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### Problems

1. Show that this system has two fixed points at N=0 and N=K.
2. Sketch out the direction sets for the logistic equation for r=.2 and K=100 and discuss the stability of the two fixed points. Hint: start with a graph where t goes from 0-40 and N goes from 0-150.
3. What do you expect to happen if we start with a population that is smaller than the carrying capacity?
4. How about larger than the carrying capacity?
5. Do you expect any qualitatively different behavior if you change the growth rate r or the carrying capacity K?
6. Now use Stella to quantitatively solve the logistic equation and test your intuition. Produce a plot showing several trajectories for r=.2, K=100 and different starting populations.
7. Do you think that this is a very interesting model? Why or why not?

#### Extra credit problems

Okay, if you liked that then here are a few more things to ponder.

1. Non-constant K There is no reason to believe that the carrying capacity needs to be constant either. If you were pessimistic, you might expect that the carrying capacity might decrease with a higher population (pollution, war, disease). Alternatively you could be an optimist and think that the more people, the more chances of finding new solutions to increase the carrying capacity (better technology, medicine whatever).For fun, pick a world that suits your personality, then formulate and analyze that model. Then go read Joel Cohen's book ``How many people can the earth support''.
2. Forced systems Another way that the carrying capacity might change is that it might be forced to change with time for reasons other than the change in populations. For example the carrying capacity might change on a seasonal or decadal time scale (you know...summertime...and the living is easy etc.). One simple model is that K oscillates with some amplitude A and period T e.g. . This one is quite tricky but try to understand the global behavior of the system in the limits of large amplitude swings and when the period is long or short compared to the growth rate of the species. In addition to solutions, it is also interesting to plot the null-clines which are the values of N and T where the growth-rate is zero. Have fun and may the force be with you.

Next: Lab 2: Life on Up: Limits to growth: the Previous: Model Formulation

marc spiegelman
Mon Sep 22 21:30:22 EDT 1997