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Limits on Growth
No population grows without bounds, so we need to modify our population
model to predict the fact that many populations have a so-called
limiting population that is determined by the carrying
capacity of their environment.
The easiest way to model a limiting population is to introduce a new
term into our population model. This term is called an
overcrowding term and the coefficient of this term is called the
coefficient of overcrowding. The simplest overcrowding term is
proportional to the square of the current population. In other words:
dP/dt = k P - A P2.
Assuming that A>0, the
negative sign in the second term indicates that this term decreases
the population. This population model is called the
If a differential equation has a solution that does not change over time
(that is, a solution of the form P(t)=C for some constant C),
then we say that the differential equation has a equilibrium solution.
How does a population change at an equilibrium solution? In
other words, at an equilibrium solution, is
dP/dt<0, dP/dt>0, dP/dt=0, or can we tell?
- Using the
Population Simulator, graphically produce
several solutions to the logistic model for a variety of initial populations.
Determine the limiting population size when
the initial population is large and
when the initial population is small for
- k=0.2; A=0.05
- k=0.2; A=0.005
- k=0.1; A=0.005
Do you see a pattern? Conjecture what the carrying capacity is
for a net birth rate of k and a coefficient of
overcrowding given by A.
- The growth curve of a population growing according to logistic growth is
typically characterized by three phases: an initial
phase in which growth is slow, a rapid expansion
phase in which the population grows relatively quickly, and a
a long entrenchment stage in which the population
is close to its limiting population due to intra-species competition.
Sketch one of the curves above and identify each phase on your plot.
How do the values of k and A affect the time
it takes for the population to progress through each phase?
- Do populations that are decreasing in number according to a logistic
model go through the same stages? If not, what stages do they go through?
Identify the stages on your plot and explain how k
and A affects each phase.
- The answer to the group discussion indicates that we can find
equilibrium solutions by solving for values of P such that
dP/dt=0. Explicitly solve for the two equilibria solutions
for the logistic model with net birth rate k and coefficient
of overcrowding A. Compare this theoretical result with
your numerical estimates of the carrying capacity for the values of
k and A given in the previous question.
- Let's name the two equilibria C and
E, with C < E. According to the logistic
equation, in terms of the quantities given in the group discussion,
how will the population change if
- P > E?
- C < P < E?
- Can the population ever satisfy P < C? Why or why not?
Invasion of the White Pine
The Bufo marinus data we worked with in the previous section
fit the exponential model well. In this section we will examine data
that indicates the prevalence of
white pine (Pinus strobus)
in the vicinity of the Lake of the Clouds, a lake in the Boundary
Waters Canoe Area of northeastern Minnesota.
The lake is deep (31 meters), calm, sheltered from wind, and devoid of
inflowing streams. Consequently, the lake's bottom is covered
with layers of annual sedimentary deposits. Each layer contains a sampling of
pollen, and by counting the pollen belonging to each species of tree and herb,
it is possible to estimate the ratio of one plant species to another.
White pine became extinct in northern Minnesota during the last period
of glaciation, although it remained in southern climates such as
Virginia. Once the glaciers began to retreat, the white pine began to
expand northward again; it reappeared in northern Minnesota about 9400
years ago (H. E. Wright, "The roles of pine and spruce in the forest
history of Minnesota and adjacent areas", Ecology, 49,
A. J. Craig (Absolute pollen analysis of laminated sediments: a
pollen diagram from northeastern Minnesota, M.S. thesis,
University of Minnesota, 1970) counted pollen in a phenomenal 9400
sedimentary laminae from a core at Lake of the Clouds. According to
his data, as white pine invaded the region surrounding the lake, it
competed with entrenched populations of jack pine (Pinus
banksiana) and red pine (Pinus resinosa), which occupy
essentially the same coarse, sandy soil as P. strobus. The
combined pine tree pollen accounted for about 60-70% of the pollen
during the period of interest; other plant species remained
essentially constant during the time period (with the exception of
spruce (Picea) which decreased).
Craig's data is condensed and analyzed by W. A. Watts ("Rates of
change and stability in vegetation in the perspective of long periods
of time", Quaternary Plant Ecology, H.J.B. Birks and
R.G. West, eds, Blackwell Scientific, 1973), and we have reproduced
portions of this data below. Note that in the second column, time is
measured in units of thousands of years.
Years 9131 P.bank/P.resin P.strobus
Ago (Thousands) percentage percentage
----- ---------- -------------- ----------
9131 0.0 53.4 3.2
8872 0.259 65.5 0.0
8491 0.640 61.8 3.7
8121 1.010 55.2 3.4
7721 1.410 60.4 1.7
7362 1.769 59.4 1.8
7005 2.126 50.6 10.6
6699 2.432 51.6 7.0
6444 2.687 40.0 21.2
5983 3.148 29.7 34.2
5513 3.618 25.0 40.4
5022 4.109 32.5 29.8
4518 4.613 22.7 46.2
4102 5.029 31.6 33.0
3624 5.507 32.5 37.6
3168 5.963 27.1 39.5
Table 2: Percentages of pollen for red/jack pine and
white pine for sedimentary layers at Lake of the Clouds, MN.
By scanning the data, it is clear that percentages of
red and jack pine decreased during the time period indicated, whereas
white pine pollen increased. If we assume that these pollen counts are
representative of the relative populations of these species,
then we have a basis for examining the population growth of
P. strobus and the simultaneous decline of P. banksiana
and P. resinosa.
- For the P. strobus data set,
use the Population Simulator
with the option to "Plot data for trees" to
model the growth of the white pine population near Lake of the Clouds.
(The green data points indicate red/jack pine pollen; the blue data points
indicate the invading white pine.)
- Find and record an initial population, a net birth rate, and a coefficient
of overcrowding that qualitatively matches the growth
curve for this species of pine.
- Identify the three phases of logistic growth for this species
of pine (if applicable). How long was the expansion phase?
- Estimate the equilibrium value for this species.
- Repeat the above analysis for the P. banksiana
and P. resinosa data set (green data points).
Again, identify phases of growth, if applicable.
- Recall that for the exponential growth model, we chose our
parameters to try to minimize the
sum of the squares of the residuals
between the data and a solution curve. Explain how one might
attempt a similar procedure to find parameter values that best fit
this data set.
- Because of your success in the mathematical modeling of population data,
you are invited to give an address at the next International Congress of
Mathematical Ecologists. You decide that the title of your talk will be
"Post-Glacial Populations of Minnesota Pines: Are the Growth Curves Logistic?"
Write an abstract (one page or less)
describing the main points of your talk. In particular, argue why you
do or do not believe that the pine data we are considering
comes from a population that obeys a logistic growth law.
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