Population Chaos: How One Equation Turns Growth Into Chaos

A one-line model of a growing population

The logistic map squeezes exponential growth and a resource limit into a single feedback loop: x_(n+1) = r · x_n · (1 - x_n). Here x_n is a population as a fraction of its maximum (between 0 and 1) in generation n, and r is a growth rate. The x_n term drives growth; the (1 - x_n) term throttles it as the population approaches its ceiling — food runs out, disease spreads, nests fill up. It was popularised by ecologist Robert May in 1976 as the discrete-time cousin of the continuous logistic differential equation.

What makes it famous is not the biology but what happens when you slide r up from 1 to 4 and watch where the population settles after many generations.

x_(n+1) = r · x_n · (1 - x_n),   0 ≤ x_0 ≤ 1, 0 ≤ r ≤ 4

The route to chaos: period doubling

For r below 3, the population converges to a single stable value — a fixed point. At r = 3 that fixed point loses stability and splits into a period-2 cycle: the population alternates between two values, high one generation, low the next. Push r further and that cycle splits again into a period-4 cycle at r ≈ 3.449, then period-8 at r ≈ 3.544, then period-16, each bifurcation arriving faster than the last. This cascade of period doubling converges at r ≈ 3.56995 — the Feigenbaum point — beyond which the population never settles into any repeating cycle at all: it is chaotic.

The bifurcation diagram

Plotting the long-run values of x against r produces the logistic map's signature image: a single curve that forks into two, then four, then a dense chaotic smear, interrupted by narrow clear periodic windows (the widest, near r ≈ 3.83, is a stable period-3 cycle). Two numbers on this diagram are astonishingly universal. Physicist Mitchell Feigenbaum discovered in 1975 that the ratio of the gaps between successive bifurcation points converges to a constant, δ ≈ 4.6692, and the ratio of successive branch widths converges to α ≈ 2.5029 — and both constants show up, unchanged, in the period-doubling route to chaos of completely unrelated systems: dripping taps, convecting fluids, driven electronic oscillators. It is one of the few universal constants in nonlinear dynamics.

r < 3.0            → stable fixed point
3.0 ≤ r < 3.449   → period-2 cycle
3.449 ≤ r < 3.544 → period-4 cycle
r → 3.56995      → accumulation point (infinite period doublings)
r > 3.56995         → chaos, interrupted by periodic windows
r = 4                → fully chaotic, provably equivalent to a coin flip

Sensitive dependence and the limits of prediction

In the chaotic regime, two populations that start a millionth of a percent apart diverge exponentially — the defining fingerprint of chaos, formalised as a positive Lyapunov exponent. At r = 4 the map is exactly solvable and provably as unpredictable as flipping a fair coin: no amount of extra decimal precision in your starting value buys you more than a few extra generations of accurate prediction. This is why weather, population booms and other logistic-like systems can be perfectly deterministic — no randomness anywhere in the equation — and still be practically unforecastable beyond a short horizon.

Lotka-Volterra: when two populations chase each other

A related but different model, the Lotka-Volterra predator-prey equations, couples two continuous populations instead of throttling one: prey grow exponentially until predators eat them down, predators starve once prey become scarce, and the pair cycles in closed loops on a phase plot rather than collapsing into chaos. It is a useful contrast: Lotka-Volterra shows oscillation from a two-species feedback loop, while the logistic map shows chaos from a single species with a resource ceiling and discrete generations — two different routes to the same headline lesson, that simple nonlinear rules can produce far richer behaviour than their linear cousins.

Frequently asked questions

Try it live

Everything above runs in your browser — open Population Chaos and change the parameters while it is running. Nothing is installed, nothing is uploaded, the whole model lives in one tab.