The point is stable since as time passes the population approaches the equilibrium. It is actually globally stable since the point is approached over time regardless of the starting conditions.

(NB - The terminology is tricky here since the total number of individuals in the system may depend on the exact starting conditions, but the ratios among all the variables reach a fixed quantity.)

There is no need to do a local stability analysis, since for linear models the local stability matrix is the same as the transition matrix (try it!). Studying the transition matrix already tells you everything you need to know about how the population will change near the equilibrium.