Equilibrium Solutions

An equilibrium solution is simply a value where the system can sit still forever: plug it in and nothing changes. The word means "balanced", like a scale with equal weights, because whatever pushes $y$ up is exactly canceled by what pushes it down. Equilibria tell you where a system ends up in the long run without solving anything. All we're really asking is: where can this system sit still, and if you nudge it, does it come back? The formula box looks like a lot, but it's just three small statements: $f(y^*) = 0$ (at $y^*$ nothing changes, so it's a resting point), $f'(y^*) < 0$ (nudges get pushed back, so it's stable) and $f'(y^*) > 0$ (nudges get pushed further, so it's unstable). Think of a marble in a bowl versus a marble balanced on top of one. It looks complex, but it's just a few simple parts, so bear with it.

You saw these in the direction fields lesson as the horizontal lines where every segment is flat. Now we'll sort them out without drawing the whole field.

Phase line. Draw the $y$-axis, mark the zeros of $f$, and put an arrow up where $f > 0$ and down where $f < 0$. That's all the information you need, because in an autonomous equation the slope only depends on $y$: every point at the same height behaves the same way.

• Arrows point toward an equilibrium on both sides: stable (an attractor).
• Arrows point away on both sides: unstable (a repeller).
• Toward on one side, away on the other: semi-stable.

It makes sense that the derivative test works. If $f'(y^*) < 0$, then $f$ goes from positive to negative as you pass $y^*$, so the arrows point up from below and down from above: toward $y^*$ on both sides.

A common mistake. It's tempting to think $f'(y^*) = 0$ means the equilibrium is semi-stable, because it's the case in between stable and unstable. It isn't necessarily. When $f'(y^*) = 0$ the test just tells you nothing, and you have to check the signs of $f$ directly. $y' = (y - 1)^2$ is semi-stable at $1$, but $y' = -(y - 1)^3$ is stable there and $y' = (y - 1)^3$ is unstable, and all three have $f'(1) = 0$.

Solutions can never cross an equilibrium (uniqueness), so each one heads monotonically toward the next equilibrium in the direction of its arrows, or off to infinity.

Fisheries management uses equilibria and their stability to set catch limits: harvest too much and a stable population can collapse past a tipping point.