*Ecological Archives* E095-148-A4

Michael Kalyuzhny, Yishai Schreiber, Rachel Chocron, Curtis H. Flather, Ronen Kadmon, David A. Kessler, Nadav M. Shnerb. 2014. Temporal fluctuation scaling in populations and communities. *Ecology* 95:1701–1709. http://dx.doi.org/10.1890/13-0326.1

Appendix D. Cleaning the Breeding Bird Survey (BBS) data from sampling errors.

To clean the BBS data from errors, we make the very reasonable and common assumption that the errors are uncorrelated in time, unlike the process (Holmes 2001). Hence, if we look at the variance of *Y* over increasing timelags Δ*t*, the sampling errors would just add a constant variance term to the var(*Y*) - Δ*t* relationship. Over short time lags we may assume that the effect of stabilizing mechanisms is minor, leading to a linear increase in the variance of *Y* with Δ*t*. This allows us to use the slope of var(*Y*) - Δ*t* over short time lags as an estimate for var(*Y*) without errors (Holmes 2001).

In order to get the general picture of the fluctuation scaling we grouped var(*Y*) by the initial population size nt for Δ*t* of 1 to 10 and calculated the slope for each group separately. This allowed us to plot the initial slope vs. the mean in the group size, as depicted in Fig. 4 of the main text.

However, it should be noted that sampling bias does affect the slope, reducing it by a factor that is proportional to the number of individuals present that are, on average, detected – causing the slopes of var(*Y*) to be smaller. This may lead to a problem if populations differ in the detection bias (caused by differences in behavior or habitat), leading to possible differences in the slopes due to this effect.

If the differences in detectability among species are significant, the bias should be stronger (lower detection rates) among rare groups. These groups consist of truly small populations and populations that seem small due to bias. The largest group, on the other hand, consists only of truly large populations. Since the bias decreases the slope, it will suppress the real value of Var(*Y*) for small groups and have almost no effect on Var(*Y*) for large groups. Accordingly, if the real growth of Var(*Y*) with m is linear (as expected by Eq. (3)), the effect of bias will make the slope superlinear (*z *> 2), not sublinear. Therefore, the sublinearity observed in Fig. 4 is real and the effect of bias, if exists, only reduces *z* even further.

Literature Cited

Holmes, E. E. 2001. Estimating risks in declining populations with poor data. Proceedings of the National Academy of Sciences USA 98:5072–5077.