Phase lines
A phase line is a one-dimensional picture of a differential equation. It shows the equilibria and the direction in which the state moves between them.
Start with an autonomous equation
Consider
\[\frac{dx}{dt}=f(x).\]The rate of change depends only on the current value of \(x\). At each value of \(x\), the sign of \(f(x)\) tells us the direction of motion.
The three possibilities
| Sign | Meaning | Phase-line direction |
|---|---|---|
| \(f(x)\gt0\) | \(x\) increases with time | toward larger \(x\) |
| \(f(x)\lt0\) | \(x\) decreases with time | toward smaller \(x\) |
| \(f(x)=0\) | instantaneous change is zero | equilibrium candidate |
Why arrows appear on the line
If \(dx/dt\gt0\), then as time passes the numerical value of \(x\) becomes larger. On a horizontal number line, larger values lie to the right, so the arrow points right.
If \(dx/dt\lt0\), the value becomes smaller, so the arrow points left.
Step 1: find the equilibria
Equilibria satisfy
\[f(x)=0.\]These values divide the phase line into intervals. Within each interval, determine the sign of \(f(x)\).
Step 2: make a sign table
Suppose the equilibria are \(x_1^*\) and \(x_2^*\). Then examine
\[x\lt x_1^*,\qquad x_1^*\lt x\lt x_2^*,\qquad x\gt x_2^*.\]Usually one convenient test value from each interval is enough to determine the sign.
Step 3: convert signs into arrows
Positive intervals receive right-pointing arrows. Negative intervals receive left-pointing arrows. The resulting diagram immediately reveals much of the qualitative behaviour.
Worked example: logistic growth
Consider
\[\frac{dN}{dt}=rN\left(1-\frac{N}{K}\right),\qquad r\gt0,\ K\gt0.\]First find the equilibria:
\[rN\left(1-\frac{N}{K}\right)=0.\]Therefore
\[N^*=0\qquad\text{and}\qquad N^*=K.\]Determine the signs
For the biologically relevant region \(N\ge0\):
If \(0\lt N\lt K\), then both \(N\) and \(1-N/K\) are positive, so
\[\frac{dN}{dt}\gt0.\]If \(N\gt K\), then \(1-N/K\lt0\), so
\[\frac{dN}{dt}\lt0.\]The logistic phase line
Read the diagram biologically
Suppose the population starts at \(N_0\) with
\[0\lt N_0\lt K.\]The arrow points toward larger populations, so \(N(t)\) increases. If instead \(N_0\gt K\), the arrow points toward smaller populations, so \(N(t)\) decreases.
Both cases suggest
\[N(t)\to K\]for positive initial populations in this model.
Phase lines reveal stability
If arrows on both sides point toward an equilibrium, it is attracting. If they point away on both sides, it is repelling.
Attracting equilibrium
The sign pattern
\[+\quad|\quad-\]means the state increases on the left and decreases on the right. Both directions lead toward the equilibrium, so it is locally asymptotically stable.
Repelling equilibrium
The sign pattern
\[-\quad|\quad+\]means trajectories move away on both sides. The equilibrium is unstable.
Semistable equilibrium
If arrows point toward the equilibrium from one side but away from it on the other, the equilibrium is sometimes called semistable.
For example, the same sign on both sides can produce
\[+\quad|\quad+\]so trajectories move right on both sides: toward the equilibrium from the left but away from it on the right.
Example with an Allee threshold
Consider the strong-Allee-effect form
\[\frac{dN}{dt}=rN\left(1-\frac{N}{K}\right)\left(\frac{N}{A}-1\right),\qquad 0\lt A\lt K.\]The equilibria are
\[N^*=0,\qquad N^*=A,\qquad N^*=K.\]For positive populations, the signs are
\[0\lt N\lt A:\quad \frac{dN}{dt}\lt0,\] \[A\lt N\lt K:\quad \frac{dN}{dt}\gt0,\] \[N\gt K:\quad \frac{dN}{dt}\lt0.\]The Allee-effect phase line
Why the threshold is unstable
Just below \(A\), the population decreases and moves away from \(A\). Just above \(A\), it increases and again moves away from \(A\).
Therefore \(A\) is an unstable equilibrium even though it is biologically important.
Phase line versus time-series graph
A phase line has state on its axis. A time-series graph has time on the horizontal axis and state on the vertical axis.
They answer different questions. The phase line summarises possible directions from many initial states at once; a time series shows one particular solution through time.
Phase line versus graph of \(f(x)\)
Another useful graph plots \(f(x)\) vertically against \(x\) horizontally. Where this curve crosses the horizontal axis, \(f(x)=0\), so those crossings correspond to equilibria.
Above the axis, \(f(x)\gt0\) and the phase-line arrows point right. Below it, \(f(x)\lt0\) and they point left.
The function graph and phase line contain the same sign information
You do not need the explicit solution
This is one of the main strengths of phase-line analysis. Without solving for \(N(t)\), we can often identify equilibria, directions of movement, stability, thresholds and likely long-term outcomes.
What phase lines cannot show
A phase line tells us direction but not directly how quickly the state moves. Two points can have arrows in the same direction while having very different values of \(|f(x)|\).
It also applies directly only to one-dimensional autonomous systems. Systems with two or more state variables require phase planes or higher-dimensional state-space methods.
Biological restrictions
Mathematically, a phase line may extend into negative values. Biologically, negative population sizes or negative compartment counts are usually meaningless.
The biologically feasible region should therefore be identified before interpreting the diagram.
A useful workflow
For \(dx/dt=f(x)\), first solve \(f(x)=0\). Put the equilibria in numerical order. Use them to divide the line into intervals. Determine the sign of \(f(x)\) in each interval. Draw right arrows for positive signs and left arrows for negative signs. Finally, classify each equilibrium and interpret the result biologically.