Mathematical Methods for Engineers 2 (MATH1064) Leslie matrix Matlab group project Due no later than 2 pm on Friday 10th October, 2014 Graduate Qualities: This project is designed to help the student achieve course objective 4: solve simple applied problems using software such as Matlab , and to develop Graduate Qualities 1 & 3, namely operating effectively with and upon a body of knowledge, and effective problem solving. Assessment: The assessment will take into account all of your documentation of the mathematical analysis of the problem, your Matlab m-file(s), your Matlab output, the correctness of the final solutions and the presentation of your whole report. Groups should contain two or three people. It will be assumed that each member of the team contributed equally and will be awarded individually the mark allocated to the report. If this is not the case, then a lesser percentage for one or more members must be agreed by the team and clearly indicated. This especially will apply to absences from the practical class or non-attendance at agreed team meetings. The University policy on plagiarism will apply between different groups. Students who wish to can submit a peer assessment form which can be found on the course webpage. How to divide the work: Each team member must participate in all aspects of the project: math- ematical calculations, Matlab work and report writing. Only one copy of your project report is required for each group. Summary: In this project you will: • Investigate the Leslie matrix model for a population • Explain how a Leslie matrix can be used to calculate the population in each age class from time to time • Use Matlab to draw plots of age class populations evolving over time • Use Matlab to study the long term behaviour of population numbers Your report must be typed, and submitted through LearnOnline by one member of your group. It should include: • Written worked answers to all questions where this is required. • Appropriately labeled figures where required. • A listing of your Matlab script file should be included at the end of your report in an appendix. • A coversheet is not needed but your report must have a title page that lists the names and student identification numbers of all members of the group. • The group’s .m file must be submitted as a saparate file via LearnOnline. Be sure to list all group members at the top of the file; only one copy per group is required. There will be marks awarded for submitting this file, so don’t forget. Your .m file may be run and checked during the marking process. 1 Leslie Matrix Model Invented by Patrick H. Leslie in the 1940s, the Leslie Matrix is a mathematical model of population growth for a species. Time is divided into discrete periods, with individual memebers of the population progressing through discrete age classes at given survival rates. Here is a simplified example: The Central Australian Budgericoot (CAB) cannot live beyond five y ...

1. Mathematical Methods for Engineers 2 (MATH1064)
Leslie matrix Matlab group project
Due no later than 2 pm on Friday 10th October, 2014
Graduate Qualities: This project is designed to help the student
achieve course objective 4: solve
simple applied problems using software such as Matlab , and to
develop Graduate Qualities 1 & 3,
namely operating effectively with and upon a body of
knowledge, and effective problem solving.
Assessment:
The assessment will take into account all of your documentation
of the mathematical analysis of the
problem, your Matlab m-file(s), your Matlab output, the
correctness of the final solutions and the
presentation of your whole report.
Groups should contain two or three people. It will be assumed
that each member of the team
contributed equally and will be awarded individually the mark
allocated to the report. If this is
not the case, then a lesser percentage for one or more members
must be agreed by the team and
clearly indicated. This especially will apply to absences from
the practical class or non-attendance
at agreed team meetings. The University policy on plagiarism
will apply between different groups.
Students who wish to can submit a peer assessment form which

2. can be found on the
course webpage.
How to divide the work: Each team member must participate in
all aspects of the project: math-
ematical calculations, Matlab work and report writing.
Only one copy of your project report is required for each group.
Summary: In this project you will:
• Investigate the Leslie matrix model for a population
• Explain how a Leslie matrix can be used to calculate the
population in each age class from time
to time
• Use Matlab to draw plots of age class populations evolving
over time
• Use Matlab to study the long term behaviour of population
numbers
Your report must be typed, and submitted through LearnOnline
by one member of your
group. It should include:
• Written worked answers to all questions where this is
required.
• Appropriately labeled figures where required.
• A listing of your Matlab script file should be included at the
end of your report in an appendix.
• A coversheet is not needed but your report must have a title

