Modeling and Differential Equations for the Life Sciences
MATH 19a

Spring 2013
Math 19a
Name
Modeling and Dierential Equations for the Life Sciences
First Practice Exam ISpring 2013
Guidelines for the test:
No books or calculators are allowed, but you may use one 4 6 index card (front and
back) of notes.
You may leave answers in s
Modeling and Differential Equations for the Life Sciences
MATH 19a

Spring 2013
Math 19a. Modeling and Dierential
Equations for the Life Sciences
Review Guide for Midterm II
John Hall
Harvard University
Spring 2013
Midterm Details
The second midterm will cover Lectures 141 , Lectures 1633, Problem Sets 59, and Readings
1020. Calculat
Modeling and Differential Equations for the Life Sciences
MATH 19a

Spring 2013
Math 19a
Name
Modeling and Dierential Equations for the Life Sciences
First Practice Exam IISpring 2013
Guidelines for the test:
No books or calculators are allowed, but you may use one 4 6 index card (front and
back) of notes.
You may leave answers in
Modeling and Differential Equations for the Life Sciences
MATH 19a

Spring 2013
Math 19a
Name
Modeling and Dierential Equations for the Life Sciences
Second Practice Exam ISpring 2013
Guidelines for the test:
No books or calculators are allowed, but you may use one 4 6 index card (front and
back) of notes.
You may leave answers in
Modeling and Differential Equations for the Life Sciences
MATH 19a

Spring 2013
Math 19a
Name
Modeling and Dierential Equations for the Life Sciences
Second Practice Exam IISpring 2013
Guidelines for the test:
No books or calculators are allowed, but you may use one 4 6 index card (front and
back) of notes.
You may leave answers in
Modeling and Differential Equations for the Life Sciences
MATH 19a

Spring 2013
Math 19a. Modeling and Dierential
Equations for the Life Sciences
Review Guide for Midterm I
John Hall
Harvard University
Spring 2013
Midterm Details
The rst midterm will cover Lectures 113 (not including Periodic Solutions), Problem
Sets 04, and Readings
Modeling and Differential Equations for the Life Sciences
MATH 19a

Spring 2013
Math 19a  Problem Set 2
due Friday, February 15
1. It is a fact that unstable equilibrium solutions are rarely observed directly in nature. Write an explanation of this fact that would convince
a reasonable member of the general public. (Your explanation
Modeling and Differential Equations for the Life Sciences
MATH 19a

Spring 2013
Modeling and Dierential Equations for the Life Sciences
Second Practice Exam I SolutionsSpring 2013
4. Suppose that a population model
dP
= f (P )
dt
has precisely four equilibrium points, at P = 0, 10, 50, and 60. Suppose further that
A population close
Modeling and Differential Equations for the Life Sciences
MATH 19a

Spring 2013
Math 19a  Problem Set 1
due Friday, February 8
1. For a certain microorganism, birth is by budding o a fully formed copy
of itself. Suppose that under reasonably favorable laboratory conditions
(plenty of food and no predation), such births occur on aver
Modeling and Differential Equations for the Life Sciences
MATH 19a

Spring 2013
Modeling and Dierential Equations for the Life Sciences
First Practice Exam I SolutionsSpring 2013
4. Suppose the population P = P (t) of deer in a forest is modeled by the logistic equation
dP
= 0.5P (4 P ) ,
dt
(1)
where the units of P are thousands of
Modeling and Differential Equations for the Life Sciences
MATH 19a

Spring 2013
Math 19a  Problem Set 4
due Friday, March 1, 2013
1. (The following problem is inspired by a Dartmouth study of tumor
growth in mice.1 ) Let x denote the volume of proliferating cells in a
tumor, and let y denote the volume of quiescent cells (also known
Modeling and Differential Equations for the Life Sciences
MATH 19a

Spring 2013
Math 19a  Problem Set 10
due Friday, May 3, 2013
1. (5 points) In this problem you will use Fourier Series to prove that the
equilibrium solution ue (x) = 0 to the lobster model
u
2u
= 2 +ru
t
x
ux (t, 0) = ux (t, R) = 0
is stable for R <
/r and unstab
Modeling and Differential Equations for the Life Sciences
MATH 19a

Spring 2013
Math 19a  Problem Set 8
due Friday, April 12
1. Consider the linear reactiondiusion model
u
2u
= 2 + ru
t
x
u(t, 0) = u(t, R) = 0,
in the case r < 0. By making the Separation of Variables substitution u(t, x) = A(t)B(x), show that the equilibrium solut
Modeling and Differential Equations for the Life Sciences
MATH 19a

Spring 2013
Math 19a  Problem Set 9
due Friday, April 12
1. For each of the following dierential equations, make the traveling wave
substitution u(t, x) = f (xct), with c > 0 being a constant, and derive
a dierential equation in one variable for the function f . (Le
Modeling and Differential Equations for the Life Sciences
MATH 19a

Spring 2013
Math 19a  Problem Set 7
due Friday, April 5
1. Recall the notrawling zone model from Lectures 22.
ut = uxx + r u
u(t, 0) = u(t, R) = 0
We showed that the lobster population grows with time provided that
R>
/r.
(a) Give an intuitive explanation (in terms
Modeling and Differential Equations for the Life Sciences
MATH 19a

Spring 2013
Math 19a  Problem Set 5
due Friday, March 15, 2013
1. Recall the FitzHughNagumo model from Lecture 7. This model is a
twodimensional simplication of the fourdimensional HodgkinHuxley
model1 of spike generation in squid giant axons. The FitzHughNagum
Modeling and Differential Equations for the Life Sciences
MATH 19a

Spring 2013
Math 19a  Problem Set 6
due Friday, March 29, 2013
1. For each of the following cases, indicate whether modeling with an
advection equation or a diusion equation is more appropriate. Explain
your answers.
(a) The concentration of snake venom in the blood
Modeling and Differential Equations for the Life Sciences
MATH 19a

Spring 2013
Math 19a  Problem Set 3
due Friday, February 22
1. The simplest possible epidemic model is an SI model, in which all
individuals are either susceptible or infected, and infected individuals
never recover. Let S be the number of susceptible individuals, a