The logistic equation P′ = r P(1 − P/K) is discussed in Examples 5 and 6. Usually the para...

The logistic equation P′ = r P(1 − P/K) is discussed in Examples 5 and 6. Usually the parameter’s r and K are constants and in Example 5 we found that for any solution P(t) which has a positive initial value we have P(t) → K as t→ ∞. For this reason K is called the carrying capacity of the system. However, in Example 6 we saw a case where the carrying capacity is not constant, yet we were able to show how the limiting behavior of the population related to the carrying capacity. In Exercise you are to examine the long term behavior of solutions, especially in comparison to the carrying capacity. In particular:

a) Use dfield6 to plot several solutions to the equation. (It is up to you to find a display window that is appropriate to the problem at hand.)

b) Based on the plot done in part a), describe the long term behavior of the solutions to the equation. In particular, compare this long term behavior to that of K. It might be helpful to plot K on the display window as we did in Example 6. In the first case the solutions will all be asymptotic to a constant. In the other two the solutions will all have the same long term behavior. Describe that behavior in comparison to the graph of K. The results of Examples 5 and 6 should be helpful.

, r = 1. This is perhaps the most interesting case. Here the carrying capacity is periodic in time with period 1, which should be considered to be one year. This models a population of insects or small animals that are affected by the seasons. You will notice that the long term behavior as t → ∞ reflects the behavior of K. The solution does not tend to a constant, but nevertheless all solutions have the same long term behavior for large values of t. In particular, you should take notice of the location of the maxima and minima of K and of P and how they are related. You can use the “zoom in” option to get a better picture of this.