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found in GPNP over the last 65 years (Brambilla & Bassano,unpublished data ).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).