3. page that lists the names and
student identification numbers of all members of the group.
• The group’s .m file must be submitted as a saparate file via
LearnOnline. Be sure to list all
group members at the top of the file; only one copy per group is
required. There will be marks
awarded for submitting this file, so don’t forget. Your .m file
may be run and checked during
the marking process.
1
Leslie Matrix Model
Invented by Patrick H. Leslie in the 1940s, the Leslie Matrix is
a mathematical model of population
growth for a species. Time is divided into discrete periods, with
individual memebers of the population
progressing through discrete age classes at given survival rates.
Here is a simplified example:
The Central Australian Budgericoot (CAB) cannot live beyond
five years of age. We shall discretise
time into years, and we shall count the number of CABs in each
of the 5 possible age classes at the
start of each year.
Only 80% survive the first year after birth. Of those that survive
the first year, only 50% live for
another year and enter their third year of life. Of those, 40%
survive the next year, and of those, 25%
make it into their fifth year, but none ever reach their fifth
birthday at the end of that year.

4. (This means that only 0.80× 0.50× 0.40× 0.25 = 0.04 = 4%, on
average, live beyond four years.)
This gives a survival rates array
p = [0.80 0.50 0.40 0.25 0.00] = [p1 p2 p3 p4 p5].
The sustainability of a species depends not only upon survival
rates, but also on the birthrate of
females of the species. Suppose that female CABs cannot give
birth during their first two years or
during their fifth year, should they live that long. On average, a
female of this species in her third
year can produce 2 female offspring, while in her fourth year on
average a female produces 1.5 female
offspring. This gives an average fertility array
f = [0 0 2 1.5 0] = [f1 f2 f3 f4 f5].
This information may be assembled into the Leslie matrix for
this species:
A =
0 0 2 1.5 0
0.8 0 0 0 0
0 0.5 0 0 0
0 0 0.4 0 0
0 0 0 0.25 0
The matrix is all zeros except for the fertility vector across the
top row, and the survival rates situated

5. below the leading diagonal.
Suppose that an investigator starts monitoring the female CABs
in a particular area in a given year.
At that time, there are 100 females between birth and their first
birthday, another 100 females in
their second year of life, another 40 females in their third year,
40 females in their fourth year, and
20 females that have survived until their last possible year.
Thus the total female population is 300.
The female population can be represented by the initial age
population vector y(in year k = 1 of the
investigation) whose elements are the numbers of females in
each of the age classes (in this case five
age classes).
y(1) =
100
100
40
40
20
In general, the population in age class j in the time period k + 1
is given by
yj(k + 1) = pj−1yj−1(k), where j = 2, 3, . . . , n,
since the population in age class j comes from survivors of the
population in age class j − 1 in the
previous time period k. The population in the age class 1
(newborns) in time period k + 1 should

6. 2
be considered separately, since it arises only by birth from the
population of other age classes in the
previous time period. In the case of n age classes,
y1(k + 1) = f1y1(k) + f2y2(k) + . . .+ fnyn(k),
where f is the average fertility array as in the example above. If
we define the population vector at
time period k as
y(k) =
y1(k)
y2(k)
...
yn(k)
then the above equations defining population at time period k +
1 can be written as
y(k + 1) = Ay(k),
where matrix A is the Leslie Matrix defined in the example
above.
Typically, in a Leslie model, the numbers in the various age
classes, and the total population itself,
exhibit fluctuations until transient effects disappear. After a

7. sufficiently long time, the population
changes at a rate r. If r > 1, the population eventually increases.
If r < 1, the population eventually
decreases and the species is in danger of extinction. After many
time periods, once the initial transient
effects are no longer relevant, it can be shown that the
proportion of the total population in each of
the age classes tends to become constant, irrespective of
whether the population is increasing or
decreasing.
The Project
1. Without using a computer, calculate y(2) = Ay(1) and y(3) =
Ay(2). By referring to the
individual steps in these matrix multiplications, explain
carefully what these represent.
What is the expected total female population in year 2, and in
year 3?
What is meant by y(k) = Ak−1y(1)?
2. The first Matlab section of the project will require one long
script M-file leslie.m.
(a) In leslie.m, enter p, f , and y(1), using y1 to represent y(1).
Find the length n of f (which
is the number of age classes in the female population under
review), and then introduce an
n× n matrix A of zeros. Find simple ways of inserting the
elements of p and f to get the
correct Leslie matrix A. Check your answers for y(2) and y(3).
(b) Add the following statements, and fill in a suitable legend:

