With independent half-counts, 43–89% of year-to-year count variance in North American and UK bird surveys is sampling error, and shared change scales with mean count at exponents 1.4–1.7, not 2

Question: do small local populations have larger relative year-to-year change after sampling error is removed, that is, is the temporal Taylor exponent of real change below 2? Approach: we divided each annual count into two halves with separate sampling of the same site and year: odd and even stops of North American Breeding Bird Survey routes (1997-2024, 184,086 route-observer-species series), early and late visits to UK Breeding Bird Survey squares (1994-2024, 112,246 series), and random halves of samples in 55 BioTIME 2.0 count studies. If the half errors are independent, the covariance of the year-to-year changes of the halves estimates the variance of the shared (real) change without a sampling model. Simulations confirmed that the estimator recovers the true exponent. Results: only 11-15% of the variance of change in UK square counts was shared between the two visits, at all abundance levels; on North American routes 18-57% was shared between odd and even stops, rising with abundance. The shared relative variance still decreased with the mean count: exponent 1.72 [95% site-bootstrap interval 1.67, 1.75] in North America and 1.41 [1.29, 1.55] in the UK; within-species medians were 1.40 (211 species) and 1.22 (77 species). Removal of sampling error thus did not bring the exponent to 2 (naive exponents 1.55 and 1.47). In a linear-plus-quadratic description of the variance of change, the shared linear (demographic-like) term was only 17% (North America) and 12% (UK) of the total linear term, but it was larger than the quadratic term below about 4 and 9 counted birds per year. Results held without flocking and colonial species, with longer series, in two periods, and with cross-fitted bins (UK upper limit 1.96). BioTIME gave 19-36% shared variance and exponent 1.77 [1.60, 1.93], but a cross-fitted check gave 2.28 [1.64, 2.65], so BioTIME does not establish an exponent below 2. Main limitation: North American halves share the survey day, the observer and some double-counted birds, so their shared part is an upper limit for real change.

Taylor's law · fluctuation scaling · population variability · sampling error · observation error · split-half · replicate counts · Breeding Bird Survey · UK Breeding Bird Survey · BioTIME · demographic stochasticity · environmental stochasticity · coefficient of variation · rarity