DATE: October 2015
UNIVERSITY OF MANITOBA
TEST 1
COURSE: MATH 2140
PAGE: 1 of 1
EXAMINATION: Modelling
TIME: 60 minutes
EXAMINER: J. Arino
This is a 60 minute exam. Please show your work clearly.
No texts, notes, calculators, cellphones, translators or an
2.6 The logistic map
We can represent parameter-dependent behavior with a bifurcation
diagram, as shown below for the logistic map. The horizontal axis
is the value of r , and the vertical axis is the value (or values) that
solutions approach in the long
Reminders and announcements
Test 1 is tomorrow, Fri Jul 22, 11am - 12pm in this room. I
have office hours this afternoon if you have any questions.
The mid-semester break begins next week on Wed Jul 27.
There are no classes or tutorials from Wed Jul 27
Section 2: Modelling Populations
2.1 Setup
In the examples we have studied so far, we knew exactly what
the mechanism was for moving from one time step to the next
(e.g., the self-replicating robot constructs one copy per hour;
the bank pays interest at
2.2 The Beverton-Holt Model
Recall:
Nt
1
1 R0NaNt
t
, R0 , a 0, K
R0 a 1 .
Fixed points:
R0
1: two distinct fixed points, 0 and K 0. The fixed
point 0 is repelling, while the fixed point K is attracting. Any
solution with positive initial value conver
3.2 Linear Systems
Recall:
We have a system of the form
X
MX ;
whereM is a constant 2 2 matrix. The point 0; 0 is an
equilibrium point.
SupposeM has real eigenvalues 1; 2 R, with corresponding
eigenvectorsV1; V2 R2. Then the system has straight-line
traje
Problems Week 7 Tutorials
Problems 3 and 4 are due at the end of tutorial on Fri Aug 19.
1. Let P be a transition matrix.
(a) Suppose P has a column that consists entirely of zeroes. Explain why P cannot be
primitive.
(b) Suppose P k is positive for some
Problems Week 6 Tutorials
Problem 2 is due at the end of tutorial on Fri Aug 12. All of the problems are relevant review
questions for the second test.
1.
In this exercise, you will analyze a second-order ODE by converting it into a system of two
first-or
Class 1,2,3 - Section 4.1
1. Euclidean vector spaces
2. Properties of Euclidean vector spaces
MATH 2090 Page 1
3. General vector spaces
MATH 2090 Page 2
MATH 2090 Page 3
MATH 2090 Page 4
MATH 2090 Page 5
MATH 2090 Page 6
MATH 2090 Page 7
MATH 2090 Page 8
Final Exam Review Selected Solutions
1. (a)
The function corresponding to the given difference equation is
f pN q N erp1N qp2N q .
The fixed points are the values of N that satisfy f pN q N . We get the equation
N erp1N qp2N q
One solution is N
N.
0. If
2.3 Ordinary Differential Equations
Let
N1
f pN q
be a first-order autonomous ODE, where f is C 1 on R. Let a P R
be an equilibrium point.
If f 1 a
p q 0, then a is stable, and for all x near to a, if
N p0q x, then N pt q converges to a.
If f 1 a
p q 0
2.3 Ordinary Differential Equations
A difference equation treats systems that change in discrete time
steps (interest paid every month, a task completed every hour). For
many populations, a better model would treat time as continuous.
Let the size of the
Announcements/Reminders
Two problems for next weeks tutorials have been posted on
UM Learn, one of which is due on Fri Jul 22.
Before Monday, two sample tests from previous years will be
posted, as well as solutions to the week 1 part 2 and week 2
tutor
2.3 Ordinary Differential Equations
Consider the autonomous first-order ODE
N1
f p N q,
where f is C 1 on R. Recall that:
p q N0, there exists an interval
containing t on which the initial value problem has a unique
solution.
For any initial condition N
Reminders/Announcements
The first test of the semester will take place Fri Jul 22, 11am
12pm, in this room. Please arrive on time if possible.
The test will cover all material up to and including the lecture
on Fri Jul 15. The new definitions we introd
DATE: 13 October 2015
UNIVERSITY OF MANITOBA
TEST 1
COURSE: MATH 2140
PAGE: 1 of 1
EXAMINATION: Modelling
TIME: 60 minutes
EXAMINER: J. Arino
This is a 60 minute exam. Please show your work clearly.
No texts, notes, calculators, cellphones, translators or
3.1 Introduction
Recall our example system:
x1
y
1
y,
x.
The solutions are
X pt q pR cos t, R sin t q
for constant R, which describe circles with radius R and
center p0, 0q.
0 and y 0.
The equilibrium point is p0, 0q. It is stable because all nearby
T
Recall:
We are studying the logistic map
Pt
1
rPt p1 Pt q.
Specifically, we are examining how the behavior of the system
depends on the parameter r , where 0 r 4.
When 0
r 1, the only relevant fixed point is 0, and it is
attracting.
When 1
r 3, there
MATH 2140 Math Modelling
Contact Information
Instructor
Email
Office
Office hours
Dr. Jennifer Vaughan
jennifer.vaughan@umanitoba.ca
449 Machray Hall
TR, 1:00pm - 2:30pm, or by appointment
Course Policies and Overview
The course syllabus can be found on
Reminders and Announcements
Tutorial-week-4 and Tutorial-week-5 problems are due in
tutorial on Fri Aug 5.
The second test will be held in class on Fri Aug 12.
Test 2 will cover all material from Mon Jul 18 Fri Aug 5.
Tutorial-week-6 problems will be
2.4 Simple population models
Recall: a basic model for a population takes the form
N1
B pN q D pN q,
where B pN q is a positive function representing the birth rate, and
D pN q is a positive function representing the death rate.
Two simple possibilities
3.2 Linear Systems
After today, there are no classes or tutorials until Tue Aug 2.
Tutorial problems for week 5 (next week) will be posted this
afternoon. Those problems and the week 4 set are due Fri
Aug 5.
Example. Consider the system
x1
y
1
y,
4x 3y
DATE: December 2015
UNIVERSITY OF MANITOBA
TEST 1
COURSE: MATH 2140 FINAL EXAMINATION
PAGE: 1 of 2
EXAMINATION: Modelling
TIME: 180 minutes
EXAMINER: J. Arino
This is a 180 minute exam. Please show your work clearly.
No texts, notes, calculators, cellphon