Stability
Stability describes what happens after a dynamical system is disturbed from an equilibrium. Does it remain nearby, return to the equilibrium, or move away?
Equilibrium and stability are different questions
For
\[\frac{dx}{dt}=f(x),\]an equilibrium \(x^*\) satisfies
\[f(x^*)=0.\]This tells us where a constant solution is possible. It does not tell us what happens if \(x\) is slightly above or below \(x^*\).
The simplest intuition
Imagine three shapes: a ball at the bottom of a bowl, a ball balanced on top of a hill, and a ball resting on a perfectly flat surface.
After a small displacement, the first tends to return, the second moves farther away, and the third can remain at its new position. These correspond to different mathematical notions of stability.
Stable versus asymptotically stable
An equilibrium is stable if sufficiently small disturbances keep the trajectory sufficiently close to it.
It is asymptotically stable if it is stable and nearby trajectories also approach the equilibrium as
\[t\to\infty.\]So asymptotic stability is stronger than stability alone.
Unstable equilibrium
An equilibrium is unstable when arbitrarily small disturbances can cause trajectories to move away from it.
The system may move away in one direction, several directions, or through growing oscillations.
See stability on a phase line
Why the arrows tell us stability
If \(f(x)\gt0\), then \(x\) increases, so the phase-line arrow points to the right. If \(f(x)\lt0\), then \(x\) decreases, so the arrow points to the left.
Therefore, if the arrows on both sides point toward \(x^*\), nearby states return. If they point away, nearby states separate from the equilibrium.
The derivative test in one dimension
Suppose \(x^*\) is an equilibrium and \(f\) is differentiable. Near \(x^*\),
\[f(x)\approx f(x^*)+f'(x^*)(x-x^*).\]Since \(f(x^*)=0\),
\[\frac{dx}{dt}\approx f'(x^*)(x-x^*).\]Let
\[u=x-x^*.\]Then approximately
\[\frac{du}{dt}=f'(x^*)u.\]Its solution behaves like
\[u(t)=u(0)e^{f'(x^*)t}.\]This gives the familiar test:
\[\boxed{f'(x^*)\lt0\Rightarrow\text{locally asymptotically stable}},\] \[\boxed{f'(x^*)\gt0\Rightarrow\text{unstable}}.\]What if \(f'(x^*)=0\)?
The derivative test is then inconclusive. Higher-order terms or a direct sign analysis of \(f(x)\) may be needed.
Example: logistic growth
Consider
\[\frac{dN}{dt}=rN\left(1-\frac{N}{K}\right),\qquad r\gt0.\]The equilibria are
\[N^*=0\qquad\text{and}\qquad N^*=K.\]Define
\[f(N)=rN\left(1-\frac{N}{K}\right).\]Then
\[f'(N)=r-\frac{2rN}{K}.\]At \(N^*=0\),
\[f'(0)=r\gt0,\]so extinction is unstable for positive \(r\). At \(N^*=K\),
\[f'(K)=-r\lt0,\]so the carrying-capacity equilibrium is locally asymptotically stable.
The biological meaning of the logistic result
If a small positive population is introduced near zero, it grows away from extinction. Near carrying capacity, a population below \(K\) grows and one above \(K\) declines.
Both directions therefore push the population toward \(K\).
Time trajectories make the same idea visible
Local stability
Local stability concerns initial conditions sufficiently close to an equilibrium. It answers a nearby-disturbance question.
This is exactly what derivative and Jacobian tests usually determine.
Global stability
An equilibrium is globally attracting within a specified biologically relevant region if trajectories from all initial states in that region approach it.
Global conclusions are stronger and generally require more than a local derivative calculation.
Why local does not imply global
A nonlinear system can contain several equilibria or attractors. A trajectory starting near one stable equilibrium may approach it, while a trajectory starting farther away may approach another state.
Therefore a locally stable equilibrium need not control the whole state space.
