Chapter 3.1, Problem 1E

Recall the model
$\begin{array}{l}\frac{dx}{dt}=ax+by\hfill \\ \frac{dy}{dt}=cx+dy\hfill \end{array}$
for Paul's and Bob's cafés, where $x(t)$ is Paul's daily profit, $y(t)$ is Bob's daily profit, and $a, b, c,$ and $d$ are parameters governing how the daily profit of each store affects the other. In Exercises $1-4,$ different choices of the parameters $a, b, c,$ and $d$ are specified. For each exercise write a brief paragraph describing the interaction between the stores, given the specified parameter values. [For example, suppose $a=1,c=-1,$ and $b=d=0.$ If Paul's store is making a profit $(x>0),$ then Paul's profit increases more quickly (because $ax>0$ ). However, if Paul makes a profit, then Bob's profits suffer (because $cx<0$ ). since $b=d=0,$ Bob's current profits have no impact on his or Paul's future profits.]

1. $a=1,b=-1,c=1,$ and $d=-1$