Problem $3.($Two Caf$$s Model$)$ After retiring from writing differential equations textbooks, Paul and Bob both decide to open small caf$$s near campus. "Paul's High Test Coffee" and "Bob's Gourmet Tea" open on the same block, and Paul and Bob soon become concerned about the effect each caf$$ has on the other. Having two beverage shops so close might make their block a more popular destination for students. On the other hand, the caf$$s might compete for a limited supply of thirsty customers. Paul and Bob argue about this until they are so sick of arguing that they hire their former colleague and famous mathematician Glen to settle the matter. Glen decides to make a differential equations model for the relationship between the profits of the two caf$$s$.$ Remembering all that he learned from Paul and Bob, Glen starts with as simple a model as possible and suggests the following system.

$x(t)=$ daily profit $of$ Paul's caf$at$ time $t$

$y(t)=$ daily profit $of$ Bob's caf$at$ time $t$

Glen formulates the simplest possible model that allows each caf$$ to affect the other $-$ a linear model. The system is

$\frac{dx}{dt}=-2x-y$

$\frac{dy}{dt}=-x-y,$

Determine the type of the equilibrium point at the origin.

Compute and sketch the straight$-$line solutions in the phase plane.

Draw a phase portrait that includes a minimum of four additional solution curves, evenly spaced across the same phase plane, to illustrate diverse system dynamics.

Describe the fates of Bob's and Paul's caf$$s$.$