Basins of attraction
The basin of attraction of a stable state is the set of initial conditions whose trajectories approach that state.
When several attractors exist, their basins describe which initial conditions lead to which long-term outcomes.
Stability in two or more dimensions
For
\[\frac{d\mathbf x}{dt}=\mathbf F(\mathbf x),\]let \(\mathbf x^*\) be an equilibrium. Write a small displacement as
\[\mathbf u=\mathbf x-\mathbf x^*.\]Near the equilibrium,
\[\frac{d\mathbf u}{dt}\approx J\mathbf u,\]where \(J\) is the Jacobian matrix evaluated at \(\mathbf x^*\).
The Jacobian
\[J=\left[\frac{\partial F_i}{\partial x_j}\right]_{\mathbf x=\mathbf x^*}.\]The Jacobian is the multidimensional version of the derivative \(f'(x^*)\). It describes how small disturbances in each variable affect the rates of change of all variables near the equilibrium.
Eigenvalues generalise the one-dimensional test
For the linearised system, disturbances can be decomposed into characteristic directions. Their growth or decay is controlled by the eigenvalues \(\lambda_i\) of \(J\).
If every eigenvalue has negative real part,
\[\boxed{\operatorname{Re}(\lambda_i)\lt0\quad\text{for all }i},\]the equilibrium is locally asymptotically stable.
If at least one eigenvalue has positive real part, the equilibrium is unstable.
Why the real part matters
A mode associated with
\[\lambda=a+bi\]contains a factor
\[e^{at}.\]The real part \(a\) therefore controls growth or decay. The imaginary part \(b\) produces oscillation.
Common two-dimensional behaviours
| Eigenvalues | Typical local behaviour |
|---|---|
| two negative real eigenvalues | stable node |
| complex pair with negative real part | stable spiral; damped oscillations |
| one positive and one negative | saddle; unstable |
| two positive real eigenvalues | unstable node |
| complex pair with positive real part | unstable spiral |
A saddle is unstable even though some trajectories approach it
At a saddle point, disturbances shrink in one characteristic direction but grow in another. Because there is a direction of escape, the equilibrium is unstable overall.
Stability does not mean monotonic return
A stable system may overshoot and oscillate many times before returning. Complex eigenvalues with negative real part produce damped oscillations.
The important feature is that the disturbance amplitude eventually decreases.
Neutral behaviour
Some systems can remain near an equilibrium without converging to it. Purely imaginary eigenvalues in a linear system can produce persistent oscillations.
For nonlinear systems, eigenvalues with zero real part require additional analysis; linearisation alone may be inconclusive.
Stability of an equilibrium versus stability of a cycle
A system can have a stable periodic orbit rather than a stable fixed point. Nearby trajectories then approach the repeating cycle instead of a constant state.
Thus “stable long-term behaviour” is broader than equilibrium stability.
Biological interpretation
In biology, a stable equilibrium can represent recovery toward a persistent population level, coexistence after a small perturbation, return to an endemic infection level, or restoration of chemical concentrations.
An unstable equilibrium can represent a threshold: a small displacement on one side may lead toward one biological outcome while a displacement on the other side leads elsewhere.
Stability and invasion
In ecological and epidemic models, the stability of a boundary equilibrium often tells us whether a rare population or infection can invade.
For example, instability of a disease-free equilibrium may indicate that introducing a small amount of infection causes infection to grow.
Deterministic stability versus stochastic behaviour
A deterministic stable equilibrium does not mean a stochastic system remains exactly at one point. Random events can continually perturb the system around it.
For small populations, sufficiently large random fluctuations can even produce extinction despite deterministic local stability.
A practical stability workflow
First find all equilibria and check biological feasibility. In one dimension, inspect the sign of \(f(x)\) around each equilibrium or calculate \(f'(x^*)\). For systems, calculate the Jacobian, evaluate it at each equilibrium, find its eigenvalues and interpret their real parts. Finally, remember that these tests are usually local and deterministic.