A few cells in fresh medium double, and double, and double — until the food runs low and the dish fills. Exponential at first, then an S that flattens at the carrying capacity: the growth curve of every culture, every colony, every population with a limit. Slide time (or tap) and watch it fill.
Logistic (Verhulst) growth, the S-curve of a bounded population. dN/dt = r·N·(1 − N/K) integrates to N(t) = K / (1 + A·e^(−r·t)) with A = (K−N₀)/N₀ — exponential while N ≪ K, slowing as resources deplete, and levelling at the carrying capacity K. The inflection (fastest growth) is exactly at N = K/2. Here K=1, N₀=0.03, r=5; the curve, the current population and the K/2 inflection are drawn live. A fail-loud self-check throws unless N(0)=N₀, N→K at large t, and the inflection sits at K/2.
A smooth deterministic model (real cultures have lag, death and stochastic phases); K, r, N₀ are fixed illustrative constants. The logistic integral and the K/2 inflection are computed exactly.