Almost all of this material refers back to stuff we did a while ago, but it still makes for interesting exploration for what a system of ordinary differential equations means.

1. The Volterra-Lotke equations for describing a model of interaction between a predator and its prey are described in Chapter 5.3. Add a term to this system of equations which would describe the prey also being hunted by human hunters. You can assume that there are game laws which prescribe the number of man hours that can be spent hunting, if this helps you decide how to make the model. (It seems reasonable to assume that few woman hours are spent on recreational hunting). Explain the terms you have added.
 

2. The following comes form Strogatz's book  titled "Non-linear Dynamics and Chaos". In (his) Chapter 5.3, he says "To arose your interest in the classification of linear systems, we now discuss a simple model for the dynamics of love affairs".
 

In short, we have Romeo and Juliet in a dynamic love affair. We give a numerical value to Romeo's love for Juliet, called R, and a numerical value to Juliet's love for Romeo, called J. We assume that each one responds to the given emotional situation by either increasing or decreasing their love according to how much love they feel for each other. For example , if both are cautious (meaning they tend to regret falling in love or hate and reverse the situation) but respond enthusiastically to the other's emotional state, and they have like responses we might model this as:

                          dR/dt = - a R + bJ
                          dJ/dt = b R - aJ.

Here we really mean for a and b to be positive. Here are two other situations for you  to model.

2.a Romeo and Juliet are still alike, but they are very self-centered. When they fall in love (or hate), these feeling tend to increase. But when another person demonstrates love or hate, it tends to decrease their own love or hate for the person. What is a reasonable model for this? Explain.

2.b Romeo and Juliet have no self-knowledge and do not even notice their own emotional states. But Romeo is very responsive to Juliet. His love (or hate) is very much inspired to change in the direction of Juliet's feelings. If she is in love with him, he tends to fall more in love, and the same with hate. On the other hand, the more Romeo loves Juliet, the more she wants to run away. But if Romeo actively seems to dislike her, she begins to find him more and more attractive. Can you also model this situation? Explain.

3.  Can you guess what the pattern to the three different types of love affairs is likely to be? You will need to either look a little ahead in the mathematics, or use the computer (Agreed, it is not your week to use the computer).

4.  Make up another scenario for the dynamics of a relationship and model it.

5.  And now for something completely different.  Taubes, page 215, problem 18.