Phase planes
A phase plane is a geometric picture of a dynamical system with two state variables. Instead of plotting each variable separately against time, we plot one variable against the other.
Start with a two-variable system
Consider
\[\frac{dx}{dt}=f(x,y),\qquad\frac{dy}{dt}=g(x,y).\]The state of the system is
\[(x(t),y(t)).\]At any instant, the pair \((x,y)\) tells us where the system is in the phase plane.
What do the axes mean?
The horizontal axis is one state variable, for example prey abundance \(N\). The vertical axis is another, for example predator abundance \(P\).
Time is not shown on an axis. Instead, time is represented by movement along the trajectory.
The velocity vector
At each point \((x,y)\), the system has instantaneous velocity
\[\boxed{\left(\frac{dx}{dt},\frac{dy}{dt}\right)=\bigl(f(x,y),g(x,y)\bigr)}.\]The first component tells us horizontal motion. The second tells us vertical motion.
| Signs | Direction |
|---|---|
| \(dx/dt\gt0\), \(dy/dt\gt0\) | right and up |
| \(dx/dt\gt0\), \(dy/dt\lt0\) | right and down |
| \(dx/dt\lt0\), \(dy/dt\gt0\) | left and up |
| \(dx/dt\lt0\), \(dy/dt\lt0\) | left and down |
One point and its direction
From vectors to a direction field
If we draw a small direction vector at many points, we obtain a vector field or direction field. This shows how the system would move from many different initial states.
The arrows do not show one particular solution. They show the local direction the differential equations prescribe at each point.
A trajectory
A trajectory is the path traced by one solution starting from a particular initial state
\[(x(0),y(0)).\]It must always follow the direction field because its tangent direction is determined by the differential equations.
Many initial conditions give many trajectories
Different initial states generally produce different trajectories. Drawing several trajectories gives a phase portrait.
A phase portrait can reveal whether states approach an equilibrium, move away, spiral, cycle, or separate into different long-term outcomes.
Equilibria in the phase plane
An equilibrium \((x^*,y^*)\) satisfies
\[f(x^*,y^*)=0,\qquad g(x^*,y^*)=0.\]At an equilibrium the velocity vector is
\[(0,0),\]so the state does not move.
Nullclines organise the phase plane
The \(x\)-nullcline is where
\[f(x,y)=0,\]and the \(y\)-nullcline is where
\[g(x,y)=0.\]Their intersections are equilibria. Between the nullclines, the signs of the two derivatives determine the broad direction of motion.
How nullclines and vectors work together
On the \(x\)-nullcline there is no horizontal motion, so vectors are vertical. On the \(y\)-nullcline there is no vertical motion, so vectors are horizontal.
A trajectory can cross a nullcline because the other component can still be changing.
Stable equilibrium in a phase plane
Consider the simple system
\[\frac{dx}{dt}=1-x,\qquad\frac{dy}{dt}=1-y.\]The equilibrium is
\[(x^*,y^*)=(1,1).\]Both variables move toward 1, so all nearby trajectories approach the equilibrium.
Why this equilibrium is stable
If \(x\lt1\), then \(dx/dt\gt0\); if \(x\gt1\), then \(dx/dt\lt0\). So horizontal motion pushes toward \(x=1\).
The same reasoning pushes \(y\) toward \(1\). Together, the two components move the state toward \((1,1)\).
Spirals and oscillations
Some systems rotate while moving toward or away from an equilibrium. In a stable spiral, trajectories circle the equilibrium while their distance from it decreases.
In time-series form, this appears as damped oscillation in the state variables.
Saddles
A saddle point is stable in one direction but unstable in another. Some special trajectories approach the equilibrium, but nearby trajectories in other directions move away.
Closed orbits
A closed trajectory represents periodic behaviour. The system repeatedly returns to the same sequence of states.
The classical Lotka–Volterra predator–prey model has closed orbits around its coexistence equilibrium under its idealised assumptions.
Predator–prey phase plane
For
\[\frac{dN}{dt}=rN-aNP,\] \[\frac{dP}{dt}=eaNP-mP,\]the positive nullclines are
\[P=\frac{r}{a},\qquad N=\frac{m}{ea}.\]These divide the phase plane into four regions with different prey and predator directions.
How to read the predator–prey orbit
When prey are abundant and predators are relatively low, both may initially increase. As predators rise, prey eventually cross their nullcline and begin to decline.
Later, prey become too scarce to sustain predator growth, so predators decline. Reduced predator pressure then permits prey recovery.
Time therefore moves around the orbit in a definite direction.
Phase plane versus two time-series graphs
A time-series plot answers “how does each variable change with time?” A phase-plane plot answers “how do the two variables change relative to one another?”
The same solution can be represented in both ways.
What is a separatrix?
A separatrix is a trajectory that separates regions with different long-term outcomes.
For systems with multiple stable states, initial conditions on different sides of a separatrix can approach different attractors.
Basins of attraction
The basin of attraction of a stable equilibrium is the set of initial states whose trajectories approach it.
A phase plane can make these basins visually clear when more than one attractor exists.
Biological feasibility
For two population variables, the biologically meaningful region is usually
\[x\ge0,\qquad y\ge0.\]The mathematical phase plane may extend into negative values, but negative populations are normally ignored.
Invariant boundaries
In many population models, axes such as \(x=0\) or \(y=0\) are invariant. If a population is absent and there is no immigration, trajectories starting on that axis remain there.
This is useful when identifying biologically feasible regions.
Phase planes and stability analysis
The phase portrait can suggest whether an equilibrium is attracting or repelling, but local stability is usually confirmed using the Jacobian matrix and eigenvalues.
The geometry and the algebra support each other: the phase plane gives intuition, while the Jacobian gives a local mathematical test.
Phase planes and numerical simulation
Many nonlinear systems cannot be solved explicitly. Numerical integration can still generate trajectories in the phase plane.
This allows us to explore how different initial conditions move through state space and whether the numerical behaviour agrees with nullcline and stability analysis.
What a phase plane cannot show directly
A trajectory shows the sequence of states but not the speed of movement unless additional information is added. Two parts of a curve may look similar while the system moves through them at very different rates.
A phase plane is also limited to two displayed variables. Higher-dimensional systems require projections, slices or other visualisations.
A practical workflow
For a two-variable biological model, first identify the biologically feasible region. Find the nullclines and equilibria. Determine derivative signs in each region. Draw or compute the vector field. Add representative trajectories from biologically relevant initial conditions. Finally, use the Jacobian and eigenvalues to classify equilibria when needed.