A community of Strue species is generated with a log-normal abundance distribution: each species' relative abundance is exp(σ·z), z ~ Normal(0,1), then normalized to sum to 1 — the standard model for real field communities (Preston 1948), where most species are rare and a few are common. Every "draw" picks one individual at random according to these true probabilities, exactly like an ecologist encountering one more organism in the field.
S(n) = number of distinct species among the first n individuals
f1 = "singletons" = species observed exactly once
f2 = "doubletons" = species observed exactly twice
Chao1 (abundance-based) estimator:
Ŝ = S_obs + f1(f1 - 1) / (2(f2 + 1))
The raw draw order is noisy — whichever species happens to appear first biases the curve. Rarefaction fixes this: the same observed multiset of n individuals is reshuffled into K independent random orders, the accumulation curve is recomputed for each shuffle, and the K curves are averaged. That average is the smooth, order-independent expected species-accumulation curve S(n) plotted below. Chao1 is recomputed at every sample size from the real singleton/doubleton counts of the actual survey, and should drift toward the dashed true-richness line as n grows — first quickly, then more slowly, since most of the "signal" is used up once common species are found and only rare ones remain to be caught.
- True species richness — the real total number of species in the hidden community (the number Ŝ should converge toward).
- Abundance evenness (σ) — 0.1 makes every species roughly equally common (accumulation is fast, Chao1 converges quickly); 3.0 makes most species rare (slow accumulation, many singletons, Chao1 stays elevated above S_obs for a long time) — the realistic case for most surveys.
- Rarefaction runs (K) — how many random re-orderings are averaged to smooth the curve; higher K gives a smoother, less noisy expected curve at the cost of more computation.
Real-world relevance: this individual-based rarefaction + Chao1 pairing is exactly what field ecologists compute from quadrat, trap, or transect data to answer "how many more samples until we've found essentially every species here?" (Chao 1984; Colwell & Coddington 1994; Gotelli & Colwell 2001).