Stat 534
Fall 2015
Homework # 3, Population models,
Due: 5 pm, Friday, 20 November 2015, to my office (2121 Snedecor) or mailbox (in 1121
Snedecor)
Reminders: Choose 3 of the 4 problems. You are not expected to do all four.
Problems 4 is a ’data question’ that requires an executive summary.
Problem 1. The marten is a medium-sized fur bearing mammal found in forests in the western
US and Canada. Vital rates have been estimated in various places at various times. The
following information collects “typical” values:
• Survival of a new born to age 1: 0.25
• Survival of adults (e.g., age 1 to 2, or age 5 to 6): 0.75
• Pregnancy rate (fraction of surviving females at age i in the spring conceiving during the
next year):
Age 1: 0, Age 2: 0.78, Age 3: 0.92, Age 4 or older: 0.96
• Litter size: (number of liveborn females per pregnant female):
Age 1: −−, Age 2: 1.64, Age 3: 1.68, Age 4 or older: 1.83
Imagine an annual census occurring in late spring. This is after birth of newborns but before
they are weaned. Because there is limited data on age-specific fecundity and survival for older
individuals, data are pooled into rates for age 4 or older. Some individuals have been known to
live 10 or more years in the wild.
You use a stage-classified matrix population model to estimate population growth rate and
related quantities. This matrix combines an age-based classification for newborn through age
3 individuals with a single stage for “age 4 or older”. Hence there should be 5 stages in your
model: newborn, age 1, age 2, age 3, and age 4 or older. Surviving newborn through age 3
individuals always progress to the next age. Surviving “age 4 or older” individuals remain in
that stage in subsequent years. As is common, this is a female-only model. Litter sizes are
expressed as females per females.
The survival probabilities are estimated from counts of surviving individuals. These counts
have independent overdispersed Binomial (quasiBinomial) distributions. Sample sizes are 300
newborns (of which 75 survived) and 400 adults (of which 300 survived). The overdispersion
parameter is 5.2. All four adult survival rates are based on the single estimated adult survival
probability.
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1. Compute the rates in the first row of the transition matrix (production of newborn females
per female in stage i the previous year).
2. Write out the 5x5 transition matrix
3. Estimate the asymptotic growth rate, λ
4. What is the stable stage distribution, expressed as proportions of the total population
size?
5. Calculate the elasticity matrix
6. Estimate the consequences for λ of increasing the survival of 4+ old individuals by 10%.
7. Calculate the estimation variances of the two survival probabilities.
Note: Ask me if you don’t know / don’t remember the variance of a the quasiBinomial.
8. The variances of the three non-zero fecundity values are 1.2, 1.6, and 1.8. These three
estimates are independent. Calculate the standard deviation of λ.
9. One potential concern is that the 5x5 stage transition matrix allows individuals to live
forever. Biological opinion is that 15 years is the effective upper age limit for martens
in the wild. Re-express the 5x5 transition as a age-classified transition matrix in which
individuals die after reaching age 15. Estimate λ for this purely age-classified model. Does
the choice of modeling strategy (5 stages or 15 ages) have much effect on the estimated λ?
10. Using what you see from the analysis of the 5 stage and the 15 age models, explain why
the [5,5] element of the elasticity matrix for the 5 stage model is so large.
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Problem 2. Two analytical issues.
2-a) Sensitivity to changes in multiple
the following form:
s1 (1 − g1 )
s1 g1
0
0
vital rates. Many stage-based transition matrices have
f2
s2 (1 − g2 )
s2 g2
0
f3
0
s3 (1 − g3 )
s3 g3
f4
0
0
s4
where si is a stage-specific survival probability and gi is a stage-specific conditional probability
that a surviving plant will grow into the next stage. For many species, the sensitivity of λ
to changes in si or gi may be more important than the sensitivities we have computed (i.e.,
∂λ/∂aij ). The sensitivities to si or gi can be computed from the “usual” sensitivities. Derive
the expressions to compute ∂λ/∂si and ∂λ/∂gi . You only have to report expressions for one
choice of i and (but not i = 4).
2-b) Evaluating why λ changes. Imagine you have constructed transition matrices to describe
population growth in two environments. To illustrate, consider a population with 3 stages with
transitions described by matrix A in environment 1 and matrix B in environment 2:
A: environment 1
0
2
4
0.31 0
0
0
0.2 0.3
B: environment 2
0
0
8
0.29 0
0
0
0.3 0.6
The asymptotic growth rates in these two environments are different: 0.990 for environment
1 and 1.138 for environment 2. The transition matrices differ in many ways. The question is
which differences are the most important. In other words, does the change in the [3,3] element
from 0.3 to 0.6 contribute more to the change in λ than does the change in the [1,3] element
from 4 to 8?
Derive an approximation to the change in λ that allows you to express (at least approximately)
the change as a sum of contributions arising from the changes in each matrix element. Calculate
the contributions from each of the 5 non-zero elements. Is the consequence of the change in [1,3]
from 4 to 10 larger than the change in [1,2] from 2 to 0?
