body, (a) Find the clearance rate of the drug, and (b) determine the
limiting concentration of drug in the patients body.
2. Suppose that N(t) denotes the size of a population at time t. The population evolves according
to the logistic equation but, in addition, predation reduces the size of the population so that
the rate of change is given by dN
dt =N1−N
50−9N
5 + N.
(a) Find the equilibrium points of the equation.
(b) Determine the stability of the equilibrium points by using linearization or phase line.
(c) Sketch the phase line. If N(0) = 35, what is the limit of N(t) as t→ ∞?
3. Consider the logistic equation N0=rN 1−N
K(t)where the carrying capacity K(t) is
time-dependent. In each of the following cases, describe the long term behavior of N(t),
using Matlab program dfield.
(a) Assume K(t) = 1 −(1/2)e−t,r= 1. This might model a human population where,
due to technological improvement, the availability of resources is increasing with time,
although ultimately limited.
(b) Assume K(t) = 1 −(1/2) cos(2πt), r= 1. Here the carrying capacity is periodic in
time with period 1. This models, for example, a population of insects or small animals
affected by the seasons.
4. Consider a population model: dx
dt =kx 1−x
N−M, x(0) = x0,where k, M , N, x0are
positive parameters.
(a) In the following table, fill in the dimensions of all parameters in terms of the dimensions
of variables.
Variable Dimension Parameter Dimension
t τ k
x λ M
N
x0
(b) Use the change of variable:
y=x
N, s =kt.
Derive the new equation (including the initial condition) in the new variables yand s.
5. If the equation in problem 1 is now
dN
dt =N1−N
50−aN
5 + N,
where a > 0 is the predation rate. When a > a0, the population will become extinct no
matter how large the initial value is. Determine this a0.