1. Power analysis on simulated data
In this section we report the outcomes obtained for a simulated decrease in population size over time, as increasing and decreasing trends showed very similar results. The results for the case of simulated population increases are presented in supplementary materials.
Sample counts were able to correctly identify the direction of a population trend if it was as strong as -40% or more over 10 or 20 years (fig.1), or for a change in abundance of -30% if the yearly trend variability was 0.1 or lower. If the population showed a change in abundance of -20% in 10 or 20 years, sample counts were able to detect the trend only under low trend variability (cvy = 0.05) and at least 15 sampled sectors. Under a 10% overall trend sample counts never reached a sufficient statistical power in identifying the direction of the population trend, even sampling the entire area. When the direction of the trend was detected, its magnitude was also correctly identified with an error lower than 5% on the yearly trend. However, sample counts were never able to reach a sufficient statistical power in estimating the 10-year trend with an annual error lower than 2%, even when sampling the entire area. Conversely, on a time span of 20 years, trends of -30% or higher were detected within the 2% error in at least 80% of the cases if the yearly trend variability was 0.05 (fig.2). With a higher annual variability in trends, the correct magnitude over 20 years was never detected with sufficient power. A number of sectors between 15 and 20 (out of 38 total) was sufficient to estimate correctly both the direction and the magnitude of the trend, corresponding to 23.4 - 60% of the total GPNP area (as the extent of the sectors is highly variable). However, while estimating actual abundance instead of the population trend, sample counts in 15-20 sectors were biased and produced a mean error of around +16% compared to total counts.
The model selection for the effect of all the parameters on the statistical power of the sample counts allowed us to choose a single best predictive model (tab.2). The strongest effects in the model were the number of sectors surveyed and the strength of the overall assigned trend (tab.3). In particular a higher number of surveyed sectors increased the statistical power of the analysis, while a higher r (thus a lower decline over time) decreased the proportion of sample counts that correctly detected the real population trend. The abundance trend variability between years had also a strong effect on the statistical power, with statistical power decreasing as the variability increases. Other parameters, such as variability between sectors and variability of detection probability, also influenced the outcome with a lower inverse effect on statistical power, increasing the number of sectors needed to be sampled. For example, with cvy = 0.05 and r = 0.982, the number of sectors required to quantify the 20-years trend passed from 12 with a low cvs to 22 for a high cvs and from 13 with a low cvd to 19 with a high cvd (see supplementary materials). Population size and detection probability did not have any effect in the models. All the graphs for the effect of the parameters are provided in supplementary materials.
When selecting the sectors to be sampled, choosing those with the highest ibex abundance in the first year of the monitoring resulted in a slightly higher statistical power, while a random selection in the first year only or in each year decreased the reliability of the analysis (tab.3), lowering the average statistical power respectively by 1.7% and 2.6% for the 10-year trend and by 0.9% and 1.2% for the 20-years trend. This difference was relevant in terms of the number of sectors to be sampled to reach the 80% power threshold, as for instance, with cvy = 0.05 and r = 0.978 the correct magnitude of a 20-year trend was detected with the 11 most abundant sectors, with 16 sectors selected at random in the first year or with 20 sectors selected at random each year (see supplementary materials). However, if abundance was estimated instead of the population trend, choosing the sectors with the highest ibex abundance in the first year produced an overestimation of population size, that was 41.7% higher on average than with a random sector selection.
To monitor the population trend of Alpine ibex in GPNP (yearly variability of 0.05 and sector variability of 0.05-0.1), sampling 15 to 20 sectors was in general sufficient (fig.3): the direction of the trend was detected in at least 80% of the cases for trends equal to -20% both in 10 and 20 years (fig.3a and fig.3b). The direction of trends of -30% or more was correctly estimated even when sampling less than 5 sectors, while a trend of -10% was never detected even performing censuses in all sectors. On a period of 20 years, surveying 10 sectors was sufficient to correctly detect the magnitude of a trend of -30% or more, while lower trends were not correctly quantified even sampling the entire area (fig.3d). However, the magnitude of a 10-years trend was impossible to detect with an error within 2% and the desired threshold of statistical power of 80% whatever the real trend was (fig.3c).
The proportion of opposite (i.e., in the wrong direction) and non-significant trends estimated with sample counts was low but increased for weak trends (fig.4). Counting animals in half of the sectors led, in the simulations, to 2.6% of non-significant and 1.2% of opposite trends estimated when the population was declining by 30% in 10 years (r = 0.965), while with the same yearly trend over 20 years (r = 0.966) the population trend was always estimated in the right direction and in 90% of the cases with the correct magnitude. Monitoring low changes in abundance, such as a 10% decline in 10 or 20 years, led in 34.4% (in 10 years) and 36.4% (in 20 years) of the cases to estimating incorrectly the direction of the population trend.