TEACHER NOTES

ISSUES RELATED TO THE PROBLEM

Here is the Mathematica code to generate the data used in the problem.

Input := 

popEq = p'[t] == .3 p[t] - .3/800 p[t]^2

Input := 

sol = DSolve[{p'[t] == .3 p[t] - .3/800 p[t]^2, p[0] == 40},
			p[t],t]

Input := 

f[t_] = p[t]/.sol[[1]] 

Input := 

Plot[f[t],{t,0,20}]

Here is the data we intend to give to students:

Input := 

data = Table[{i, Floor[f[i]]},{i,0,20}];

In questions 1-3 we ask a few short questions on the qualitative behavior of the logistic equation and attempt to infer some ecological and semantic meaning to the constant K - carrying capacity.

The major question is really question 4.

In question 4 the objective is to estimate parameters in a problem, in this case the growth rate r and the carrying capacity K, using observed data where the model may not be in closed form originally. We also give several other opportunities for estimating parameters in a differential equation given some observed data by using numerical derivatives as estimates of the derivative and fitting data to these approximations to the differential equation.

So for p'(t) = r p(t) (K - p(t))/K we are given some data and we seek to estimate r and K, first using a number of different ways without resorting to solving the differential equation, and second with a solution of the differential equation in hand.

We give several suggestions in the question itself about how to proceed, but it is up to the students to discover the approaches and use the software appropriately, e.g. manipulation of data and parts of data, the use of symmetric differences or non-symmetric differences as numerical derivatives to approximate the derivative, and the solution of the differential equation followed by setting up a non-linear least square function with the data, in the parameters r and K.

Questions (1) - (3), being warm-up questions can be done in class in about 45 minutes (maximum), small group efforts will assure all understand the issues in that time.

Question (4) should be given considerable time, with in class group efforts and suggestions by the teacher, e.g., on numerical derivatives which are not often done.

The class may determine a final comparison method, but we offer comparing the sum of the squares between the population values from the analytic solution to the differential equation with the parameters used and the observed population data for each time population data pair. The class may use how the data looks when plotted over the curve with the estimated parameters as a comparison.

This data is very clean and should give rise to an excellent fit. One could offer up dirty data by using the function Pop(t) offered in the solution to Question (4), generating the numbers for a particular set of parameters r and K and then adding some noise to that data. But getting data which is too dirty at this stage would be detrimental as the students would not see the good fits.

Effectively one could use p(t) = (Exp[r * t] * k)/(-1 + Exp[r * t] + k/p(0)) to compute new data. where r and K are the parameters to be estimated (selected to generate the problem by the teacher) and p(0) is the initial population level at time t = 0.

Prerequisites

Some fitting of functions to dasta, solution of first-order differential equation.

Time allotment - time management

This could be a several days project, for it demands data fitting, solutions to differential equation, and discovery of technique by students. Suggesting one of the techniques proposed would shorten the time.

Expectations

Future payoffs

Extensions

References and Sources