1
BCOR 102
Exam 1
Name: KEY
Be sure to show your work and provide units for your answer wherever possible.
1. You are studying a population of protists that doubles in size every 4 days.
a) Calculate r for this population. (5 pts)
r=ln(2)/4 =0.173
b) If you began with a population of 100, what would the population size be after 2 days? (5
pts)
N=N0 exp(rt)
=100 * exp(0.173 * 2)
=141.4
2. Imagine a population of black bears that is growing according to the logistic equation, with K =
200 and r = 0.05 individuals / individual / year.
a) What is the maximum population growth rate (dN/dt)? (5 pts)
dN/dt = rN(1-N/K)
=0.05 * 100 * (1-100/200)
=2.5
b) What is the population size for which this growth rate occurs? (5 pts)
Max growth rate occurs at K/2 = 100
c) What is the population growth rate (dN/dt) when the population size is 210 bears? (5 pts)
=0.05 * 210 * (1-210/200)
= - 0.525
3. Let’s assume we are studying a population of birds and have a cohort of 400 hatchlings.
Mortality due to predation and disease is very high and 80% of the birds die every year.
The maximum age observed for this species is 4 yrs (i.e. the number of 5 yr olds is 0).
Reproduction begins at age 1. First time breeders have an average of 3 female offspring
per female. In subsequent years the fecundity is a little bit higher, with an average of 4
female offspring per female for the remainder of her life.
Note that this question gives mortality rate (80%). The life table uses survivorship, which is
20% per year.
2
Age (x)
0
1
2
3
4
5
Lx
1
0.2
0.04
0.008
0.0016
0
Bx
0
3
4
4
4
0
Lxbx
0
0.6
0.16
0.032
0.0064
0
0.7984
xlxbx
0
0.6
0.32
0.096
0.0256
0
1.0416
1.304609218
a.
Calculate R0 and G (10 pts)
R0=sum( lx bx) = 0.7984
G=sum(x lx bx)/R0 = 1.3046
b.
Is this population growing or declining? (2 pts)
c.
How many babies will an average newborn have over her entire life? (2 pts) 0.798
d.
Bonus pts: what kind of survivorship curve does this
population have? (+2 pts)
type 2, because constant mortality rate
Sketch a graph of
N
populaton size (N) vs
time for this model.
(3 pts)
e.
declining
time
4. Carduus nutans is an invasive
thistle in New Zealand.
Katriona Shea calculated the following transition matrix. She wanted to simulate two
possible kinds of control for this invasive plant: insects and grazing. The biological
control insects attack flowers and reduce fecundity by about 30%. Grazing reduces the
establishment of seedlings.
Next year
Buried
Seeds
Seedlings
Mediuim
Large
Buried
Seeds
0.038
0.18
0
0
This year
Seedlings
Medium
Large
8.25
179.4
503.1
1.09
0.01
0.005
22.2
0
0.02
62.2
0
0
3
a. If the population starts with 1000 seeds, 100 seedlings, 20 medium and 10 large
plants, how many seedlings will there be next year? (8 pts)
=0.18*1000 + 1.09*100 +22.2*20 + 62.2 * 10
= 180 + 109 + 444 + 622
=1355
b. Suggest modifications to her observed matrix that could be used to project the
effects of biocontrol insects on population growth. (you don’t necessarily have to
calculate anything for this part) (8 pts)
She would alter the fecundity terms in the model. Normally that would just be the numbers in the
top row, showing the number of seeds produced by small, medium, and large plants. In particular,
each of those numbers would be decreased by about 30%.
(Note: In this particular example, she also altered the second row, because some seeds germinate
immediately to become seedlings and don’t enter the buried seed pool. But that is a detail you
don’t have to include that detail to get credit).
5. How do “Environmental Stochasticity” and ”Demographic stochasticity” affect the risk of
extinction for populations growing exponentially? (8 pts)
Environmental stochasticity (randomness) comes from external variation in the environment and
causes r to vary among years. Good and bad years affect all individuals in the same way. There
is a possibility that a population will decline to extinction if variation in r is too high, because you
may get a string of bad years.
Demographic stochasticity comes from variation in the particular pattern of births and deaths
among individuals. For example, even if the individual risk of mortality is low, there is always a
chance that the individual will die before reproducing. In large populations that source of
variation usually averages out, but in very small population there can be a real risk of extinction
from demographic stochasticity. (To use a coin-tossing expt, it is easier to get all heads if you
toss two coins then if you toss 100 coins).
6. Imagine two populations both growing with r = 0.1. One of the populations has density
dependent growth with K=1000. The other is growing without density dependence.
Sketch a graph that shows dN/dt vs N for both populations (you should not need to
calculate anything for this). Be sure to label which population is which and identify the
point K. (12 pts)
dN/dt
exponential growth
logistic growth
0-
4
N
K
7. This simple transition matrix has an asymptotic growth rate of lambda =1.13.
Small
Med
Large
Small
0
0.1
0
Med
4
0
0.4
Large
25
0
0
N0
100
10
1
a. Is the population projected to grow or decline? (2 pts)grow, because lambda > 1
b. If the population starts with 111 individuals (100 small, 10 medium and 1 large), What
will be the total population size next year? (4 pts)
100*0 + 10*4 + 1*25 = 65 small
100*0.1 + 10*0 + 1*0 = 10 medium
100*0 + 10*0.4 + 1*0 = 4 large
total =79
c. How can you reconcile a and b? (4 pts)
Lambda is the asymptotic growth rate, which means that after projecting the population for
several years lambda will stabilize at that value. But initially, the population growth rate can
fluctuate a lot. Here , the initial population is mostly newborns, which have low survival so
the number declines.
8. Here is a graph of population size vs time.
Population size
Time
a. Does this population show evidence of density dependent population regulation?
How do you know? (6 pts)
Yes, it is regulated, because the population size eventually stabilizes at an intermediate size (K)
b. What are two explanations for the shape of the curve? (6 pts)
5
1. A very high r, with discrete generations (or time lags), which causes the population to
“overshoot” and “undershoot” the carrying capacity.
2. Variation in the environment (good and bad years), so the population size just tracks
the available resources.