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 rec
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 car
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 , Lect
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
Modeling and Differential Equations for the Life Sciences
MATH 19a

Spring 2013
Math 19a
Name
.
Modeling and Differential Equations for the Life Sciences
Exam I Fall 2014
Guidelines for the test:
No books or calculators are allowed, but you may use one 4 6 index card (front and
Modeling and Differential Equations for the Life Sciences
MATH 19a

Spring 2013
Chapter 5
Modeling Interacting
Populations
During the first week of this course, we learned about the Malthus (exponential) and logistic population models for a single species for which interactions
w
Modeling and Differential Equations for the Life Sciences
MATH 19a

Spring 2013
Math 19a. Lecture 5
Qualitative Analysis, continued
Discussion of Reading 3
Eulers Method
John Hall
February 6, 2013
1
The Effects of Harvesting, continued
On Monday we considered a population of fish
Modeling and Differential Equations for the Life Sciences
MATH 19a

Spring 2013
Modeling and Differential Equations for the Life Sciences
First Practice Exam I SolutionsFall 2013
4. Suppose the population P = P (t) of deer in a forest is modeled by the logistic equation
dP
= 0.5P
Modeling and Differential Equations for the Life Sciences
MATH 19a

Spring 2013
Chapter 7
Basics of Vectors and
Matrices
7.1
Motivation
When we sketched phase plane diagrams or direction fields for planar systems of ODEs, we drew vectors to indicate what the flow would look like.
Modeling and Differential Equations for the Life Sciences
MATH 19a

Spring 2013
Chapter 9
Eigenvalues and Eigenvectors
9.1
Recap
Last time we explained that in order to understand the stability of equilibria
of planar systems, we will need to understand the behavior of constantco
Modeling and Differential Equations for the Life Sciences
MATH 19a

Spring 2013
Math 19a. Lecture 6
Eulers Method
FirstOrder Systems
Eulers Method for Systems
John Hall
February 8, 2013
1
Eulers Method
Many important differential equations cannot be solved analytically. Let us
s
Modeling and Differential Equations for the Life Sciences
MATH 19a

Spring 2013
Part IV. Linear Differential Equations
Section 3. Two Dimensional Systems
A second order differential equation in normal form
y' = f( t, y, y' )
can always be converted into an equivalent system of tw
Modeling and Differential Equations for the Life Sciences
MATH 19a

Spring 2013
Chapter 14
Limit Cycles and Hopf
Bifurcation
14.1
Academic Example Revisited
Last time we gave the following purely academic example of a previouslyunencountered type of bifurcation:
dx
= x y
dt
dy
=
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 car
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 ca
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 ans
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 sp
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
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 diere
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 substitut
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
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
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
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
(plen
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, 1
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 rea
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 inclu
Modeling and Differential Equations for the Life Sciences
MATH 19a

Spring 2013
Math19a Problem Session
Part I
Ethan Addicott
February 5, 2014
This guide covers Lectures 14
Outline This guide will cover the mathematical techniques covered in lectures 14.
Definitions and Abbrevi