Answer key: Spreadsheet exercises 7-8
Spreadsheet exercise 7
1.
This is exactly what your initial assumptions were for the graphs you made. Note the
values you put for b and d, or for r (=b-d) in both key equations used. Thus, you would
expect geometric or exponential growth under these circumstances. You’ve already tested
it.
2.
Think about this logically: If birth rates are less than death rates, you ought to see the
population decline. If you changed b and d so that b is less than d (or r<0) into your
spreadsheet, you will get what is called an “exponential decay” curve.
Nt
120
100
Population size
80
60
Nt
40
20
0
0
5
10
15
20
25
Time
3.
Think about this logically. If b=d, the births equal deaths and thus the population size
remains stable.
4.
It is the difference between birth rate and death rate that determines ΔNt. This should
make intuitive sense, but should also be clear from the geometric growth rate equation,
and the fact that r = b-d in the model.
5.
The population growth rate clearly increases over time. This is related to the increase in
population size. This is depicted by the graph of ΔNt over time for an increasing
population (i.e. your original graphs made in this exercise). If the population itself was
decreasing, then the population growth rate would decrease. Note that the population
growth rate is depicted in the exponential model as dN/dt, the instantaneous slope of the
curve. The curve keeps getting ever-steeper as the population grows.
6.
Total births and deaths, as well as ΔNt, all increase with an increase in population size.
This makes sense mathematically as well as intuitively, since the term for total births in
the geometric growth equation is b*Nt, or “birth rate multiplied by population size”.
Similarly, total deaths in the geometric growth equation is d*Nt “death rate multiplied by
population size”. See also the answer to the previous question for a discussion of ΔNt.
7.
In this model, the per capita rate of population growth remains constant. Think about the
fact that b, d and r (or R) are all constants.
8.
The continuous time population grows faster because it responds to its increases in
numbers immediately, rather than having to wait a generation to gain the reproductive
benefits from new members added to the population. Obviously, this is much more like
bacteria than humans.
9.
For the growth rates to be the same, λ = er, since Nt = N0ert compared to Nt = λt N0. (See
problem set 2.) Note that the t is excluded because it is a variable, and we are determining the
value of constants.
We can then convert this to obtain a value of R:
λ = 1+R = er
R = er - 1
10.
If you graph this data, you will obtain an approximately exponential growth curve.
Spreadsheet exercise 8.
1.
The logistic model takes into account that population size itself affects per capita birth and death
rates, rather than those rates being constant (as the other models assume). Specifically, the model
assumes that per capita birth rates decrease and per capita death rates increase as the population
grows, until an equilibrium population size (K) is reached.
2.
Population stabilizes at carrying capacity because at the point where total births equal total deaths,
the population size will remain constant. There are no further changes to the birth and death rates
because N doesn’t change.
3.
The values for ΔNt (or dN/dt) first increase with increasing population size, up until K/2 is
reached. They then decline to 0. The values for ΔNt/Nt decline linearly with population
size. (See Figs. 4 and 11).
4.
The y-intercept of ΔNt/Nt as a function of population size (i.e. graphed against Nt) is R. The
x-intercept is K, the carrying capacity.
5.
Intuitively: When the population size is larger than its carrying capacity, the death rate will be
greater than the birth rate. This will cause the population to decline until the birth and death rates
are equal, which is at the carrying capacity. Depending upon the difference between the old and
new carrying capacities, the population might undershoot the new carrying capacity before
stabilizing at it, declining so precipitously that it will end up at 0 (extinction)
With the spreadsheet: You can somewhat simulate this by changing Nt at time 0 to a number
greater than 50 (in other words, a starting population above the carrying capacity) in the first
model. See what happens. Notice what happens if you start with an ever-higher population size.
6.
The carrying capacity is a stable equilibrium. The population will tend to stabilize at this value
from below it or above it. But also see #4. If the population begins too far above K, it may
undershoot so severely it cannot recover.
7.
Intuitively: If per capita birth rates increase rather than decrease with population size, whether or
not the population reaching a carrying capacity or not depends on how much they increase
compared to the death rate. If you increase them equal to or above the death rate, then the
population will continue to grow, but if the per capita birth rate is still below the death rate,
equilibrium will be reached, because births and deaths will have to balance at some point. You
can follow this same logic for a decrease in death rate with population.
With the spreadsheet (for an increase in birth rate with population): Simply change b’ from a
negative to a positive number in the first spreadsheet and see what happens to the population size.
If you change it to a positive b’, but <d’, you should see that a carrying capacity is reached, albeit
a higher one than before. If you make b’ = d’, you get exponential growth. Above values for b’ >
d’ give you incredibly rapid growth without a carrying capacity.
8.
Intuitively: An equal change in per capita birth rate and per capita death rate would mean that b-d
is a constant. Hmmm, looks kind like the exponential growth equation, where b-d = r, a constant!
Thus, if this is true, you will see exponential (or, if discrete generations, geometric) growth.
With the spreadsheet: What this is really saying is the b’ = d’, so you just have to make them
equal. If you make the values positive, then you get exponential growth. If you make both values
negative, you get exponential decay (see graph in this answer key for exercise 7).
9.
Intuitively: If birth rates keep increasing relative to death rates, the population has to grow
because there are continuously evermore individuals added to the population than leaving it, and
this becomes evermore pronounced as the population grows.
With the spreadsheet: This is really the same question as #7 for an increase in birth rates with
population size, and you set b’> d’. Watch that population grow and grow!
10.
If the population grows more quickly, it will take less time to reach its carrying capacity. In the
discrete time model, increasing rates lead to a cycling around K. This basically is due to the time
lag. If the growth rate, and thus the time lag, is too severe, the population will crash. In the
continuous time model, this doesn’t occur, and K is simply reached more quickly.
With the spreadsheet: In the discrete model, experiment with higher values of b, without changing
anything else. In the continuous time model, simply change r and see what happens.
11.
If you graph the census data from the site and use the “add trendline” function, you find the
growth rate is approximately linear, suggesting that while we are still growing, we are on the >K/2
side of the logistic growth curve.. You can then estimate r and K by graphing ΔNt/Nt as a
function of population size (i.e. graphed against Nt)—see question 4. The values obtained
are:
Intrinsic rate of growth (r) = approximately 0.031
Carrying capacity (K) = approximately 11 billion