MATH111: Applied Mathematical Modelling I
This page contains material for the course: MATH 111
Applied Mathematical Modelling I.
This course emphasises the physical, mathematical and computational
aspects of applied mathematics in science, engineering
and industry (including financial applications). The aim of the course is
to provide students with some fundamental mathematical modelling
skills that are useful in a range of scientific and commercial
activities. Students will learn how to model a realworld problem as
an idealised mathematical system, how to solve the representative
mathematical problem and how to interpret the results.
Student learning
 Unless you are an exceptional student you should expect
to find the course difficult and timeconsuming: there is not time for
all material
to be presented and reinforced in lectures.
 Students should not expect to grasp all material from the lecture
presentation. They should anticipate spending time outside the
lecture to attain the necessary level of understanding.
 The main feature distinguishing university from school is that students
are expected to learn on their own, outside the classroom.
 Ultimately, students are responsible for their own development, progress
and understanding.
 A student's failure to master something is a failure of the student, not
the educational system or the lecturer.
Your final mark in MATH111 will be determined
as follows (*). Four marks will be calculated using
scheme one (S1), scheme two (S2),
scheme three (S3) and
scheme four (S4).
Scheme  S1 
S2 
S3 
S4 
Final Exam 
60%  50%  50%  50% 
Midsession Exam 
20%  20%  20%  10% 
Computer assignments 
15%  25%  20%  30% 
Assignments 
5%  5%  10%  10% 
 The mark for computer assignments is based upon your best
four assignments.
 Your final result will be the higher of the marks calculated
using schemes one through four.
 (*) Attendance at lab sessions, lectures and
tutorial classes may be taken into account.
 Scaling of marks is
not a standard procedure in this subject.
Note that you are not required to ``pass'' each individual
component to receive a Pass grade in MATH111. However, you
would seriously jeopardise your chances of passing this
subject if you do not aim to be successful in every component
of the assessment.
PDF file showing your: midsession
test score (out of 40);
your maple lab score (out of 400); your
assignment mark (our of 5).
PDF file showing your mark on each
maple assignment (out of 100).
The student who receives the best mark in MATH111 is awarded the
Broadbridge Prize.
Year  Winner 
2005  Glen Macdonald 
2006  Andrew Holder & Oliver Towers 
2007  Scott Harris 
 Display the information sheet for
this course.
(Last updated 1st August 2007).
I will make my overhead transparencies available.
If you miss a lecture you can download these notes and fill in
most of the blanks in your handouts by copying from the
OHTs.
 Introduction, Definitions
and Basic Concepts
 Aims.
 Overview of Discrete Systems.
 Examples of Difference Equations
 Formal Definitions.
 Revision of key ideas.
 Concept Map.
 Questions.
 FirstOrder Linear
Difference Equations
 Aims.
 Example of a FirstOrder Difference Equation:
Carp Population in a Lake.
 Solving the FirstOrder Difference Equation: The Case
b(n)=0.
 Solving the FirstOrder Difference Equation: The Case
a=1.
 Solving the FirstOrder Difference Equation: The General Case.
 Revision of key ideas.
 Concept map.
 Questions.

Applications of FirstOrder Difference Equations: Finance
 Aims.
 Interest.
 Loan Repayments (Amortization Scheme).
 Annuities.
 Revisionn of Key Ideas.
 Concept map.
 Questions.
 First Order
NonLinear Difference Equations: An Introduction.
 Introduction.
 Aims.
 A linear model for population growth.
 The failure of the linear model: The need for nonlinear models.
 Cobwebbing: A graphical solution.
 Fixed points.
 The logistic equation.
 Cobwebing and fixed points: An example of putting them together.
(This is a new section to be included in the
2007 notes).
 Revision of key ideas.
 Concept map.
 Linear stability
analysis of fixedpoints.
 Aims.
 Physical motivation of `stability'.
 Linear stability analysis.
 Stability of the fixed points in the logistic equation.
 Fixed points, their stability and bifurcations.
