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What are the assumptions of exponential growth? To what situations,
generally, does it best apply?
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Four deer are introduced to an island on which there were previously no
deer. The resources for deer are excellent and the population grows
at an instantaneous per capita rate of 0.2 per year. It has been
shown that if the deer population on the island exceeds 500 then the deer
population will deplete all the resources and go extinct. If the
deer population keeps growing at the same per capita rate, how many years
will it take before the population reaches 500 individuals?
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If r = 0.3 per month how long will it take for a population to double in
size?
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You are studying the growth of a population of fruit flies living in the
garbage can in your kitchen. Last week, there were 100 flies in your population.
This week, there are 1,000 flies. (a) What is the finite per capita rate
of increase R? (b) What is the instantaneous per capita rate of increase
r? (c) How many flies will there be one week from now? (d) What is doubling
time for this population? (e) Isn't it about time to take the trash out?
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A species of finch has been introduced to the big island of Hawaii and
the population is growing exponentially. Genetic analysis suggests that
all the finches on the island were derived from a single pair of birds.
You want to know when these birds were introduced to the island.
You observe that one year, there are 500 finches. The next year,
there are 600 finches. Assuming the per capita instantaneous growth
rate has been constant, how many years ago were finches introduced to the
big island?
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Reindeer are introduced to a nature reserve where there are no predators
and no hunting is allowed. You observe the following change in population
number over five years:
| year |
number of reindeer |
| 1 |
6 |
| 2 |
8 |
| 3 |
11 |
| 4 |
14 |
| 5 |
19 |
Make the appropriate graph to most clearly test whether the population
is growing exponentially. Is it? If so, estimate r from the
graph.
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What is density dependence? What part of the logistic growth equation makes
it density dependent? Demonstrate mathematically, by using appropriate
values in the logistic growth equation, that this part of the equation
makes it density dependent. What ecological phenomenon causes density
dependence?
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A population is growing according to the simple model of logistic growth
with a carrying capacity of 648. At what population size is the population
growth rate maximum? Explain why, either in words or with a mathematical
derivation. At what population size is per capita growth rate the
highest? Why?
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A population of rabbits grows according to the logistic equation with a
time lag of 2 time units. The population cycles regularly. (a) Explain
how a time lag can lead to population cycling. (b) How might each of the
following changes in the population affect the pattern of population growth:
(i) A new kind of plant, which the rabbits eat, colonizes the area.
This plant re-grows after being eaten much faster than do the plants that
were originally in the area. (ii) The rabbits evolve a slower reproductive
rate which lowers the value of rm for the population.
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What value would you assign to theta (i.e. greater than 1, 1, or less than
1) for each of the following graphs? Explain in words what each graph means
with respect to the intensity of intraspecific competition at different
population densities.
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A population of trout in a lake has a carrying capacity of 200 and
a maximum per capita growth rate of 0.75. If the population grows
according to the simple logistic growth model, how many fish should be
harvested from the lake when the trout are at carrying capacity in order
to maintain the population at its maximum growth rate? Would the
number of fish that should be harvested be larger, smaller or the same
under any of the following conditions: (a) if the population grows according
to a modified form of the logistic with a theta value of 1.5? (b)
if the maximum per capita growth rate were 0.25? (c) If K were increased
to 300?
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Explain in words what is meant by each of the following: rm,
K, t, q, N, dN/dt, dN/Ndt, t.
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You study population growth in a laboratory population of Daphnia
("water fleas"; small aquatic crustaceans). You start a population
in a jar with 2 individuals, and count individuals in the jar each week,
to obtain the following data:
| Week # |
Population Size |
| 1 |
2 |
| 2 |
4 |
| 3 |
6 |
| 4 |
10 |
| 5 |
16 |
| 6 |
24 |
| 7 |
32 |
| 8 |
37 |
| 9 |
39 |
| 10 |
40 |
| 11 |
40 |
| 12 |
40 |
Plot population size versus time. Which version of logistic growth (e.g.
simple, with time lag, non-linear...) does this population appear to fit?
Use the graph to estimate K. Finally, estimate rm (to do so,
remember that initially a population undergoing logistic growth will grow
exponentially).
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What is the traditional definition of "carrying capacity" as used for most
animal species, and, according to Barrett and Odum (2000), why does it
not apply well to humans? What must be considered in addition to
density to develop a meaningful definition of the maximum carrying capacity
for humans?
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Barrett and Odum (2000) state that the "overshoot pattern" occurs in nature
and in human affairs. What is the "overshoot pattern"? What
are examples of causes for the overshoot pattern in nature? What
are causes of the overshoot pattern in human affairs? How does the
"ecological footprint" or a city such as Vancouver BC provide evidence
of an overshoot pattern?
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According to Barrett and Odum (2000), what is the difference between economic
development and economic growth? What is the ecological argument
for focusing on economic development rather than economic growth?
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What is the economic welfare threshold? Why do Barrett and Odum (2000)
argue that it is equivalent to the optimum carrying capacity?
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What do you think of the Barrett and Odum (2000) recommendations for an
integrated (ecologic/economic) capitalism? Are they desirable?
Are they practical? Why/why not?