The following system of differential equations arises in the modeling of a stirred tank reactor with an autocatalyic reaction:

\[\begin{aligned}& \frac{d x}{d t}=x-x y \\& \frac{d y}{d t}=-y+x y+y^{2}\end{aligned}\]

Find the steady states by setting the LHS to zero and solving the set of resulting algebraic equations. Note that the system admits two steady states.
Examine the nature of the solution around these steady states. Show a phase-plane plot of the system (namely a plot of \(x\) vs. \(y\) at each instant of time).