ENVR 325: Fundamentals of Ecology
Homework Exercise #1: Population
Dynamics of Mountain Goats on
Mountain Goats (Oeramnos americanus) are found in mountainous terrain
throughout the northwestern portions of
Native populations of
Mountain Goats in
TOOLS: Although you
could complete this homework assignment using nothing more than a pencil and
paper, I would STRONGLY recommend that you use Microsoft Excel. This piece of software will be extremely useful
to you in many of your courses.
Computational tasks that would take hours or days to complete with a
pencil and paper can be completed in minutes.
If you do not already know how to use Excel, now is a good time to
learn. I am assuming that you already
know how to use Excel. If you do not,
ask a friend for help or go to the Student
Technology Center. The folks there
should be able to get you started.
LIFE TABLES: In class, we discussed the use of life
tables. Your text also provides a
detailed discussion of life tables. Here
is a life table for female Mountain Goats:

Copy this table and
paste it into an Excel workbook. Or to make
it even easier for you, you can start with this file: goat_pop_template.xls. Using the formula presented in class and in
your text, calculate the following parameters from these data:
R_{o }– the Net Reproductive rate
T – the Average Generation time
r – the intrinsic rate of increase for the population (also
known as the Malthusian parameter)
As you will see,
completing these calculations in Excel is very straightforward.
POPULATION
PROJECTIONS FOR THE
POPULATION PROJECTION
USING A LESLIE MATRIX: The computational approach taken above assumes that the
population has reached a stable age distribution. This may or may not be a good
assumption. The use of a Leslie Matrix
allows you to evaluate this assumption.
Read about the Leslie Matrix by going to this web site and read sections
7.1 and 7.2:
The other sections on
this site (sections 7.37.5) are interesting but are not critical for
completing the rest of this exercise. In section 7.2, grab a copy of the Excel file, “leslie.xls.” Use this file as a model for creating your
own Leslie matrix in Excel. Here is the
transition matrix for the Mountain Goat:

kid 
yearling 
2 
3 
4 
5 
6 
7 
8 
9 
10 
11 
12 
13 
Kid 
0 
0 
0 
0.323 
0.368 
0.368 
0.368 
0.368 
0.368 
0.368 
0.288 
0.288 
0.288 
0.288 
yearling 
0.619 
0 
0 
0 
0 
0 
0 
0 
0 
0 
0 
0 
0 
0 
2 
0 
0.718 
0 
0 
0 
0 
0 
0 
0 
0 
0 
0 
0 
0 
3 
0 
0 
0.945 
0 
0 
0 
0 
0 
0 
0 
0 
0 
0 
0 
4 
0 
0 
0 
0.945 
0 
0 
0 
0 
0 
0 
0 
0 
0 
0 
5 
0 
0 
0 
0 
0.945 
0 
0 
0 
0 
0 
0 
0 
0 
0 
6 
0 
0 
0 
0 
0 
0.945 
0 
0 
0 
0 
0 
0 
0 
0 
7 
0 
0 
0 
0 
0 
0 
0.945 
0 
0 
0 
0 
0 
0 
0 
8 
0 
0 
0 
0 
0 
0 
0 
0.945 
0 
0 
0 
0 
0 
0 
9 
0 
0 
0 
0 
0 
0 
0 
0 
0.945 
0 
0 
0 
0 
0 
10 
0 
0 
0 
0 
0 
0 
0 
0 
0 
0.68 
0 
0 
0 
0 
11 
0 
0 
0 
0 
0 
0 
0 
0 
0 
0 
0.68 
0 
0 
0 
12 
0 
0 
0 
0 
0 
0 
0 
0 
0 
0 
0 
0.68 
0 
0 
13 
0 
0 
0 
0 
0 
0 
0 
0 
0 
0 
0 
0 
0.68 
0 
This table is another
way of presenting data on agespecific mortality and natality. The column and row labels are that age of the
Mountain Goat in years. A kid represents
an age of zero; this is a newborn animal.
A yearling is one year old, etc..
This is a transition matrix. Each
cell in the table represents the probability of making the transition from one
age class to another during any given year.
As explained in section 7.1 of the Model of Leslie, the first
row in this table indicates the probability that a female of any given age will
produce a female offspring. Note that
goats do not reach sexual maturity until their third year. The other rows (below the first row) present
the probability that an animal of a given age will survive for another
year. For example, the probability that
a kid will survive to become a yearling is 0.619. The probability that a yearling will survive
to become a twoyear old is 0.718. Doing
population projections using a leslie matrix involves
something called “matrix multiplication.”
We will multiply our leslie matrix (above) by
a vector of data that represents the number of individuals in each age class in
a particular year. We will start with
the 1960 Mt. Baker Mountain Goat population.
The vector of data that represents this population is:

Population
Size (N females) 