 Revision of key ideas.
 Questions.
 Harvesting.
 Introduction.
 Fixed harvesting.
 Proportional harvest.
 Comparison of harvesting strategies.
 Advanced Topics.
 Revision of key ideas.
 Concept map.
 First Order Difference
Equations: Concluding Thoughts.
 Review of key ideas.
 Concept map.
 Differential Equations:
Introduction, definitions and basic concepts.
 Aims.
 Overview of Continuous Systems.
 What does dy/dx mean?
 Examples of Differential Equations.
 Formal Definitions.
 Some comments on modelling.
 Revision of key ideas.
 Concep map.
 FirstOrder Differential
Equations.
 Aims.
 Solving the firstorder differential equation: Integrable equations.
 Solving the firstorder differential equation: Separable equations.
 Solving the firstorder differential equation: Integrating Factors.
 Revision of key ideas.
 Concept map.
 Applications of firstorder
differential equations: Lake Pollution.
 Aims.
 The lake pollution model.
 Pollutant dumped into a clean lake: Fixed flowrate.
 Pollutant dumped into a clean lake: Seasonal flowrate.
 Pollutant flowing into a lake.
 A degradable pollutant flowing into a lake.
 Extensions to the basic model.
 Two lakes in series.
 Rivers flowing into lake one only.
 Rivers flowing into lakes one and two.
 Revision of key ideas.
 Concept map.
 First Order
NonLinear Differential Equations.
 Introduction.
 Aims.
 A linear model for population growth.
 The failure of the linear model: The need for nonlinear models.
 Firstorder Ordinary Differential Equations: Graphical Insights.
 Steadystate solutions.
 Revision of key ideas.
 Concept map.
 Linear Stability Analysis
of steadystate solutions.
 Physical motivation of `stability'.
 Linear stability analysis.
 Stability of the steadystates in the logistic equation.
 Stability of the steadystates in the Gompertz model.
 Steadystate solutions: Stability, bifurcations and
steadystate diagrams.
 Revision of key ideas.
 Concept map.
 Harvesting.
 Introduction.
 Aims.
 The logistic differential equation: A quick recap.
 Constant yield harvesting.
 Constant effort harvesting.
 Comparison of harvesting strategies.
 Revision of key ideas.
 Concept map.
 First Order Differential
Equations: Concluding Thoughts.
 Review of key ideas.
 Concept map.
 Concept maps.
 What is a concept map?
 A concept map of concept maps.
 Further Information.
 Sigma
Notation.
 What is Sigma notation?
 Questions.
 Geometric
Progression.
 What is a Geometric Progression?
 Questions.
 Taylor
series.
 Taylor series.
 Taylor series expansion.
In this section of the webpage I will try to summarise the main
ideas of each lecture.
 Week 7, Lecture 1 (Chapter 8).
 If you solve an ODE, then check that you solution is correct by:
 substituting it back into the ODE and showing that LHS=RHS.
 checking that is satisfies the initial conditions.
This is a good tip for other courses.
 Week 7, Lecture 2 (Chapter 9).
 Solving firstorder differential equations,
particularly a firstorder linear differential equation.
The latter will be frequently used.
 Week 7, Lecture 3 (Chapter 10).

 Lake pollution models are firstorder linear differential
equations which we can solve using an integrating factor!
 Concentration = Mass/Volume.
 Week 8, Lectures 1 & 2 (MidSession Test).
 Week 8, Lecture 3 (Chapter 10).
 Lake pollution models are firstorder linear differential
equations which we can solve using an integrating factor!
 It's important to be able to look at the solution of a differential
equation and determine the limit is t approaches infinity.
 Week 9, Lecture 1 & 2 (Chapter 11).
 Consider the differential equation
dx/dt =f(x).
Everything you want to know about the solution of this equation,
including how the longterm behaviour depends upon the initial
conditions, can be determined from the graph
y=f(x)
 Steadystate solutions are values of x for which
dx/dt = f(x) =0.
