Oscillations
Oscillations occur when biological quantities rise and fall repeatedly through time. They appear in ecological populations, epidemics, biochemical systems, physiology and many other biological models.
What does an oscillation look like?
A simple periodic signal can be written as
\[x(t)=A\cos(\omega t+\phi).\]| Quantity | Meaning |
|---|---|
| \(A\) | amplitude: maximum distance from the central level |
| \(\omega\) | angular frequency |
| \(\phi\) | phase: where in the cycle the oscillation begins |
The period is
\[\boxed{T=\frac{2\pi}{\omega}},\]and the ordinary frequency is
\[\boxed{f=\frac{1}{T}}.\]Amplitude, period and phase
Why oscillations occur in biological systems
Oscillations usually arise from feedback, delay, external forcing or interaction between variables. A biological quantity changes, that change affects another process, and the response feeds back later.
Examples include predator–prey feedback, delayed density dependence, immune–pathogen interactions, hormone regulation and seasonal transmission.
Feedback and delay
Negative feedback can stabilise a system, but if the response is delayed or sufficiently strong, the system may overshoot equilibrium repeatedly.
For example, a prey population can increase first. Predator abundance responds later, then predation becomes strong enough to reduce prey. Predator decline follows only after prey have already become scarce.
Damped oscillations
Damped oscillations decrease in amplitude and eventually approach an equilibrium.
A simple form is
\[x(t)=A e^{-ct}\cos(\omega t),\qquad c\gt0.\]The factor \(e^{-ct}\) shrinks the oscillation envelope through time.
Connection with complex eigenvalues
Near an equilibrium, a linearised two-dimensional system may have eigenvalues
\[\lambda_{1,2}=a\pm ib.\]If
\[a\lt0,\]the oscillation amplitude decays. The imaginary part \(b\) produces the oscillatory rotation, while the negative real part \(a\) produces damping.
Growing oscillations
If the eigenvalues have positive real part,
\[a\gt0,\]small oscillations grow as the trajectory spirals away from the equilibrium.
In a nonlinear model, this growth does not necessarily continue without bound. Nonlinear effects may eventually produce a stable limit cycle or another attractor.
Sustained oscillations
A sustained oscillation repeats indefinitely with approximately constant amplitude and period.
In a nonlinear autonomous system, sustained periodic behaviour is represented by a closed periodic orbit in the phase plane.
Limit cycles
A limit cycle is an isolated periodic orbit. Nearby trajectories may approach it or move away from it.
If nearby trajectories approach the cycle, it is a stable limit cycle. The system can therefore settle into persistent oscillation rather than a fixed equilibrium.
Limit cycle versus centre
A centre and a stable limit cycle can both show closed curves, but they are different.
Near a centre, there is typically a continuous family of closed orbits, and nearby trajectories do not converge to one special orbit. A limit cycle is isolated and can attract or repel neighbouring trajectories.
Predator–prey oscillations
The classical Lotka–Volterra model can generate closed predator–prey cycles:
\[\frac{dN}{dt}=rN-aNP,\qquad\frac{dP}{dt}=eaNP-mP.\]Prey rise first. Predator growth then increases because more food is available. Higher predator abundance eventually drives prey downward. Predators decline after prey have become scarce, allowing prey to recover.
Two oscillating populations
Seasonally forced oscillations
Not all oscillations are generated internally. A parameter can vary periodically with the environment, for example
\[\beta(t)=\beta_0\bigl(1+\varepsilon\cos(2\pi t/T)\bigr).\]This can represent seasonal transmission, breeding, resource supply or temperature effects.
The resulting oscillation is externally forced rather than arising solely from autonomous internal feedback.
Forced versus self-sustained oscillations
| Type | Main cause |
|---|---|
| damped oscillation | stable equilibrium with oscillatory return |
| self-sustained oscillation | nonlinear internal dynamics, often a stable limit cycle |
| forced oscillation | periodic external input or parameter variation |
| growing oscillation | unstable equilibrium with oscillatory departure |
Hopf bifurcation and the onset of oscillation
A Hopf bifurcation can occur when a complex pair of eigenvalues crosses from negative to positive real part as a parameter changes.
In a supercritical Hopf bifurcation, a stable equilibrium can lose stability and a small stable limit cycle can appear.
This provides a mathematical mechanism by which a biological system changes from steady behaviour to persistent oscillation.
Oscillations in epidemic models
Epidemic incidence can oscillate because susceptible individuals are depleted and later replenished, immunity wanes, births introduce new susceptible individuals, behaviour changes, or transmission is seasonal.
Observed epidemic waves are therefore not automatically evidence of a stable mathematical limit cycle. External forcing, interventions and stochastic effects can also produce repeated waves.
Oscillations in delayed systems
If a regulatory process depends on the past state rather than only the current state, a delay differential equation may be appropriate.
For example, a delayed density-regulation model might use a term involving
\[N(t-\tau),\]where \(\tau\) is the delay. Sufficient delay can destabilise an otherwise stable equilibrium and generate oscillatory behaviour.
Stochastic oscillations
Randomness can generate irregular cycles even when the deterministic system has a stable equilibrium. Noise can repeatedly excite damped oscillatory modes, producing visible fluctuations around the equilibrium.
This is sometimes called quasi-cycling or noise-sustained oscillation.
Oscillation versus random fluctuation
A finite data series that rises and falls does not necessarily contain a true periodic mechanism. Random variation can produce apparent waves.
Evidence for oscillation may involve repeated timing, spectral structure, phase relationships between variables, or support from a mechanistic model.
Phase relationships
When two biological variables oscillate, their peaks may not occur at the same time. The shift between them is called a phase difference.
Phase relationships can reveal mechanism. Predator lag behind prey is one example; hormonal or biochemical feedback networks can show similar delayed responses.
Period can depend on parameters
The oscillation period is not always fixed by a simple formula. In nonlinear systems it can change with model parameters and sometimes with amplitude.
Near a linear equilibrium with eigenvalues \(a\pm ib\), the local angular oscillation frequency is approximately \(|b|\), giving approximate local period
\[T\approx\frac{2\pi}{|b|}.\]Why oscillations matter biologically
Average population size can hide important peaks and troughs. A system may have a moderate mean but repeatedly approach dangerous low levels, exceed hospital capacity, or generate large epidemic peaks.
Understanding amplitude and timing can therefore be as important as understanding the average state.
A practical workflow
First determine whether the observed or simulated oscillation decays, grows or persists. Examine equilibria and Jacobian eigenvalues. Use the phase plane to look for spirals or closed orbits. Check whether periodic parameters or time delays are present. If the system is stochastic, compare repeated simulations rather than interpreting one trajectory alone.