Proportional variability relative to lower abundances: An interpretable and efficient index of deep lows in population time series

Variability in population abundances is usually measured with the coefficient of variation (CV) or the standard deviation of log-transformed data (SDL), both of which are biased for non-Gaussian dynamics and zero counts. Proportional variability (PV_hi), which scales each pairwise difference relative to the larger of the two abundances, avoids reference to the mean but responds only weakly to isolated crashes and to the depth of declines. I introduce PV_lo, an index of proportional variability relative to lower abundances, based on all pairwise comparisons and bounded between 0 and 1. Zero counts are handled without data transformation. A sorting-based algorithm reduces the computation from O(n^2) to O(n log n), making the index applicable to long time series. In example series and simulations (Gaussian, Cauchy, rare-crash, bimodal and gradual-decline regimes), PV_lo remained sensitive to crashes and to the contrast between abundance levels, whereas PV_hi levelled off, but PV_hi was more robust against heavy-tailed extremes. In a comparison with series of ordinary fluctuations, PV_lo and SDL separated crashes better than PV_hi and the CV only if the ordinary variability of a population was unknown, and PV_lo did not outperform SDL; during a sustained decline the indices differed little. PV_lo is a computationally efficient index that responds to deep lows on an interpretable ratio scale without transformation of the data, an alternative to SDL that is relevant for screening many populations in a conservation context.