Finding steadystate solutions are an important took in
understanding the behaviour of the differential equation.
 Week 9, Lecture 3 (Review of midsession test).
 Week 10, Lecture 1 (Chapter 12).
 Let x^{*} be a steadystate of the differential
equation
dx/dt = f(x)
Then its eigenvalue is given by
λ = f'(x^{*}).
 Week 10, Lecture 2 (Chapter 12).
 A steady state solution of the differential equation
dx/dt = f(x)
is stable if its eigenvalue is negative.
It is unstable if its eigenvalue is positive.
If the eigenvalue is zero it may be either stable or unstable.
 The stability of a steadystate solution of a differential equation
dx/dt = f(x)
can be determined in two ways.
 Algebraically by determining
the sign of the eigenvalue.
 Graphically by drawing the
function y=f(x) and looking at the sign of the
function f(x) on either side of the steadystate
solution.
 Week 10, Lecture 3 (Chapter 12).
 You need to know how to draw a steadystate diagram for the
differential equation
dx/dt = f(x,λ),
remembering that stable and unstable steadystate solutions are
indicated by solid and dashed lines respectively.
 There are three kinds of bifurcation points: limitpoint,
transcritical and pitchfork.
 Week 11, Lecture 1 (Chapter 13).
 If you can draw the graph
y=f(x)
and understand how to obtain the graph
y=f(x)H, H>0
by translating the first graph down H units then you
already understand constant yield harvesting!
 Week 11, Lecture 2 (Chapter 13).
 For the logistic equation we fixed harvesting the maximum
sustainable yield is
H_{cr} = rK/4. This corresponds to a
limit point bifurcation on the steadystate diagram.
 Week 11, Lecture 2 (Chapter 13).
Constant effort harvesting.
 Consider the differential equation with constant effort harvesting
dx/dt = f(x) Ex.
The yield function is given by
Y = E*x(E)
where x(E) represents the steadystate solution(s) as a
function of the effort.
 Find the value of E that maximise the yield and
hence the maximum yield.
 Week One: MAPLE Crash
Course

 Basics
 Graphics
 Differentiation
 Integration
 Substitution
 Week Two: Iterating a simple
difference equation
 Population of carp in a lake.
 Week Three: Marked
Assignment.
 Investigating the discrete Ricker model
 The article that you are required to read to complete this
assignment is available as an electronic reading
in the library.
 Week Five: Marked
assignment.
 Bifurcations in firstorder difference equations.
.
 Download the
maple bifurcation code.
 Download the
maple Ricker code.
 The article that you are required to read to complete this
assignment is available as an electronic reading in the
library.
 Week Seven: Marked
assignment.
 Fixed harvesting in the Ricker model.
 Week Nine: Marked
assignment.
 Solving firstorder autonomous differential equations in Maple.
.
 Download the
spruce budworm code.
 The article that you are required to read to complete this
assignment is available as an electronic reading in the
library.
 Week Eleven: Marked
assignment.
 SIR Endemics with and without vaccination.
 Download the maple code.
Note that the chapter number is a guideline.
What was chapter 11 in 2004 may not be the same as chapter 11 this year!
 The midsession test contributes either 20%
(S1S3) or 10%
(S4) of your final mark for this
course.
 Your mark on the midsession test will be divided by two (four)
to give you a final mark out of twenty (ten).
 Appendix A: Concept maps
 There are many web pages on concept maps. One that I find
particularly useful is
www.coun.uvic.ca/learn/program/hndouts/map_ho.html.
If you are reading for a degree in biotechnology (or related areas)
you might find the following article of interest.
James E. Bailey. 1998. Mathematical Modeling and Analysis in Biochemical
Engineering: Past Accomplishments and Future Opportunities.
Biotechnology Progress, 14, 820.
You can access this article via the library catalogue under
ereading/short loans for MATH111.
<< Return to my start page.
<< Return to my teaching home page.
Page Created: 27th June 2003.
Last Updated: 1st November 2008.