Year 

1960 
kid 
70 
yearling 
25 
2 
20 
3 
10 
4 
10 
5 
10 
6 
10 
7 
10 
8 
10 
9 
5 
10 
5 
11 
5 
12 
5 
13 
5 
To calculate the number
of yearlings in 1961, we multiply the number of kids in 1960 (70) by the
probability of a kid surviving to be a yearling (0.619). So 70*0.619=43.3. To predict the number of 2 year olds in 1961,
we multiply the number of yearlings in 1960 (25), by the probability of a
yearling surviving to be a 2 year old (0.718). So, 25*0.718=17.95. The same logic is used to calculate the 1961
population of 3, 4, ….13 year olds.
Predicting the number of kids in 1961 is a bit more complicated. Doing so involves calculating the number of
female kids produced by 3year old females (10*0.323) PLUS the number of female
kids produced by 4year old females (10*.368) PLUS ….and so on up to the female
kids produced by 13year old females.
To calculate the age
class distribution for 1962, you multiply the transition matrix by the vector
of data that represents the age class distribution for 1961. The same idea applies for 19632000.
As you can see, you
could write formula in Excel to do all of these calculations, BUT THERE IS AN EASIER WAY!
Fortunately, Excel
has a function that will do the matrix multiplication for you. This is the “MMULT” function. The use of this function is explained in the
leslie.xls file that I mentioned above.
Getting this function to work properly takes some fiddling and you must
follow the instructions (on the leslie.xls spreadsheet available in section 7.2
in the Model of Leslie ) very, very carefully.
To make your life a bit easier, I’ve entered the MMULT
function to do the calculation for 1961 in the
goat_pop_template.xls
spreadsheet. To do the calculations for
19622000. You simply need to highlight the cells for 1961 (cells C31:C44) and
copy these to cover the full 19622000 range.
Note that in your
population size and % of population tables will have one column for each year
from 1960 through 2000. I have entered
formula to do virtually all of the calculations for you.
How does your Leslie Matrix estimate of the population in
2000 compare with the population estimate derived from the life table approach
used above? Why do these estimates
differ? From
the Leslie Matrix, when does this population reach a stable age class
distribution? For the purposes of
this exercise, you may conclude that a stable age class distribution has been
reached when the percentage of the population in each age class remains
unchanged when rounded off to the nearest full percentage point. Note that section 7.4 in the Model of Leslie web pages
describes a way to calculate this age class distribution based only on the
transition matrix. You DO NOT need to
use the approach described in section 7.4.
Simply projecting the population forward from 1960 to 2000 will enable
you to determine when you have reach a stable age class distribution by
inspection.
REALWORLD DATA VS.
MODEL PREDICTIONS: Recall that in the introduction
I indicated that our survey data suggests that the Mt. Baker Mountain Goat
population has declined from about 400 animals in 1960 to about 100200 animals
in the year 2000. Assuming that 50% of
the total population is female, this means that our field surveys suggest that
there were only about 50100 females on Mt. Baker in the year 2000. How does this compare
with the prediction that you made for the female Mountain Goat population
during this time? If you did your
calculations correctly, your Leslie Matrix should indicate an increase in the
population during this time.
We don’t really know
why the Mt. Baker population has declined over this time period. We suspect that our survival data are
incorrect. Play around with the survival
data a bit while leaving the natality data unchanged. To do so, I have included another worksheet
in your Excel workbook that is called “Altered Survivorship Leslie.” Follow the instructions on this worksheet
(text in red on the worksheet) to explore the effect of reducing the survival
rate for each age class by 1%. Do this
by multiplying the ALL of the original survival rates by 0.99. How does this change the total population size
in the year 2000? Now reduce all
survival rates by 2% (Multiply ALL of the original survival rates by
0.98). Note that I have set this up for
you so that making this change should be very easy. Continue reducing the survival rates until
you obtain a population of about 50 females in the year 2000. By how much do you
need to reduce the survival rates to obtain a population of about 50 females in
the year 2000?
You should also try exploring the
effect of altering the natality rates.
To do so I have also included a worksheet called “Altered Natality
Leslie.” As before, follow the instructions (in red) on this worksheet to
explore the effect of leaving the survival rates unchanged but multiplying ALL
of the natality rates by 0.99. How does
this change the population size in the year 2000. Then multiply ALL of the natality rates by
0.98 and so on. By how much do you need to
reduce the natality rates to obtain a population of about 50 females in the
year 2000?
This exercise should
give you an appreciation of how sensitive populations can be to very minor
changes in demographic factors.
WHAT SHOULD YOU TURN IN? Submit your homework assignment via email directly to the Teaching Assistant for this class Chloe Cason at: casonc@wwu.edu THE SUBJECT LINE OF YOUR EMAIL SHOULD INCLUDE THE FOLLOWING TEXT AND NOTHING ELSE!
ESCI325: Homework #1
As an attachment to
this email, you should include your Excel worksheet. Be sure to delete any extra worksheets from
this file. Neatness counts. In the body of your email message, include
the answers to each of the items listed in red above. You may also include a brief narrative to
explain your answers.
The due date for this
assignment is listed on the syllabus.
Assignments that are turned in late will be penalized as described on
the class web page.
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