Interactive logistic growth equation visualization: explore the S-curve, inflection point, carrying capacity, and population dynamics
The logistic growth equation dN/dt = rN(1 − N/K) was proposed by Pierre-François Verhulst in 1838. It is the fundamental model for population growth in resource-limited environments. When N ≪ K, it approximates exponential growth dN/dt ≈ rN; as N → K, the growth term (1−N/K) → 0 and the population stabilizes. The analytical solution is N(t) = K / (1 + ((K−N₀)/N₀)·e^(−rt)), producing the classic S-shaped (sigmoid) curve.
The logistic curve has an inflection point at N = K/2 where the growth rate dN/dt reaches its maximum rK/4. Before the inflection point, growth acceleration is positive; after it, acceleration becomes negative and growth decelerates. The inflection time is t* = (1/r)·ln(K/N₀ − 1). This point marks when resource constraints begin to dominate — in epidemiology, it corresponds to the peak of daily new cases.
Comparing exponential growth dN/dt = rN (solution: N = N₀·e^(rt)) with logistic growth: in the early phase (N ≪ K), they are nearly identical; as N grows, exponential growth continues unbounded while logistic growth is constrained by carrying capacity K. Nearly all real-world growth processes eventually face resource constraints — the logistic model is far more realistic than pure exponential growth.
Setting dN/dt = 0 yields two equilibria: N* = 0 (extinction, unstable) and N* = K (carrying capacity, globally stable). For N < K, dN/dt > 0 (population grows); for N > K, dN/dt < 0 (population declines). Thus K is a globally stable equilibrium — regardless of initial N₀ > 0, the system converges to K.
Growth rate r controls how fast N approaches K: larger r → steeper S-curve, earlier inflection. Carrying capacity K sets the final steady-state value: increasing K only changes height, not shape (normalized curves overlap). Initial N₀ affects the starting position and inflection time: smaller N₀ → longer exponential phase, later inflection.
Important generalizations: (1) Gompertz model: dN/dt = rN·ln(K/N), used for tumor growth; (2) Richards model: unifies logistic and Gompertz with a shape parameter; (3) Time-delay model: dN/dt = rN(t)(1 − N(t−τ)/K), producing oscillations; (4) Allee effect: growth rate decreases at low density; (5) Discrete logistic map: x_{n+1} = rx_n(1−x_n), chaotic for r > 3.57.
The logistic equation is the cornerstone of population ecology. Classic case: Gause (1934) verified logistic growth with Paramecium experiments. Maximum Sustainable Yield (MSY) occurs at N = K/2 — the inflection point — with harvest rate ≤ rK/4.
In SIR models, cumulative cases follow an S-curve. Early exponential growth → inflection (peak daily cases) → saturation. Fitting early data to estimate r and K enables forecasting. COVID-19 cumulative case curves in many countries showed classic S-shaped growth.
Everett Rogers' Diffusion of Innovations (1962) describes adoption as an S-curve. Smartphone, internet, and social media user growth all follow logistic patterns. The inflection point marks 'Crossing the Chasm' from early adopters to the mass market.