Solved: Invasion of the Marine Toads* In 1935, the poisonous American marine toad (Bufo

Chapter 2, Problem 51

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In 1935, the poisonous American marine toad (Bufo marinus) was introduced, against the advice of ecologists, into some of the coastal sugar cane districts in Queensland, Australia, as a means of controlling sugar cane beetles. Due to lack of natural predators and the existence of an abundant food supply, the toad population grew and spread into regions far from the original districts. The survey data given in the accompanying table indicate how the toads expanded their territorial bounds within a 40-year period. Our goal in this problem is to find a population model of the form \(P\left(t_{i}\right)\) but we want to construct the model that best fits the given data. Note that the data are not given as number of toads at 5-year intervals since this kind of information would be virtually impossible to obtain.

(a) For ease of computation, let’s assume that, on the average, there is one toad per square kilometer. We will also count the toads in units of thousands and measure time in years with t = 0 corresponding to 1939. One way to model the data in the table is to use the initial condition \(P_{0}=32.8\) and to search for a value of k so that the graph of \(P_{0} e^{k t}\) appears to fit the data points. Experiment, using a graphic calculator or a CAS, by varying the birth rate k until the graph of \(P_{0} e^{k t}\) appears to fit the data well over the time period \(0 \leq t \leq 35\).

Alternatively, it is also possible to solve analytically for a value of k that will guarantee that the curve passes through exactly two of the data points. Find a value of k so that \(P(5)=55.8\). Find a different value of k so that \(P(35)=584\).

(b) In practice, a mathematical model rarely passes through every experimentally obtained data point, and so statistical methods must be used to find values of the model’s parameters that best fit experimental data. Specifically, we will use linear regression to find a value of k that describes the given data points: • Use the table to obtain a new data set of the form \(\left(t_{i}, \ln P\left(t_{i}\right)\right)\), where \(P\left(t_{i}\right)\) is the given population at the times \(t_{1}=0\), \(t_{2}=5\), . . . .

• Most graphic calculators have a built-in routine to find the line of least squares that fits this data. The routine gives an equation of the form ln \(P(t)=m t+b\), where m and b are, respectively, the slope and intercept corresponding to the line of best fit. (Most calculators also give 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. A correlation coefficient near 0 may mean that the data do not appear to be fit by an exponential model.)

• Solving ln \(P(t)=m t+b\) gives \(P(t)=e^{m t+b}\) or \(P(t)=\) \(e^{b} e^{m t}\). Matching the last form with \(P_{0} e^{k t}\), we see that \(e^{b}\)  is an approximate initial population, and m is the value of the birth rate that best fits the given data.

(c) So far you have produced four different values of the birth rate k. 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 the toad population 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 \(7,619,000 \mathrm{~km}^{2}\), how confident are you of this prediction? Explain your reasoning.