Picture: a map of who-eats-whom over time: A phase portrait shows two interacting quantities on one map.
Picture: a map of who-eats-whom
- Imagine tracking two interacting populations — say lynx (x) and hares (y).
- Instead of two separate graphs against time, draw ONE map with x across and y up.
- Every point is a possible "state" of the system; the arrows show which way the state moves next.
- The curves the state traces out are trajectories, and the whole picture is the phase portrait.
- For a linear system we write the two rates as a matrix product:
The equilibrium sits at the origin: An equilibrium is where nothing changes — both rates are zero at once. For a linear system M(x, y) = 0 has the single solution (x, y) = (0, 0) (whenever M is invertible).
Stable or unstable?: So the only equilibrium is the origin, and the key question is: do nearby trajectories move TOWARD it (stable) or AWAY (unstable)? The eigenvalues of M answer that.
IB-style question — classify a two-species system
A reserve holds two interacting species. With x = thousands of grazers and y = thousands of browsers, the model is dx/dt = x + y and dy/dt = 4x + y (rates in thousands per year).
(a) Write the system as d/dt(x, y) = M(x, y) and state the equilibrium.
(b) Find the eigenvalues of M and classify the equilibrium.
Step by step
- (a) Read the coefficients of x and y straight off the two rate equations to build M.
- Both rates are zero only at the origin, so that is the equilibrium.
- (b) Form the characteristic equation using tr M = 2 and det M = (1)(1) − (1)(4) = −3.
- Factorise (or use the GDC's eigenvalue tool).
Final answer
(a) M = (1, 1; 4, 1), equilibrium at the origin (0, 0). (b) Eigenvalues are 3 and −1 — two REAL eigenvalues of OPPOSITE sign, so the origin is a SADDLE point: unstable. Trajectories are pulled in along one direction but flung out along the other, so the two populations cannot settle at zero.
The classification rule in one line: Read the eigenvalues:
• Two real, same sign → node (both negative = stable sink; both positive = unstable source).
• Two real, opposite signs → saddle (always unstable).
• Complex a ± bi → spiral (stable if a < 0, unstable if a > 0); if a = 0 (purely imaginary) → centre (closed loops).
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Real eigenvectors are the straight-line trajectories: Real eigenvectors give the straight-line trajectories through the origin.
Along an eigenvector
- Along an eigenvector direction the system just scales — it doesn't twist.
- So a trajectory that starts on an eigenvector stays on that straight line through the origin.
- Its eigenvalue decides the motion along that line.
NEGATIVE eigenvalue
- Pulls the state IN toward the origin
- eλt → 0
POSITIVE eigenvalue
- Pushes it OUT
- eλt → ∞
Everything else: Every other trajectory is a blend of the two eigen-directions:
IB-style question — coupled tanks settling to empty
Brine flows between two connected tanks. With x and y the salt mass (kg) in each tank, the model is dx/dt = −2x and dy/dt = x − 4y (per hour).
(a) Find the eigenvalues and a corresponding eigenvector for each.
(b) Write the general solution and describe the long-term behaviour.
Step by step
- (a) Build M from the coefficients, then find the eigenvalues from the characteristic equation (tr M = −6, det M = 8).
- For λ = −2 solve (M + 2I)v = 0: the bottom row gives x − 2y = 0, so v₁ = (2, 1).
- For λ = −4 solve (M + 4I)v = 0: the top row gives 2x = 0, so x = 0 and v₂ = (0, 1).
- (b) Combine into the general solution.
Final answer
(a) λ = −2 with v₁ = (2, 1) and λ = −4 with v₂ = (0, 1). (b) BOTH eigenvalues are negative, so e−2t and e−4t → 0: the origin is a stable node (sink) and every trajectory drains to (0, 0). In context, the salt in both tanks tends to zero — the tanks flush clean. (Assumes the linear mixing model stays valid as masses get small.)
Which way do the arrows point near the origin?
Draw the lines
Draw the eigenvector lines through the origin.
Add arrows
Put arrows IN on a line whose eigenvalue is negative and arrows OUT on a line whose eigenvalue is positive.
Node
The bigger-magnitude eigenvalue is the "fast" direction; curved trajectories leave/arrive tangent to the slow (smaller-magnitude) eigenvector.
Saddle
Trajectories come in along the negative direction and swing out along the positive one.
Lines first, then arrows, then curves.