8. y=y1;
M=[y];
for i=1:29
y=A*y;
M=[M,y];
end
plot(M’)
hold on
legend( )
xlabel(’time in years’)
ylabel(’number predicted in each age group’)
title(’predicted populations in the five age groups over 30
years’)
hold off
3
In your report, you must include a section that explains exactly
what the for loop accom-
plishes, and how this different use of the plot command works.
(c) For year 30 (29 years after the study begins), use the Matlab

9. sum function to find the
expected total size of the female population, as well as the
relative distribution within the
n age groups (i.e. the proportion in each age group).
(d) Draw a plot (as above) of the populations within the age
groups for years 1 to 100 and
comment.
3. (a) Show that y(k) = xi(λi)
k−1 is a solution of y(k + 1) = Ay(k), where λi and xi are the i
th
eigenvalue and a corresponding eigenvector, respectively, of the
matrix A.
(b) Assume that the real eigenvalues are ranked in the order λ1
> λ2 > · · · > λm. In this case
λ1 is called the dominant eigenvalue of the matrix A. When the
time period k becomes
sufficiently large, that is, when k → ∞, the behaviour of the
population vector y(k) is
determined by the term with the dominant eigenvalue λ1, while
the other terms are much
smaller. It can be shown that
y(k) ∼ C1x1λk−11 when k →∞
or in other words
lim
k→∞
1

10. C1λ
k−1
1
y(k) = x1.
Use Matlab to find y(100) and y(101), and calculate
yj(101)
yj(100)
for j = 1, 2, 3, 4, 5. Compare
these five ratios with each other and with λ1 for matrix A.
Find the the proportions of the total population for each of the n
age classes for k = 100
and k = 101 and compare.
What do you think is the value of r for this population?
4. In the fertility rate array f , suppose that f4 is changed from
1.5 to 1.0. Repeat 2 parts (b)-(d)
above, with suitable comments. Find the value of r in this case.
4
5. The aphid midge, Aphidoletes aphidimiza, is a type of small
fly whose larvae are very effective
predators of aphids, and are actually used for biological control
of aphids. You can read about
the important aphid midge at the Cornell University Biological
Control site:
http://www.biocontrol.entomology.cornell.edu/predators/Aphido

11. letes.php
The midge can be introduced to help control aphid infestations.
It is an alternative to the use
of poison sprays. For this exercise, assume the aphid midge
does not survive more than 23 days
from the laying of its egg, and that the life table of the aphid
midge is the following:
Age Stage Numbers Eggs laid
(days) per female
0 egg 1000 0
1 egg 1000 0
2 larva 700 0
3 larva 650 0
4 larva 600 0
5 larva 550 0
6 larva 530 0
7 cocoon 400 0
8 cocoon 400 0
9 cocoon 400 0
10 cocoon 400 0
11 cocoon 400 0
12 cocoon 400 0
13 adult midge 350 5
14 adult midge 340 15
15 adult midge 330 20
16 adult midge 300 20
17 adult midge 260 15
18 adult midge 200 12
19 adult midge 120 8
20 adult midge 40 5
21 adult midge 20 2
22 adult midge 5 0

12. Hence, there are 23 age classes, and so the Leslie matrix A will
be 23 × 23. Because it is
impossible to differentiate the sexes at each stage, you should
assume the population is equally
divided between males and females, which is in any case quite
accurate. In this exercise, you
will be monitoring the size of the whole population, not just
females. Hence, in finding f and
A, you will need to consider that only half the population is
laying the given number of eggs per
individual.
Suppose that an aphid outbreak has occurred, and that 100
midge cocoons, at the 10 day stage,
are introduced to the area on day 1 of the trial. Make
adjustments to your previous code to
accomplish the following:
(a) Create appropriate p, f and y(1) arrays, and then generate the
Leslie matrix. Run the trial
for a further twenty one days, to generate a 23× 22 matrix M .
(b) Create a plot (similar to that in Question 2), with four
graphs on it, representing the total
numbers of eggs, larvae, cocoons and adult midges over the 22
days of the trial. This should
be accomplished by using the Matlab sum command on sub-
matrices of M to produce a
new 4× 22 matrix, prior to plotting. Make sure you include a
suitable legend.
(c) Suppose the trial continues for years. Find the expected
proportions in each of the 23 age
classes on days 900 and 950. Find the value of r for aphid
midges and eigenvalues of the

13. corresponding Leslie matrix, and comment.
5