stability of equilibria
A marble resting at the bottom of a bowl and a marble balanced on top of a dome are both, technically, at rest — but nudge each one and they behave completely differently. The marble in the bowl rolls back; the marble on the dome rolls away. Stability is the study of that difference: it asks whether a steady state, once disturbed a little, returns to itself or runs away from itself.
For an autonomous equation dy/dt = f(y), an equilibrium y* is a value where f(y*) = 0, so a solution sitting exactly there never moves. The question is what happens to a solution that starts slightly off. If nearby solutions are pulled back toward y* over time, the equilibrium is stable (also called asymptotically stable, or a sink). If even the tiniest displacement grows and solutions flee, it is unstable (a source). There is a quick test: look at the slope f'(y*). If f'(y*) < 0, the equilibrium is stable; if f'(y*) > 0, it is unstable; if f'(y*) = 0, the linear test is inconclusive and you fall back on the sign of f on either side (the phase line).
Stability is the single most important qualitative property of a model, because the stable equilibria are the states you actually see in the long run — terminal velocity, the carrying capacity, the steady temperature of a cooling object. Unstable equilibria are real but invisibly fragile: the system can sit at one only in perfect, disturbance-free theory. This first-order idea is the seed of the much richer stability theory for systems and nonlinear dynamics.
For dy/dt = y(1 − y), check slopes with f'(y) = 1 − 2y. At y = 0, f'(0) = 1 > 0, so y = 0 is unstable; at y = 1, f'(1) = −1 < 0, so y = 1 is stable — agreeing with the phase line.
f'(y*) < 0 means stable (sink); f'(y*) > 0 means unstable (source).
The slope test is inconclusive when f'(y*) = 0; then you must inspect the sign of f on both sides, because the equilibrium might be semistable or a degenerate case the linearization cannot see.