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 initial population, and
- the net birth rate.

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 (*Bufo marinus*, in Australia and
its effects on indigenous vertebrates," *Mem. Queensland Mus*,
**17**: 305-310*Bufo marinus* in Queensland, Australia
from 1935 to 1974", *Copeia*, no. 3, 1981,
pp. 676-680

Year Area Occupied (square km) ------- ---------------------------- 1939 32,800 1944 55,800 1949 73,600 1954 138,000 1959 202,000 1964 257,000 1969 301,000 1974 584,000

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?

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 the 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 (*t*, ln(_{i}*P(t*)._{i}) - 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.

- Take the natural log of the dependent quantity (in our case,
the population,
- 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.

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
question?

What does this tell us? Comment on the importance of knowing the exact average density of the toad population.

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Created: May 15 1996 ---
Last modified: May 15 1996

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