Paul and Bob decide to open up a store that sells CD's in part of their town. They open up two shops, respectively, called "Pauls Shop" and "Bob's Shop". They worry about competition, and obviously want to attract as many customers as possible. Two stores may attract more customers, or it may cause immense competition due to limited customers. They do some math. Paul lets his daily profit be denoted as x(t). And Bob lets his daily profit be denoted as y(t). So, for example, if x(t) > 0, that means Paul is making money. And if x(t) < 0, then he's obviously losing money.They create the linear model below that shows how each store has an impact on the other:

$\displaystyle \frac{dx}{dt} = ax + by$
$\displaystyle \frac{dy}{dt} = cx + dy$

1.) Focusing on Paul's profits, explain what $\displaystyle ax$ and $\displaystyle by$ mean. Separately explain the signs of $\displaystyle a$ and $\displaystyle b$ while $\displaystyle x > 0, y > 0$. (NOTE: YOU HAVE TO CONSIDER 4 CASES). Explain what might be happening to business.

2.) A model for this situation is given by:

X' = ([[-2, -3],[-3, -2]])X

a.) What does the model say about Paul's and Bob's profits given that we know Paul's profits are positive? Explain why this might happen.

b.) Solve the system and illustrate a phase portrait.

c.) Explain what happens with profits of the two stores assuming this model is accurate. Consider cases, in particular, $\displaystyle x(0) = y(0), x(0) < y(0), x(0) > y(0)$. (Note: suppose that x and y differ by a tiny amount at first)