2- Power analysis on simulated trends
We built simulated populations of Alpine ibex in GPNP, with a given total abundance (A) randomly distributed among the 38 sectors of the Park. In each simulation, A was drawn from a Poisson distribution under three different abundance scenarios, with expected values of A of 2,500, 3,500 or 5,000 individuals (tab.1), thus a low, medium and high abundance (based on the historical census data for Alpine ibex in GPNP, Mignatti et al., 2012). The probability for an individual to be assigned to a specific sector was given by the average Alpine ibex density of the sector (i.e., the mean number of individuals observed in that area during actual censuses in GPNP conducted between 2000 and 2020). Therefore, the simulated ibex distribution was not dependent only on the extent of the sectors (as it would occur assuming a constant density across the target area), but was determined by unknown multiple factors that in the real censuses affect the presence of animals and their density in each sector.
For each random population we simulated a growth of the population with a yearly overall abundance trend (r ) over a 10 or 20-year period, either with a decrease or increase in population size. The different values of r that were used are reported in tab.1 and correspond to a total change of 10%, 20%, 30%, 40% and 50% in 10 or in 20 years.
We simulated different scenarios of year-to-year and sector-to-sector variability in the overall trend r , assigning a random trend,Ty,s , to each y-th year and s-thsector, sampled around the value of r with a specified coefficient of variation (cvy for the year and cvs for the sector). Further details about the mathematical formulation of the simulations are provided in supplementary materials.
We simulated different scenarios of variability in the overall trend by assigning four different possible values (0.05, 0.1, 0.15 and 0.2) to the coefficient of variation between years (cvy) and sectors (cvs). A coefficient of variation of around 0.05 between years and of 0.05-0.10 between sectors was indeed historically found in GPNP over the last 65 years (Brambilla & Bassano,unpublished data ). We also included higher variability to potentially extend the results to other species.
Simulated censuses were run on the random-built population in onlyn out of the 38 total sectors and population growth rates were estimated from the total number of individuals counted in the sample sectors and compared to the real assigned growth rate r .
We included in the simulations different values of detectability for the animals, where detection probability varied in each repetition (between years) and was different between the sectors. The coefficient of variation of the detectability from one sector to the other was either 0.1 (low variation, thus most sectors have a similar detectability), 0.3 (medium variation, detectability is different between sectors) and 0.5 (high variation, detectability is very different between sectors) around an estimated detection probability for Alpine ibex ranging from 0.4 to 0.8 (Gaillard et al. 2003). Further details are provided in supplementary materials.
The n survey sectors were selected as: i) n sectors selected at random the first year and then sampled each following year; ii) n randomly selected sectors where the random selection was repeated each year; iii) a biased selection with only the ninitial sectors with the highest number of individuals detected in the first year of monitoring.
We estimated the growth rate in each simulation with a Generalized Linear Model (GLM) with a Poisson data distribution (O’Hara and Kotze, 2010).
We ran the simulation 500 times for each of the 656,640 combinations of our seven parameters: assigned overall trend (r), method to select the survey sectors, total population size (A), cv between areas (cvs), cv between years (cvy), cv for detectability between sectors (cvd), and the number of sectors in which the surveys were conducted (n). Over the total of 328,320,000 simulations, statistical power was calculated as the proportion of simulations in which a significant trend was estimated and was in the same direction of the assigned one (Weiser et al. 2019), with a threshold of 0.8 (80%). A statistical power of 0.8 (i.e. an 80% probability of detecting the effect of interest) is conventionally considered sufficient to conclude that the sampling design is able to detect the true population trend (Cohen 1992).
We also calculated the power in detecting the trend with an error on yearly trend lower than 5% and 2% in magnitude (i.e. |(restimated - r) / r | < 0.05 or 0.02), thus the performance of sample counts in correctly detecting the magnitude of the trend. Similar tolerance levels were used by Wauchope et al. (2019) in simulations to estimate the required length of a time series of counts.
All the population and censuses simulations, together with the growth rates estimation, were performed in R version 4.1.3 (R Core Team 2022) and the full script is provided in supplementary materials.
To determine the parameters with a higher effect on statistical power for sample counts to correctly detect the population trend, we built linear models (LMs) with combinations of all the parameters used in the simulations, and selected the best predictive model as the one with the lowest AIC (Akaike 1974).