BIOL 463 Homework 1 -- Demography and Population Growth
How to complete this assignment. This assignment is divided into two parts.
There are 9 questions in Part 1, 2 questions in Part 2, and one extra credit question.
There is an associated Excel spreadsheet file that you will be directed to use to
complete the assignment. There are also two journal articles that you may read and
consult to complete the assignment. Please type your answers in a separate file (that
contains your answers only – any text format will do) and submit them to me in the
Homework 1 dropbox on the course ANGEL website. In Part 1, you will also be
asked to save and submit a filled-in copy of the Excel spreadsheet. Please name both
the answer file and the Excel spreadsheet with your PSU ID – e.g. ABC1234.doc and
ABC1234.xls.
Please use complete sentences for all answers.
This assignment must be uploaded to ANGEL by 5PM on February 5.
Part 1. Life Tables and Reproductive Value
Elk are a dominant herbivore in the Greater Yellowstone Ecosystem, numbering
>120,000 animals living in the area that includes Yellowstone National Park itself
and the contiguous national and state forests in Montana, Wyoming and Idaho.
Prior to 1995 the major source of mortality in the elk population was harvest due to
human hunting and mortality due severe winter conditions in the area. In 1995, 31
wolves were reintroduced into Yellowstone National Park. The wolf population
quickly grew to 132 animals by 2001, prompting concern among tourism and
hunting groups about the impact of wolf predation on the growth and maintenance
of the elk population. In 2006, Wright et al. published a life table analysis of the elk
population with the goal of assessing the impact of wolf predation relative to other
sources of mortality in the elk population. The results of that life table analysis are
presented in the excel spreadsheet associated with this assignment (sheet 1). In the
first part of this assignment, review both the analysis in the excel spreadsheet and
the Wright et al (2006) paper itself, and answer the associated questions. In the
sections below, I provide calculations for two additional measures: life expectancy
and reproductive value.
Reproductive value: A useful statistic that can be calculated from the life table is
the reproductive value (Fisher 1930). The reproductive value is the relative number
of offspring that remain to be born to individuals of a given age. Though it is
intuitive to assume that the reproductive value is maximized in newborns (you have
your whole life to reproduce), a newborn’s reproductive value must be discounted
against the fact that a newborn may not achieve its full reproductive lifespan. Thus,
an individual that has made it past the initial developmental stages may have higher
reproductive value because it is guaranteed to have survived long enough to
reproduce. Let v(x) be the reproductive value of an individual of age x, then the
reproductive value is defined as:
𝑣(𝑥) =
# 𝑜𝑓 𝑜𝑓𝑓𝑠𝑝𝑟𝑖𝑛𝑔 𝑝𝑟𝑜𝑑𝑢𝑐𝑒𝑑 𝑏𝑦 𝑎𝑔𝑒 𝑥 𝑜𝑟 𝑜𝑙𝑑𝑒𝑟
𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑖𝑛𝑑𝑖𝑣𝑖𝑑𝑢𝑎𝑙𝑠 𝑜𝑓 𝑎𝑔𝑒 𝑥
The numerator accounts for all the offspring left to be produced in ones lifetime, and
the denominator discounts for the probability of achieving the rest of ones lifetime.
Using the notation of the life table, we can write v(x) as
𝑘
𝑒 𝑟𝑥
𝑣(𝑥) =
∑ 𝑒 −𝑟𝑦 𝑙𝑥 𝑚𝑥
𝑙𝑥
𝑦=𝑥+1
The Elk Life Table: On sheet 1 of the Homework 1 Excel file (the tab called “Life
Table”) you will find the elk life table from the Wright et al paper. The cells in grey
are the sampled values for elk survival and fecundity from that paper. The cells in
blue are blank, and you will be asked to fill these values in. The cells in green
contain the formulas to calculate the reproductive value (above) – ignore the initial
values in these cells, they will fill in with the correct values as you fill in the life table.
Below the table are cells in yellow where you will be asked to fill in the summary
values for the life table. (Note: I realize that all these values exist in the paper and
you could simply fill them in without performing any calculations –I changed a few
values so the calculated values won’t match the paper exactly. Since there’s a
reasonable chance you’ll be asked to do these kind of calculations on an exam, it might
be worth proving to yourself that you can do it for real.)
Complete the following questions/task that pertain to sheet 1 of the excel file. Note:
you may calculate the necessary values by hand, with a calculator, or using function
capabilities in Excel. For the questions below, present give all answers to 2 decimal
places unless otherwise specified.
1. Calculate and fill in the values for the blue columns (lxmx and xlxmx).
2. Using those values, calculate the basic reproductive value, R0 and fill it in the
corresponding yellow box at the bottom of the table.
3. Calculate the generation time, Tc, and fill it in the corresponding yellow box
at the bottom of the table.
4. Calculate the intrinsic rate of increase, r, and fill it in the corresponding
yellow box at the bottom of the table (provide this answer to 4 decimal
places.).
When you have completed questions 1-4, save a copy of the Excel worksheet.
Rename the file with your PSU ID (i.e. ABC1234) and submit the file (make sure it
has the values for 1-4 filled in!) to the Homework 1 Dropbox.
As you complete questions 1-4, the values in the green columns should start
changing (because their functions depend on the values you are calculating).
5. On average, how many female offspring is one cow elk expected to have in
the course of her lifetime.
