(This exercise is taken from the preliminary edition of Differential Equations by Blanchard, Devaney, and Hall, PWS Publishing Co.)

In this exercise, a logistic model of population growth is developed and finally includes a term that accounts for "harvesting". You should imagine a fish population subject to various degrees and types of fishing.

1. A differential equation model which is a first approximation of the fish population is

where k is a constant of proportionality and p is the time-dependent fish population. What does k represent physically? Interpret this equation in words. (Make k a slider.)

2. There are limits on the fish's environment such that the population is limited to a certain population, N. We expect the growth of the population to be greatest when p is small and to be zero when p = N. For populations greater than N, we expect the population to asymptotically approach N. Populations less than or equal to zero are either uninteresting or physically impossible. These characteristics may be accounted for by introducing a logistic term into the above equation.

Discuss the behavior of p(t) for different values of p₀ = p(t=0) for a given N. (Make p₀ a slider.)

3. The equation

represents a logistic model of population growth with harvesting at a constant rate a. What will happen to the fish population for various initial conditions? Give results for several values a. (Make a a slider.)

4. The equation

represents a periodic harvesting model. What do the parameters a and b represent? (Make b a slider.) Discuss the behavior of this equation for various initial conditions.