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, In this case K(t) is monotone increasing, and K(t) is asymptotic to 1. This might model a situation of a human population where, due to technological improvement, the availability of resources is increasing with time, although ultimately limited.