Phase lines
A phase line is a one-dimensional diagram showing the direction in which a variable changes between equilibria.
Sign of the derivative
For
\[\frac{dx}{dt}=f(x),\]the sign of \(f(x)\) determines the direction of motion. If \(f(x)>0\), then \(x\) increases. If \(f(x)<0\), then \(x\) decreases.
Example
For logistic growth,
\[\frac{dN}{dt}=rN\left(1-\frac{N}{K}\right),\]with \(r>0\), the equilibria are \(0\) and \(K\). For \(0<N<K\), the population increases; for \(N>K\), it decreases.
0 → → → K ← ← ←
Key idea. A phase line gives a qualitative picture of one-dimensional dynamics without solving the differential equation explicitly.