Note: A historical aside. Those of you who know Malthus’s An essay on the principle of
population (1798) should recognize in these results why Malthus argued for later marriage (and
hence later age of first child, in principle if not so much in practice) as an effective way to reduce
population growth rates.
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Problem 3. Mountain Golden Heather is a federally threatened plant growing on rocky outcrops
in North Carolina. If you’re interested in the plant, a picture and a summary of its biology is at http://www.fws.gov/raleigh/species/es_mountain_golden_heather.html. Pictures of the habitat are at: http://www.fs.fed.us/wildflowers/Rare_Plants/profiles/
TEP/hudsonia_montana/index.shtml.
H. montana is a long-lived perennial dwarf shrub with a persistent seed bank. The researchers
who collected the demographic data divided the population into two seed/seedling stages and
four adult stages based on the cross-sectional area of the plant. The six stages are “in seed
bank”, ”seedling”, 0 − 25cm2 , 25 − 50cm2 , 50 − 100cm2 , and > 100cm2 . The demography in
non-fire years is dominated by transitions among the four “adult” stages. This problem explores
the relationship between a matrix model and an integral projection model. Ignore the “in seed
bank” and “seedling” stages in this problem.
The data in mghsize.csv contains fate and size information for 318 plants measured in 1990 and
remeasured in 1991. The numbers are the cross-sectional area in cm2 . I have simulated these
data based on field data from the 1980’s. An NA indicates a plant that died between 1990
and 1991. To help me evaluate your work, please consider four “reporting” plants. These are
hypothetical plants with sizes of 20cm2 , 40cm2 , 60cm2 , and 80cm2 . Where asked for below, give
me the requested information for these four plants.
1. Construct a model for survival probability as some function of size in 1990. Report your
model and the survival probability for the four “reporting” plants.
2. Construct a model for the distribution of size in 1991 conditional on size in 1990. Report
your model and the distribution of 1991 size for each of the four “reporting” plants.
3. Construct the transition matrix that is the numerical approximation to the integral projection operator. I suggest you consider sizes from 2cm to 100cm in steps of 1cm. It
is traditional to use the midpoint rule to construct the matrix. That is, the value for
each matrix element is computed from the survival and growth functions evaluated at the
midpoint of the cell boundaries. If there is a substantial conditional probability of a size
beyond 100cm, increase the upper limit of the approximating matrix. Use an image plot,
a contour plot, or some other graphic to visualize the transition matrix.
4. Use the numerical approximation to the IPM to calculate the asymptotic population
growth rate.
5. Use the numerical approximation to the IPM to calculate the stable size distribution. Plot
relative frequency as a function of size.
6. Previous analyses of the demography have used matrix projection models with 4 categories
(0-25 cm, 25-50cm, 50-75 cm, and 75-100cm). Use the size data to estimate the transition
matrix among these 4 categories. Calculate and report the asymptotic growth rate, λ, and
stable size distribution.
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Problem 4. Another Mountain Golden Heather (MGH) problem, although completely unrelated
to problem 3. The demography of MGH was studied in detail in the mid-late 1980’s. Marked
plants were followed for up to 5 years. The researchers who collected the demographic data
divided the population into two seed/seedling stages and four adult stages based on the crosssectional area of the plant. The six stages are “in seed bank”, ”seedling”, 0 − 25cm2 , 25 − 50cm2 ,
50 − 100cm2 , and > 100cm2 . The annual transition matrices are in mghfire.txt, where the first
two columns give the row and column of the matrix element and columns 3 through 6 have the
expected number of size j plants per size i plant. Since these data were collected, it has become
clear that fire substantially increases seed germination. I presume that fire also reduces adult
survival. Column 7 contains hypothetical data for a “fire year”. The data in columns 3 through
6 come from non-fire years. You have been hired to construct a stochastic matrix projection
model for MGH and evaluate different choices of fire return frequency.
The endangered species biologists suggest the following:
• It is impossible for a fire year to be followed by another fire year.
• The four transition matrices for non-fire years are equally probable. All transitions from
one of those non-fire matrices to any of the non-fire matrices (including no change) are
equally probable.
• The probabilities of transition to the fire year from any non-fire environment are equal.
• You will need to determine that non-fire to fire transition probability so that the marginal
probability of a fire year is one of three values: 0.02, 0.05, and 0.10, corresponding to fire
return times of 50 years, 20 years, and 10 years. Also consider a fourth ’no-fire’ scenario,
where the fire year environment is removed from the set of potential environments.
For each of the four fire return times (never, 50, 20, and 10 years) compute the stochastic growth
rate, its standard deviation (variability across years, not estimation variability), and the average
population stage distribution.
You need to recommend what you believe is the most appropriate fire return time, with an
explanation for your recommendation. (Note: I just want you to think about the problem, the
results of your analysis, and common sense about what might be practical; you are not expected
to know about fire ecology or fire management).
Include as an appendix to your report the environmental transition matrices you used in your
analysis.
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