Field ecologists rarely see a whole community at once — a citizen-science bioblitz, a long-term LTER transect, or a network of sensors each contributes one imperfect "visit" that may detect one individual. Plotting cumulative species found (Sobs) against cumulative visits gives the species accumulation curve: steep early on, then flattening as easy, common species are already logged and only rare ones remain to be found.
Each visit here draws one encounter from a simulated community whose true richness is S and whose relative abundances follow a geometric series, then applies an imperfect detection probability p (even a present species can be missed):
p_i ∝ (1 − k)^i, i = 0..S−1 (dominance k → skewed vs. even community)
visit: draw species i ~ p_i, record only with prob. p (imperfect detection)
Because the true richness is never directly observable, the curve is used to estimate it. The non-parametric Chao1 estimator uses only the rarest classes — species seen exactly once (f₁, "singletons") and exactly twice (f₂, "doubletons") — because they carry almost all the information about what hasn't been found yet:
Chao1 = S_obs + f1² / (2·f2) (bias-corrected form when f2 = 0:
S_obs + f1·(f1−1)/2)
Completeness = S_obs / Chao1 × 100%
- Sampling speed — how many survey visits run per second once sampling is started.
- True community richness — the actual number of species present at the site (never shown directly — you only ever see S_obs and the estimate).
- Community dominance — how skewed the abundance distribution is; a high value means a few common species dominate encounters and rare species take far longer to detect.
- Detection probability — the chance a present, encountered individual is actually recorded, modeling observer error, camera-trap misses, or sensor noise.
This is the same statistic behind real monitoring programs — LTER networks, camera-trap grids, eDNA transects and citizen-science apps like iNaturalist all use accumulation curves and richness estimators like Chao1 to decide whether "no new species in the last 50 visits" means the site is fully surveyed, or just under-sampled.