Next: Limits on Growth
The very first differential equation that one typically encounters is
the equation that models the change of a population as being
proportional to the number of individuals in the population. In
symbols, if P(t) represents the number of individuals in a
population at time t, then the so-called exponential
growth model is
dP/dt = k P.
Recall that the general solution of this differential equation is
of the form C exp(kt) where C is the population at
the time that we first consider it, and where "exp" is just
another way to write "ee-to-the." Recall also that in order
to get a particular solution, we must have some sort of experimental
observations that tell us
The sign of k determines whether the population will
grow without limit, or whether it will become extinct.
- the initial population, and
- the net birth rate.
Invasion of the Toads
The next question is based on a question from the
Boston University Differential Equations Project.
As is often the case in mathematical modeling, there is no "best
answer" for this question. There are, however, "better answers":
the correctness of a mathematical model is
determined by how well it agrees with the reality of experimental
From 1935-37, the American marine toad (Bufo marinus) was
introduced into Queensland, Australia in eight coastal sugar cane
districts. Due to lack of natural predators and an abundant food
supply, the population grew and the poisonous toads began to be found
far from the region in which they were originally introduced. Survey
data presented by J. Covacevich and M. Archer ("The
distribution of the cane toad, Bufo marinus, in Australia and
its effects on indigenous vertebrates," Mem. Queensland Mus,
shows how the toads expanded their territorial
bounds within a forty-year period. This data is reproduced below;
it was mathematically analyzed by
M. Sabath, W. Boughton, and S. Easteal ("Cumulative
Geographical Range of Bufo marinus in Queensland, Australia
from 1935 to 1974", Copeia, no. 3, 1981,
Year Area Occupied (square km)
Table 1: Cumulative geographical range of Bufo Marinus
in Queensland, Australia.
Our goal is to construct a mathematical model that best fits the given
data. Note that the data is not given to us as "number of toads at
five year intervals," and, in fact, this is often the case. For the
toads in question, this would be virtually impossible data to obtain,
although statistical methods may be used to estimate this value.
- What assumptions could one make in order to convert the given data
into population data? How realistic are these assumptions?
- Suppose you are writing a small grant to study the population of this
toad. You want to ask for money to hire two undergraduates to gather
additional data for three months. What additional data would you
propose to obtain that would give you additional insight into the
toad's total population?
For ease of computation, we will assume that, on the average,
there is one toad per square kilometer. (Of course, some
fields are more densely populated, the middle of cities and lakes
don't have any toads, and so on.) We will also count the toads in units
of thousands, and time in units of years, beginning with 1939
as "time zero."
In your report, please include sketches of four solution curves,
for differing values of the birth rate, as described below.
You can generate these solution curves by using
Population Simulator with an initial population of "32.8 units,"
and choosing the option to "Show Toad Data."
- Vary the birth-rate k until the solution to
the differential equation appears to fit the data well
over the time period [0, 35].
Record this value of k.
- It also is possible to solve analytically for a value of k
that will guarantee that the curve passes through exactly
two of the data points. If P(0) = 32.8,
find and record a value of k
so that P(5) = 55.8. Find a different value of
k that will give P(35) = 584.
- In practice, a mathematical model rarely passes through every
experimentally obtained data point,
and so statistical methods must be used to "tune" the
model's parameters to find the curve of "best fit" that describes
the data. In this question we will use linear regression to
find a value for k that describes the given data points.
If we suspect that data fits an exponential model, we can
proceed as follows:
- Take the natural log of the dependent quantity (in our case,
the population, P) so that we get a new data set of the form
- Using almost any scientific calculator, it is easy to find the
line of least squares that fits this data. This gives us
an equation of the form ln(P(t)) = m t + b
where m and b are the slope and intercept
corresponding to the line of best fit. Most calculators
also gives the value of the correlation coefficient that
indicates how well the data is approximated by a line
(a correlation coefficient of 1 or -1 means perfect correlation).
- Exponentiating both sides allows us to find an initial
population and value of the birth rate that best fits
the given data.
Using the table you were given,
carry out this program for the given data.
- Do your four values of k agree closely with each other?
Should they? Which of the four values do you think is the best
model for the growth of Bufo marinus during the years
for which we have data? Use this birth-rate to predict the
toad's range in the year 2039. Given that the area of Australia
is 7,619,000 square kilometers,
how confident are you of this prediction? Explain your reasoning.
You may question the validity of the previous question's assumption
that there is an average of one toad per square kilometer.
Suppose we were wrong and there were actually an average of
two toads per square kilometer.
As before, solve analytically for a value of k that will
guarantee that the curve passes through exactly two of the data
points. In particular, if we now assume that P(0) = 65.6, find
and record a value of k so that P(5) = 111.6, and a
different value of k so that P(35) = 1168. How do these
values of k compare with the values you found in the previous
What does this tell us? Comment on the importance of knowing the
exact average density of the toad population.
For more recent information about the ever growing population of
Bufo marinus, see the
December 1995 issue of FROGLOG (number 15), the Newsletter of the World
Conservation Union (IUCN), Species Survival Commission Declining
Amphibian Populations Task Force (DAPTF). (No joke, it exists!)
Next: Limits on Growth
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