6. Wright et al. constructed this life table using demographic rates and
population counts from animals harvested by hunters between (1996-2001);
i.e. this is a static rather than a cohort life table. Describe how using hunterkilled elk may bias the estimate of the age distribution of the elk population.
7. What age has the highest reproductive value? What is it?
8. Wright et al. concluded that wolf predation should have a smaller impact on
elk population growth than mortality due to hunters based on an analysis of
the age-specific reproductive value. Describe the quantitative basis for this
conclusion.
Part 2. Environmental Variability and Population Growth
The calculation of basic reproductive rate and the intrinsic rate of growth provide a
measure of the expected rate of population growth. In practice, that growth rate is
likely to fluctuate from year to year as a consequence of environmental variability
(e.g. El Nino years, droughts, hurricanes, severe winters). Thus the growth rate of
any particular population may best be described as a distribution with a mean equal
to the life table value, and some variance determined by how much the local
environmental conditions fluctuate. Brian Dennis (1991) and colleagues considered
the impact of this environmental fluctuation on the growth populations of
endangered species – and in particular considered the probability that a population
with, on average, positive population growth rate (i.e. as might be calculated from a
life table), might go extinct due to random environmental fluctuations. Using the
associated excel spreadsheet, I would like you to explore the impact of
environmental variability on population growth and extinction probability for
endangered populations.
Deterministic and Stochastic Growth models: The exponential growth model that
we discussed in class is deterministic (i.e. there is no random fluctuation in values) –
as such the population at any one time can be calculated directly from the initial
value (N0). In the real world, population will be higher or lower than expected
following particularly good or bad year, so the value of the population size at time t
depends on the value in time t-1, not just on the initial conditions. The second sheet
of the Excel file (“Population Projections”) calculates and plots the projected
exponential growth for a deterministic model, and one where the annual growth
rate fluctuates from year to year. Below I will ask you to manipulate this sheet and
summarize the results to investigate how environmental variability may affect the
growth of a population.
Orienting to the Excel sheet:
A. On the left of the sheet are 5 number that correspond to the important
parameters of the population model:
i. Initial population size is the starting number of animals
ii. Basic reproductive rate, R0.
iii. The generation time, Tc.
iv. The intrinsic growth rate, r.
v. The environmental variability in the growth rate, V. This indicates
how much the intrinsic growth rate, r, varies from year to year.
Technically, the growth rate each year is drawn from a normal
distribution with variance equal to V. Basically, the bigger this
number is, the bigger the chance of really good or really bad years.
Note, the chance of good years are equally as likely as bad years.
B. To the right of that are 3 columns of numbers:
i. year is the time in years.
ii. “Deterministic Population size” is the projected population size each
year for the deterministic exponential growth model given the
parameters and starting population size (at left).
iii. “Mean of Stochastic Runs” gives the average projected population size
each year for 100 randomly generated population trajectories given
the parameters and starting population size (at left).
iv. If there are enough bad years, the population size might drop below 1
individual, thus going extinct. The “Proportion of Stochastic runs still
alive” indicates the proportion of those random simulations where the
population has stayed above 1 individual (i.e. the probability of the
population not going extinct) for each year.
C. To the right of that is a figure that shows the projected population models
over time. The thick solid red line indicates the deterministic model. All the
other lines are the individual random simulations (these are all calculated
using the remaining sheets). Notice that the lines that drop to 0 and stop are
the simulations in which this population went extinct. Note: the y-axis is on
the log scale, so exponential growth looks linear.
D. NOTE: Every time you calculate something new on the sheet, it will generate
new random simulations – so the two right most columns of numbers and the
figure will change. This is supposed to happen! You can make the sheet
calculate a new set of stochastic simulations by pressing F9 (or function+F9
depending on your keyboard set up) – you’ll need to do this to complete the
assignment.
Using this Excel sheet, I would like you to do a simulation experiment to quantify
how extinction probability responds to 3 variables (initial population size,
generation time, and environmental variability) when the basic reproductive rate is
held constant. Please fill in the values of N0, Tc, and V for each of the 8 scenarios in
Table 1 and fill in the proportion of the simulations in which the population
survived for 25 years. Using F9, recalculate this value for each set of parameters 3
times and fill it in the table. NOTE: keep R0 constant at 3; the intrinsic rate of
increase with calculate automatically.
Table 1. Proportion of populations surviving for different parameter values.
1
2
3
4
5
6
7
8
Initial
Population
2
2
2
2
10
10
10
10
Generation Environmental
Length
Variability
5
.2
10
.2
5
.4
10
.4
5
.2
10
.2
5
.4
10
.4
1
Iteration
2
3
Questions:
1. Fill in the values for the proportion of the simulations in which the
population survived for 25 years for three iterations of each of the 8
parameter combinations in Table 1. Please copy and paste the table, into the
file with your responses.
2. Describe the relationship between the 3 variables tested in the simulation
experiment above (Table 1) and the extinction probability. Describe the
direction of the relationship (i.e. extinction probability goes up as X goes
down) and provide a justification for why this relationship makes sense.
a. Initial population size:
b. Generation time.
c. Environmental variability.
Extra Credit:
3. Consider the re-introduction of wolves from section 1 of this homework.
Following re-introduction, the Yellowstone wolf population was very
successful and clearly did not go extinct – in part due to the successful efforts
of the wildlife managers overseeing the reintroduction. Referring to the
patterns in from this simulation experiment, suggest 2 specific steps that
wildlife managers could have taken to increase the probability of successful
reintroduction of wolves to Yellowstone.