Applications of nonlinear dynamics
Nonlinear models explain population cycles, epidemics, chemical clocks, and the limits of weather prediction.
Predator-prey models
The Lotka-Volterra system \(x' = ax - bxy\), \(y' = -cy + dxy\) has an equilibrium at \(\left(\frac{c}{d}, \frac{a}{b}\right)\) whose linearisation is a centre. A conserved quantity confirms that solutions are closed orbits: predator and prey populations cycle forever, with predator peaks lagging prey peaks.
More realistic ecology
Adding prey self-limitation, \(x' = ax(1 - x/K) - bxy\), turns the centre into a stable spiral: cycles damp toward coexistence. Other refinements, like predator satiation, can produce stable limit cycles. Model choices change predictions qualitatively, which is why ecologists test models against data.
Epidemic models
The SIR model \(S' = -\beta SI\), \(I' = \beta SI - \gamma I\) predicts an outbreak only if \(R_0 = \frac{\beta S_0}{\gamma} > 1\). The epidemic peaks when \(S\) falls to \(\frac{\gamma}{\beta}\), and herd immunity is reached when the susceptible fraction drops below \(\frac{1}{R_0}\). Vaccination aims to push the susceptible fraction under this threshold.
R_0 = \frac{\beta S_0}{\gamma}Chemical oscillators
Some chemical reactions, like the Belousov-Zhabotinsky reaction, oscillate in colour for minutes, a chemical clock. Their models have limit cycles arising from Hopf bifurcations, showing that sustained oscillation does not require external forcing.
Neurons
The FitzHugh-Nagumo model reduces nerve-cell firing to two equations. A small stimulus decays, but one above a threshold triggers a large excursion, a spike, before returning to rest. Continuous stimulation produces a limit cycle of repeated firing.
Weather and chaos
Edward Lorenz discovered in 1963 that a three-variable convection model is chaotic: rounding initial values changed long-term forecasts completely. This sensitivity limits detailed weather prediction to roughly two weeks, no matter how good the computers, while climate statistics remain predictable.
Worked example: SIR threshold
A disease with \(\beta = 0.3\) per day and \(\gamma = 0.1\) per day in a fully susceptible population has \(R_0 = 3\). An outbreak occurs, and herd immunity requires the susceptible fraction to fall below \(\frac{1}{3}\), so about two-thirds of people must be immune.
Common mistakes
Treating neutrally stable cycles as robust; small model changes can destroy them. Confusing chaos with randomness; chaotic systems are deterministic. Ignoring units of rates when computing \(R_0\).
Key takeaways
Lotka-Volterra gives predator-prey cycles, and refinements change their stability. SIR models predict outbreaks when \(R_0 > 1\) and herd immunity below \(1/R_0\) susceptible. Limit cycles explain chemical clocks and neuron firing. Chaos limits detailed weather forecasting.
Flashcard
Predator-prey model
Check your understanding
In the SIR model, an epidemic grows initially when: