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\begin{document}

\title{Monitoring wildlife population trends with sample counts: A case study
on the Alpine ibex (Capra ibex)}



\author[1]{Matteo Panaccio}%
\author[2]{Alice Brambilla}%
\author[3]{Bruno Bassano}%
\author[1]{Tessa Smith}%
\author[1]{Achaz von Hardenberg}%
\affil[1]{University of Chester}%
\affil[2]{University of Zurich}%
\affil[3]{Ente Parco Nazionale Gran Paradiso}%


\vspace{-1em}



  
  \date{June 17, 2024}


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\selectlanguage{english}
\begin{abstract}
Monitoring population dynamics is of fundamental importance in
conservation but assessing trends in abundance can be costly, especially
in large and rough areas. Obtaining trend estimations from counts
performed in only a portion of the total area (sample counts) can be a
cost-effective method to improve the monitoring and conservation of
species difficult to count. We tested the effectiveness of sample counts
in monitoring population trends of wild animals, using as a model
population the Alpine ibex (Capra ibex) in Gran Paradiso National Park
(Italy), both with computer simulations and using historical census data
collected over the last 65 years. Despite sample counts could fail to
correctly estimate the true population abundance, sampling half of the
target area could reliably monitor the trend of the target population.
In case of strong changes in abundance, an even lower proportion of the
total area could be sufficient to identify the direction of the
population trend. However, when there is a high yearly trend
variability, the required number of samples increases and even counting
in the entire area can be ineffective to monitor population dynamics.
Lastly, the effect of other parameters (such as which portion of the
area is sampled or the detectability) was marginal, but these should be
tested case by case. Sample counts could therefore constitute a viable
alternative to assess population trends, allowing for important,
cost-effective improvements in the monitoring of wild animals of
conservation interest.%
\end{abstract}%



\sloppy


\textbf{INTRODUCTION}

Monitoring population dynamics is a key goal for conservationists, as it
allows to investigate the effects of ecological variability on wildlife
populations (Jacobson et al. 2004, Jonas et al. 2008, Mignatti et al.
2012), to quantify the efficacy of management interventions (Hinkson and
Richter 2016, Saunders et al. 2018) and to evaluate the conservation
status of a species (Vors and Boyce 2009, Brambilla et al. 2020a). For
instance, the assessment of population trends is used in the IUCN
framework to evaluate the risk of extinction of a species, which under
criterion A (IUCN 2022) is considered threatened if it suffered a
population decline of 30\% or more over the last 10 years or three
generations.

Trends in abundance are generally estimated through multiple counts over
time (Magurran et al. 2010), but this process implicates several
challenges and can be subject to different sources of errors (Yoccoz et
al. 2001). Indeed, the true long-term population trends are often hidden
by short-term random fluctuations, due to demographic stochasticity and
environmental year-to-year variability (McCain et al. 2016). Surveys too
limited in time can therefore fail to identify the sign and magnitude of
a population trend (Wauchope et al. 2019). Moreover, population dynamics
can also vary across sites, with different subpopulations exhibiting
different trends (Urquhart 2012). If this variability is not taken into
account, biases in determining the real population trend can easily
arise (Palmer 1993, Weiser et al. 2019, Fournier et al. 2019). Finally,
imperfect detection during counts (Lahoz-Monfort et al. 2014, Kellner
and Swihart 2014) can influence the ability to detect population trends
(K\selectlanguage{ngerman}éry et al. 2009, Ficetola et al. 2018, Wood et al. 2019). The design
of population monitoring programs should try to minimize the above
mentioned potential biases, but also reduce survey costs in order to
make long term monitoring financially sustainable (Caughlan and Oakley
2001, White 2019). Power analysis is the tool generally used to quantify
the minimum sampling effort required to correctly detect the true
population trend (Gerrodette 1987, Taylor and Gerrodette 1993, Thomas
1997, Vallecillo et al. 2021). In a power analysis, multiple replicates
of the data are built, either with simulations or subsets of real
measures, and the statistical power is estimated as the proportion of
replicates in which a statistical model fitted to the data is able to
detect the change in population size over time (Steidl and Thomas 2001).
Many authors have used power analysis to estimate the minimum number of
years required in a monitoring project to detect a significant trend
(Luymes and Chow-Fraser 2019, Wauchope et al. 2019), showing that a
minimum of 5-35 years is needed depending on the species (Reynolds et
al. 2011, White 2019). However, fewer studies considered the use of
censuses conducted only in a sample of the target area (defined here as
sample counts) in monitoring population trends and estimated the
proportion of sites of the total area required to detect trends with
similar power to total counts. Sample counts have been described as
imprecise (Stoddard et al. 1998, Yoccoz et al. 2001), as it has been
shown for instance that a limited number of sites generally enables
detection of only strong declines in the population (Sewell et al. 2012,
Wagner et al. 2022), unless the species is particularly abundant and
easy to detect (Ficetola et al. 2018). However, the reliability of a
particular sampling design (such as sample counts) to detect population
dynamics is strictly linked to the study system (White 2019, Weiser et
al. 2019) and the efficacy of sampling only a portion of the target area
to monitor trends needs therefore to be analyzed for each system.

Monitoring certain taxa such as mountain ungulates, can present several
technical constraints due to difficult terrain, climatic conditions,
remote habitats and required logistics (Singh and Milner-Gulland 2011).
Mountain ungulates are usually counted with helicopter censuses
(Gonzalez-Voyer et al. 2001, Rice et al. 2009) or ground counts
(Jacobson et al. 2004, Largo et al. 2008, Suryawanshi et al. 2012), but
both these methods require big financial and human efforts, especially
in large areas and rough terrain conditions. Reducing the effort in
terms of time or number of surveyed sectors, while maintaining enough
power to detects trends, is therefore of critical importance in
monitoring species living in remote areas (Lindenmayer and Likens 2010).

In some cases, the survey effort can be reduced by sampling the
population less often (interval sampling), for instance every 3 or 5
years as suggested by some authors (Urquhart et al. 1998, Singh and
Milner-Gulland 2011, Reynolds et al. 2011). However, interval sampling
has proved to be less effective than sampling every year in detecting
the magnitude of a trend (e.g. Wauchope et al. 2019) and could therefore
not be adequate if the goal is to analyze the drivers of population
dynamics in a long-term monitoring project or for endangered species for
which declines must be detected quickly in order to provide appropriate
conservation strategies. Consequently, surveying only a sample of the
target area every year could be the only viable and cost-effective
alternative. To the best of our knowledge, a complete assessment of the
statistical power of sample counts for monitoring has not been performed
before.

In this study, we investigated the reliability of sample counts in
estimating population trends, using as a case study the Alpine ibex
(\emph{Capra ibex} ) population of Gran Paradiso National Park (GPNP,
North-West Italian Alps), for which an exceptionally long-term series of
65 years of count data, collected with block counts over the whole
territory of the Park, is available (Jacobson et al., 2004; Mignatti et
al., 2012). With a real-case range of parameters, we simulated
populations under various scenarios of abundance trends over 10 and 20
years and calculated the power of population models based on sample
counts in detecting the real trend. We also estimated the minimum
proportion of sectors (count areas in which the Park is divided) needed
to correctly detect the population trend of Alpine ibex population in
the whole area from which the samples are drawn. Moreover, we explored
the effect of different strategies for selecting which sectors to
monitor: i) random selection each year; ii) random selection in the
first year, then the same sectors are surveyed each of the following
years and iii) biased selection towards the sectors with the highest
abundance. We also included different magnitudes of both trend
variability between years and sectors and variability of detection
probability, a phenomenon that heavily affects the species and possibly
leads to severe underestimates of abundance (Gaillard et al. 2003,
Morellet et al. 2007, Largo et al. 2008).

Finally, we tested the predictions obtained with simulations analyzing
the real count data collected in GPNP over the last 65 years and
comparing trends estimated from the entire area with trend estimations
that would have been obtained from sample counts.

\textbf{METHODS}

\textbf{1- Study species and area}

The Alpine ibex is a mountain ungulate endemic to the European Alps,
where it lives mostly in alpine meadows and rocky cliffs between 1,600
and 3,200 m asl (Brambilla et al. 2020b). The species recovered from
near extinction during the last century (Biebach and Keller 2009) and
thanks to reintroduction programs, is now present again on the entire
Alpine arc with a population of around 55,000 individuals (Brambilla et
al. 2020a). However, low genetic variability (Biebach and Keller 2010)
and climate change (Brivio et al. 2019) may constitute a severe threat
for the species in the future.

The Alpine ibex population in Gran Paradiso National Park ranged from
\textasciitilde{}2,600 to \textasciitilde{}5,000 individuals over the
last 65 years, with a fluctuating population dynamic, showing a recent
strong decline after a population peak in the 1990's (Jacobson et al.
2004, Mignatti et al. 2012). Ibex abundance in GPNP has been estimated
with annual block counts performed by Park rangers every year since 1956
over the whole territory of the Park (with a total area of 720
km\textsuperscript{2}). Block counts are performed by walking over
established routes and from vantage points at the beginning of
September, dividing the entire Park area into 38 sectors (or
surveillance areas) that range from 5.24 to 37.21
km\textsuperscript{2}each. Alpine ibex of both sexes show a rather high
site fidelity (Parrini et al. 2003, Grignolio et al. 2004), and are
assumed to remain in the same sector during the summer season.

\textbf{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 (\emph{r} ) over a 10 or 20-year
period, either with a decrease or increase in population size. The
different values of \emph{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 \emph{r} , assigning a random
trend,\emph{T\textsubscript{y,s}} , to each \emph{y-th} year and
\emph{s-th}sector, sampled around the value of \emph{r} with a specified
coefficient of variation (cv\textsubscript{y} for the year and
cv\textsubscript{s} 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 (cv\textsubscript{y}) and
sectors (cv\textsubscript{s}). 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,\emph{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
only\emph{n} 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 \emph{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 \emph{n} survey sectors were selected as: i) \emph{n} sectors
selected at random the first year and then sampled each following year;
ii) \emph{n} randomly selected sectors where the random selection was
repeated each year; iii) a biased selection with only the
\emph{n}initial 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
(cv\textsubscript{s}), cv between years (cv\textsubscript{y}), cv for
detectability between sectors (cv\textsubscript{d}), 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.
\textbar{}(r\textsubscript{estimated} - r) / r \textbar{} \textless{}
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).

\textbf{3- Power analysis on real ibex trends in GPNP}

We randomly sampled 112,000 total time intervals each of a specified
length (5, 10, 15 or 20 years) from the original dataset of Alpine ibex
counts in GPNP and estimated the population trend with a Generalized
Linear Model (GLM) for the Poisson distribution. If a trend was detected
(with P\textless{}0.05) in the target interval using the full number of
sectors (complete trend), we randomly selected n sectors (with 1
\textless{} n \textless{} 38) and estimated again the trend using only
the sum of the individuals counted in the n sectors (sample trend). We
compared the two trends and, following Wauchope et al. (2019),
classified the sample trend as matching or opposing the complete trend,
missed (sample counts that did not detect the complete trend) or false
trends (sample counts that erroneously detected a stable complete
trend). As for our simulations, we considered the sample trend correct
in magnitude if the difference with the complete trend was of 0.02 or
lower (2\% error each year). The procedure was repeated 1,000 times for
each possible number of sampled sectors (1 to 38) and we assessed the
statistical power as the proportion of samples in which the sample trend
matched the complete trend, in terms of direction (increase or decrease
in the population) or magnitude, with a threshold power of 0.8 (80\%).
All the analyses were performed in R, version 4.1.3 (R Core Team 2022)
with the code provided in supplementary materials.

\textbf{RESULTS}

\textbf{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 (cv\textsubscript{y} = 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 cv\textsubscript{y} = 0.05 and r = 0.982,
the number of sectors required to quantify the 20-years trend passed
from 12 with a low cv\textsubscript{s} to 22 for a high
cv\textsubscript{s} and from 13 with a low cv\textsubscript{d} to 19
with a high cv\textsubscript{d} (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
cv\textsubscript{y} = 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.

\textbf{2. Power analysis on real ibex trends in GPNP}

If the complete trend observed in the ibex count data in GPNP was
significant (86.9\% of the time intervals), sample counts were able to
correctly detect its direction with a sufficient statistical power (fig.
5a) using a number of sectors between 5 (for a 20-year trend) and 28
(for a 5-year trend). To correctly measure the magnitude of the trend, a
higher number of sectors was required (fig.5b). The complete trend was
detected with an error lower than 2\% and in at least 80\% of the cases
if 10 (for a 20-years trend) to 33 (for a 5-years trend) sectors were
surveyed. To detect a trend over 10 years, 15 sectors were needed to
correctly assess the direction of the trend and 22 sectors to also
detect its magnitude, while for a 20-year trend 10 sectors were
sufficient to accurately quantify the trend. Sampling more sectors
reduced the number of false, missed and opposing trends (fig.6). Indeed,
counting in the full area compared to half of sectors led to false
trends declining from 14\% to 6\%, missed trends from 10\% to 3\% and
opposing trends from 5\% to 0\%.

\textbf{DISCUSSION}

Our results show that sample counts can be adequate to monitor
population trends of mountain ungulates in a large area. Sampling half
of the total area allowed us to correctly identify medium to strong
population trends. In addition, in most of the cases in which sample
counts were not able to estimate the population trend, also complete
counts failed in detecting the direction or magnitude of the trend. The
retrospective power analysis confirmed our results, as the trends
historically estimated with complete counts would have been inferred
equally well, with a sufficient statistical power, using sample counts
with only half of the sectors monitored. Given that the results obtained
with sample counts are very similar to those obtained with a complete
count, we suggest that sample counts (in particular monitoring half of
the target area) can be used as a viable alternative when monitoring
trends, hence allowing for important, cost-effective improvements in the
monitoring of wild animals of conservation interest. Time and budget
constraints can indeed disincentivize wildlife managers to implement
monitoring programs (McDonald-Madden et al. 2010), while potentially
reducing the survey effort could lead to undertake monitoring projects
that are otherwise impossible to perform.

In the case of Alpine ibex, the species is currently monitored
throughout its entire area of extent in the Alps without a standardized
protocol and, while in GPNP a long time series is available, the lack of
data in many other areas makes impossible to draw reliable conclusions
about the population trends (Brambilla et al. 2020a). Reducing the costs
of monitoring by using sample counts, could therefore be of critical
importance to allow more conservation and management authorities to
perform periodic counts. Assessing population trends in Alpine ibex is
essential as the species may be sensitive to future declines (Toïgo et
al. 2020) because of low recolonization rates, low genetic diversity and
high inbreeding levels (Biebach and Keller 2009, Brambilla et al. 2014,
2020b) and heat-sensitivity (Aublet et al. 2008, Mason et al. 2017,
Semenzato et al. 2021) in an environment that suffers strong global
warming effects (Giorgi 2006, Gobiet et al. 2014, Rogora et al. 2018).
However, as the reliability of sample counts depends on the variability
between years and sectors\emph{,} wildlife managers must carefully
evaluate the expected reliability of censuses in their specific case.

Sample counts can also be useful to perform a quick and highly
cost-effective preliminary monitoring in new areas to collect
information on the status of a population in the short term, as we
showed that the direction of a strong trend over 10 years can be
detected even with very few sectors sampled (less than 5 out of 38). For
instance, the method can be useful to detect recovery after epidemic
outbreaks with high mortality, common in mountain ungulates as the
Alpine ibex (Giacometti et al. 2002, Garnier et al. 2016, Pérez et al.
2021), or to identify a strong response after management actions such as
reintroductions (Giacometti 1991, Stüwe and Nievergelt 1991, Brambilla
et al. 2020a).

However, if there are sufficient resources, sampling the entire area may
still be preferable than using sample counts. Sampling all sectors,
indeed, reduced the occurrence of errors in trend detection that, even
if at a low frequency, were present with sample counts. Furthermore,
sampling the entire area would allow for less biased abundance
estimates, as shown in our results and as pointed out by Sutherland
(2006). An accurate estimate of population size is important for
instance to measure carrying capacity (Holzgang 1997, Terry Bowyer et
al. 2014) or for management purposes such as reintroductions (Peracino
and Bassano 1990) and hunting (e.g., Carvalho et al. 2018). Therefore,
conservation authorities must weigh up the costs and benefits of using
sample counts, using them to reliably detect population trends at a
lower cost or performing complete counts to measure abundance with
higher financial effort but also reduced errors in the estimation of
trends and abundance.

We also showed that neither sample counts nor complete counts could
reliably monitor the magnitude of short-term trends and that for such
analysis at least 15 or 20 years of censuses data are needed. This
result is consistent with several other studies pointing out that more
than 10 years are required to draw reliable conclusions on population
trends (e.g., Gerber et al. 1999, Hatch 2003, White 2019). Long-term
wildlife monitoring projects are indeed of critical importance for
conservation (Nichols and Williams 2006, Magurran et al. 2010,
Giron-Nava et al. 2017), but series of yearly counts are lacking for
most mountain ungulates (Singh and Milner-Gulland 2011, Brambilla et al.
2020a, Nuttall et al. 2022). Therefore, we further recommend that
stakeholders plan long-term monitoring projects to correctly evaluate
population dynamics at a local and global scale. Sample counts, that
proved in this study to be as reliable as complete counts in detecting
the long-term trends if half of the sectors are sampled, can constitute
a viable method to reduce the required census effort for such projects,
but further studies are needed before they can also be applied in
abundance estimations.

Our results show that the strongest constraint for accurate trend
estimations is the variability between years, for which Alpine ibex in
GPNP showed a coefficient of variation around 0.05 in the last 65 years.
Keith et al. (2015) found that, among nine species of terrestrial
mammals for which population trend was estimated counting the total
number of individuals in a specified area (mostly Cricetidae and
Muridae), the mean variability between years was 0.052, and \emph{Saiga
tatarica} , the only ungulate species in the study, showed a yearly
variation of 0.058. Therefore, sample counts would likely have been
efficient for estimating the trend also in those species and other with
similar trend variation over time. Under a greater variability,
estimating the population dynamics could be subject to severe errors
even with complete counts and especially for weak trends, as also
pointed out by other studies (Wilson et al. 2011, Rhodes and Jonzén
2011, Rueda-Cediel et al. 2015).

In general, in Keith et al. (2015) species with lower yearly variation
of abundance were mostly those with a higher generation time
(coefficient of variation was 0.088 for species that give birth to a new
generation within 2 years and 0.026 for species with longer generation
time, with a mean variability of 0.014 if the generation length was
higher than 5 years). Species with a long life-history can indeed show
less extreme yearly abundance fluctuations, as for example they are
slower to react to fast environmental changes (Berteaux et al. 2004).
However, White (2019) showed that life history traits (such as
generation length) have only a very weak influence on the minimum time
required to accurately monitor the species, while time series
characteristics (e.g., trend strength) seems to be a much more important
driver. Our results could therefore potentially be relevant to any
animal species with a sufficient generation length (possibly more than 2
years), but the applicability to other species with a different
life-history must be tested.

In our simulations, sampling half of the sectors allowed to achieve a
sufficient statistical power even with high values for
cv\textsubscript{s} and cv\textsubscript{d}. A large number of species
is likely to exhibit a trend variability between sectors in the range we
used in our simulations (i.e., 0.05-0.20): Weiser et al. (2019) reported
a sector-specific variation between 0.05 and 0.1 in many species of
different taxa (from 0.02 to 0.024 in small mammals). Additionally, the
wide range of cv\textsubscript{d} we used in the simulations (0.1-0.5)
suggests that our results can be expanded to many study areas in which
spatial variation of parameters that affect detectability is high.

We also show that the overall detectability in the target area did not
have any effect on the statistical power, thus trend estimation could be
performed with cost-effective methods such as block counts, that are
believed to underestimate population size (Loison et al. 2006, Morellet
et al. 2007, Corlatti et al. 2015). In such case, the costs of trend
estimations could be further reduced by the use of block counts instead
of other techniques, that account for imperfect detection but require a
higher field or data analysis effort, such as Capture-Mark-Resight
(Schwarz and Seber 1999) or Distance Sampling (Buckland et al. 2001).
Besides, also several sources of errors in abundance estimation, such as
underestimations of group size (Vallecillo et al. 2021) or missed
detection of small groups (Samuel and Pollock 1981), could possibly be
ignored while performing sample counts to estimate direction and
magnitude of population trends. Our results about detectability are in
contrast to other studies, for which a low detectability critically
influenced the ability of detecting a trend (e.g. Newson et al. 2013,
Ficetola et al. 2018, Sanz-Pérez et al. 2020). However, some of these
studies included the possibility of sites with very few animals and a
detectability (p) close to zero (e.g. abundance of 7.5 per site and
p=0.05 in Ficetola et al. 2018), thus analyzed the situation in which no
animals were detected in most of the sites. This phenomenon, not present
in our Alpine ibex population where the population per sector was
higher, could have been therefore the major cause of a difficult trend
estimation in such studies. Besides, in other of the above mentioned
researches, the detectability was also variable across the sites (e.g.
Sanz-Pérez et al. 2020), and such effect could have been the actual
driver of a difficult trend estimation, rather than the value of
detectability, as we show here.

When using sample counts, the monitoring costs can be further reduced by
performing censuses only in the sectors with the highest abundance (that
can be easier to sample). However, as the method of sector selection had
a weak effect on statistical power of trend estimations, also selecting
sectors based on other needs could lead to a reliable analysis. This
result is in contrast with Fournier et al. (2019) who claimed that
counting in the best sites (i.e., those with the highest abundance)
leads to detect false trends (Fournier et al. 2019). However, as also
pointed out by Fournier and colleagues, not every real population
exhibits this bias. The advantage of selecting the most abundant sectors
however disappears if abundance is estimated instead of the population
trend, with a considerable overestimation compared to a random sector
selection.

\textbf{CONCLUSIONS}

In this study we tested if sample counts allow the detection of
population trends in a wild mountain ungulate, using simulated and real
census data. We show that in most of the cases in which sampling the
entire area would be able to detect population trends, sample counts in
only half of the sectors have a sufficient statistical power to draw
reliable conclusions about the population trend. Furthermore, a very
small proportion of the total area is required to detect a strong trend
even in a short period. Therefore, in taxa such as mountain ungulates,
for which the cost of monitoring is high, sample counts could constitute
a viable alternative to reduce the amount of resources needed for
censuses, to monitor a higher number of populations of the species and
to detect rapid changes of conservation interest. The results can be
potentially extended to other species of mammals, especially those who
exhibit a long generation time, and sample counts could be used
successfully and extensively to reduce costs of monitoring.
Nevertheless, the applicability of the method to species that exhibit
high trend variability or with a short generation time should be
carefully evaluated. Furthermore, as estimating population trends under
high trend variability and weak trends on a short term is challenging,
long-term monitoring is fundamental to correctly assess population
dynamics of wild animals. At last, abundance extrapolated from sample
counts could be biased compared to that estimated with complete counts
in the entire target area, and for this reason sampling the entire area
is probably still a preferable option (where there are adequate
resources) to estimate abundance instead of the population trend.

Future developments of this study can include evaluating the
effectiveness of sample counts with real census data from other ungulate
and non-ungulate species, testing the applicability of the method to the
wider possible number of species, or investigating in other systems the
effect of detectability and of a biased sector selection, in order to
identify cheaper census procedures also in other species.

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\textbf{Tab.1 .} Parameters used in the simulations and their values\selectlanguage{english}
\begin{longtable}[]{@{}lllll@{}}
\toprule
\begin{minipage}[b]{0.17\columnwidth}\raggedright\strut
\textbf{Parameter}\strut
\end{minipage} & \begin{minipage}[b]{0.17\columnwidth}\raggedright\strut
\textbf{Meaning}\strut
\end{minipage} & \begin{minipage}[b]{0.17\columnwidth}\raggedright\strut
\textbf{N of possible values}\strut
\end{minipage} & \begin{minipage}[b]{0.17\columnwidth}\raggedright\strut
\textbf{Values}\strut
\end{minipage} & \begin{minipage}[b]{0.17\columnwidth}\raggedright\strut
\textbf{Meaning}\strut
\end{minipage}\tabularnewline
\midrule
\endhead
\begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
A\strut
\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
Abundance\strut
\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
3\strut
\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
3000, 4000, 5000\strut
\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
Low, medium or high population size in GPNP\strut
\end{minipage}\tabularnewline
\begin{minipage}[t]{0.19\columnwidth}\raggedright\strut
r\strut
\end{minipage} & \begin{minipage}[t]{0.19\columnwidth}\raggedright\strut
Overall yearly trend over 10 years\strut
\end{minipage} & \begin{minipage}[t]{0.19\columnwidth}\raggedright\strut
5 5\strut
\end{minipage} & \begin{minipage}[t]{0.19\columnwidth}\raggedright\strut
1.01, 1.018, 1.027, 1.034, 1.041 0.989, 0.978, 0.965, 0.95, 0.933\strut
\end{minipage} & \begin{minipage}[t]{0.19\columnwidth}\raggedright\strut
10\%,20\%,30\%,40\% or 50\% total population increase in 10 years
-10\%,-20\%,-30\%,-40\% or -50\% total population increase in 10
years\strut
\end{minipage}\tabularnewline
\begin{minipage}[t]{0.19\columnwidth}\raggedright\strut
\strut
\end{minipage} & \begin{minipage}[t]{0.19\columnwidth}\raggedright\strut
Overall yearly trend over 20 years\strut
\end{minipage} & \begin{minipage}[t]{0.19\columnwidth}\raggedright\strut
5 5\strut
\end{minipage} & \begin{minipage}[t]{0.19\columnwidth}\raggedright\strut
1.005, 1.009, 1.013, 1.017, 1.025 0.995, 0.989, 0.982, 0.975,
0.966\strut
\end{minipage} & \begin{minipage}[t]{0.19\columnwidth}\raggedright\strut
10\%,20\%,30\%,40\% or 50\% total population increase in 20 years 10\%,
20\%, 30\%, 40\% or 50\% total population increase in 20 years\strut
\end{minipage}\tabularnewline
\begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
cvy\strut
\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
Coefficient of variation of the trend between years\strut
\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
4\strut
\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
0.05, 0.1, 0.15, 0.2\strut
\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
Low, medium and high variability\strut
\end{minipage}\tabularnewline
\begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
cvs\strut
\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
Coefficient of variation of the trend between sectors\strut
\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
4\strut
\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
0.05, 0.1, 0.15, 0.2\strut
\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
Low, medium and high variability\strut
\end{minipage}\tabularnewline
\begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
P\strut
\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
Detection probability\strut
\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
3\strut
\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
0.4, 0.6, 0.8\strut
\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
Low, medium and high detectability\strut
\end{minipage}\tabularnewline
\begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
cvd\strut
\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
Coefficient of variation for detection probability\strut
\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
3\strut
\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
0.1, 0.3, 0.5\strut
\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
Low, medium or high variability\strut
\end{minipage}\tabularnewline
\begin{minipage}[t]{0.19\columnwidth}\raggedright\strut
\strut
\end{minipage} & \begin{minipage}[t]{0.19\columnwidth}\raggedright\strut
Sector-selection (method to select the survey sectors)\strut
\end{minipage} & \begin{minipage}[t]{0.19\columnwidth}\raggedright\strut
3\strut
\end{minipage} & \begin{minipage}[t]{0.19\columnwidth}\raggedright\strut
``random each'' ``random first'' ``best''\strut
\end{minipage} & \begin{minipage}[t]{0.19\columnwidth}\raggedright\strut
Sectors are selected randomly each year Sectors are selected randomly
the first year Only the sectors with the highest abundance are
selected\strut
\end{minipage}\tabularnewline
\begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
n\strut
\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
Number of sectors used for the estimation\strut
\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
38\strut
\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
1 to 38\strut
\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
\strut
\end{minipage}\tabularnewline
\bottomrule
\end{longtable}

\textbf{Tab.2} Model selection for the effect of simulation parameters
on the statistical power of sample counts in detecting the true
population trend\selectlanguage{english}
\begin{longtable}[]{@{}lllll@{}}
\toprule
\begin{minipage}[b]{0.17\columnwidth}\raggedright\strut
\textbf{Response}\strut
\end{minipage} & \begin{minipage}[b]{0.17\columnwidth}\raggedright\strut
\textbf{Models}\strut
\end{minipage} & \begin{minipage}[b]{0.17\columnwidth}\raggedright\strut
\textbf{dF}\strut
\end{minipage} & \begin{minipage}[b]{0.17\columnwidth}\raggedright\strut
\textbf{AIC}\strut
\end{minipage} & \begin{minipage}[b]{0.17\columnwidth}\raggedright\strut
\selectlanguage{greek}\textbf{ΔAIC}\selectlanguage{english}\strut
\end{minipage}\tabularnewline
\midrule
\endhead
\begin{minipage}[t]{0.19\columnwidth}\raggedright\strut
Power 10-years decreasing trend\strut
\end{minipage} & \begin{minipage}[t]{0.19\columnwidth}\raggedright\strut
\textbf{\textasciitilde{} r + n + cvy + cvd + cvs + selection}\strut
\end{minipage} & \begin{minipage}[t]{0.19\columnwidth}\raggedright\strut
9\strut
\end{minipage} & \begin{minipage}[t]{0.19\columnwidth}\raggedright\strut
- 83178.6\strut
\end{minipage} & \begin{minipage}[t]{0.19\columnwidth}\raggedright\strut
0\strut
\end{minipage}\tabularnewline
\begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
\strut
\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
\textasciitilde{} r + n + cvy + cvd + selection\strut
\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
8\strut
\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
- 83155.7\strut
\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
22.9\strut
\end{minipage}\tabularnewline
\begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
\strut
\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
\textasciitilde{} r + n + cvy + cvs + selection\strut
\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
8\strut
\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
- 82995.1\strut
\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
183.5\strut
\end{minipage}\tabularnewline
\begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
\strut
\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
\textasciitilde{} r + n + cvy + selection\strut
\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
7\strut
\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
- 82972.4\strut
\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
206.2\strut
\end{minipage}\tabularnewline
\begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
\strut
\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
\textasciitilde{} r + n + cvy + cvd + cvs\strut
\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
7\strut
\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
- 82105.0\strut
\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
1073.5\strut
\end{minipage}\tabularnewline
\begin{minipage}[t]{0.19\columnwidth}\raggedright\strut
Power 20-years decreasing trend\strut
\end{minipage} & \begin{minipage}[t]{0.19\columnwidth}\raggedright\strut
\textbf{\textasciitilde{} r + n + cvy + cvd + cvs + selection}\strut
\end{minipage} & \begin{minipage}[t]{0.19\columnwidth}\raggedright\strut
9\strut
\end{minipage} & \begin{minipage}[t]{0.19\columnwidth}\raggedright\strut
- 90655.7\strut
\end{minipage} & \begin{minipage}[t]{0.19\columnwidth}\raggedright\strut
0\strut
\end{minipage}\tabularnewline
\begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
\strut
\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
\textasciitilde{} r + n + cvy + cvs + selection\strut
\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
8\strut
\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
- 90624.5\strut
\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
31.2\strut
\end{minipage}\tabularnewline
\begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
\strut
\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
\textasciitilde{} r + n + cvy + cvd + selection\strut
\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
8\strut
\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
- 90590.7\strut
\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
65.0\strut
\end{minipage}\tabularnewline
\begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
\strut
\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
\textasciitilde{} r + n + cvy + selection\strut
\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
7\strut
\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
- 90559.6\strut
\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
96.1\strut
\end{minipage}\tabularnewline
\begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
\strut
\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
\textasciitilde{} r + n + cvy + cvd + cvs\strut
\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
7\strut
\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
- 90318.4\strut
\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
337.3\strut
\end{minipage}\tabularnewline
\bottomrule
\end{longtable}

\textbf{Tab 3.} Summary of the best models for the effect of simulation
parameters on the statistical power. All variables were standardized\selectlanguage{english}
\begin{longtable}[]{@{}lllll@{}}
\toprule
\begin{minipage}[b]{0.17\columnwidth}\raggedright\strut
\textbf{Model}\strut
\end{minipage} & \begin{minipage}[b]{0.17\columnwidth}\raggedright\strut
\textbf{Parameter}\strut
\end{minipage} & \begin{minipage}[b]{0.17\columnwidth}\raggedright\strut
\textbf{Estimate}\strut
\end{minipage} & \begin{minipage}[b]{0.17\columnwidth}\raggedright\strut
\textbf{Std. error}\strut
\end{minipage} & \begin{minipage}[b]{0.17\columnwidth}\raggedright\strut
\textbf{P value}\strut
\end{minipage}\tabularnewline
\midrule
\endhead
\begin{minipage}[t]{0.19\columnwidth}\raggedright\strut
10y trend (decrease)\strut
\end{minipage} & \begin{minipage}[t]{0.19\columnwidth}\raggedright\strut
Intercept\strut
\end{minipage} & \begin{minipage}[t]{0.19\columnwidth}\raggedright\strut
0.793\strut
\end{minipage} & \begin{minipage}[t]{0.19\columnwidth}\raggedright\strut
\textless{}0.001\strut
\end{minipage} & \begin{minipage}[t]{0.19\columnwidth}\raggedright\strut
P\textless{}0.001\strut
\end{minipage}\tabularnewline
\begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
\strut
\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
r\strut
\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
-0.113\strut
\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
\textless{}0.001\strut
\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
P\textless{}0.001\strut
\end{minipage}\tabularnewline
\begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
\strut
\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
n\strut
\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
0.034\strut
\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
\textless{}0.001\strut
\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
P\textless{}0.001\strut
\end{minipage}\tabularnewline
\begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
\strut
\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
cvy\strut
\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
-0.037\strut
\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
\textless{}0.001\strut
\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
P\textless{}0.001\strut
\end{minipage}\tabularnewline
\begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
\strut
\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
cvd\strut
\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
-0.004\strut
\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
\textless{}0.001\strut
\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
P\textless{}0.001\strut
\end{minipage}\tabularnewline
\begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
\strut
\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
cvs\strut
\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
-0.002\strut
\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
\textless{}0.001\strut
\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
P\textless{}0.001\strut
\end{minipage}\tabularnewline
\begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
\strut
\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
selection random each\strut
\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
-0.026\strut
\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
\textless{}0.001\strut
\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
P\textless{}0.001\strut
\end{minipage}\tabularnewline
\begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
\strut
\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
selection random first\strut
\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
-0.016\strut
\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
\textless{}0.001\strut
\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
P\textless{}0.001\strut
\end{minipage}\tabularnewline
\begin{minipage}[t]{0.19\columnwidth}\raggedright\strut
20y trend (decrease)\strut
\end{minipage} & \begin{minipage}[t]{0.19\columnwidth}\raggedright\strut
Intercept\strut
\end{minipage} & \begin{minipage}[t]{0.19\columnwidth}\raggedright\strut
0.786\strut
\end{minipage} & \begin{minipage}[t]{0.19\columnwidth}\raggedright\strut
\textless{}0.001\strut
\end{minipage} & \begin{minipage}[t]{0.19\columnwidth}\raggedright\strut
P\textless{}0.001\strut
\end{minipage}\tabularnewline
\begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
\strut
\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
r\strut
\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
-0.094\strut
\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
\textless{}0.001\strut
\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
P\textless{}0.001\strut
\end{minipage}\tabularnewline
\begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
\strut
\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
n\strut
\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
0.018\strut
\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
\textless{}0.001\strut
\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
P\textless{}0.001\strut
\end{minipage}\tabularnewline
\begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
\strut
\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
cvy\strut
\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
-0.028\strut
\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
\textless{}0.001\strut
\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
P\textless{}0.001\strut
\end{minipage}\tabularnewline
\begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
\strut
\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
cvd\strut
\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
-0.002\strut
\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
\textless{}0.001\strut
\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
P\textless{}0.001\strut
\end{minipage}\tabularnewline
\begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
\strut
\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
cvs\strut
\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
-0.002\strut
\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
\textless{}0.001\strut
\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
P\textless{}0.001\strut
\end{minipage}\tabularnewline
\begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
\strut
\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
selection random each\strut
\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
-0.012\strut
\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
\textless{}0.001\strut
\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
P\textless{}0.001\strut
\end{minipage}\tabularnewline
\begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
\strut
\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
selection random first\strut
\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
-0.009\strut
\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
\textless{}0.001\strut
\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
P\textless{}0.001\strut
\end{minipage}\tabularnewline
\bottomrule
\end{longtable}\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image1/image1}

\end{center}
\end{figure}

\textbf{Fig.1} Statistical power of sample counts in correctly detecting
the direction of the real population trend over 10 years (left) and 20
years (right). r is the total population decline over the time period
and cvy is the yearly coefficient of variation in the trend. Statistical
power is averaged between all the combinations of the other parameters.\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image2/image2}

\end{center}
\end{figure}\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image11/image11}

\end{center}
\end{figure}

\textbf{Fig.2} Statistical power of sample counts in correctly detecting
the magnitude (with an yearly error lower than 2\%) of the real
population trend over 10 years (left) and 20 years (right). r is the
total population decline over the time period and cvy is the yearly
coefficient of variation in the trend. Statistical power is averaged
between all the combinations of the other parameters.\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image15/image15}

\end{center}
\end{figure}

\textbf{Fig.3} Statistical power for sample counts in detecting the
direction (fig. 3a) and magnitude (fig. 3b) of the population trend for
a 10 years trend (on the left) and 20 years (on the right). The hashed
line represents the threshold for statistical power (0.8). Trend
variability between years and sectors is the one measured for ibex in
GPNP (cvy=0.05, cvs=0.10) and statistical power is averaged between all
the combinations of the other parameters.\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image3/image3}

\end{center}
\end{figure}

\textbf{Fig.4} Proportion of correct and erroneous trends detected
surveying half of the sectors in 10 years (left) and 20 years (right) in
the simulated populations of Alpine ibex in GPNP (cvy = 0.05 and cvs =
0.1). The correct magnitude correspond to the simulations in which
yearly trend is estimated with less than 2\% error, the biased magnitude
to all the other cases in which the direction of the trend is correctly
estimated but with an error higher than 2\% yearly. Irrelevant trend
correspond to the cases in which sample counts erroneously estimate a
trend that is non-significative (P\textgreater{}0.05) and irrelevant
trend are the simulations in which a trend in the opposite direction
(increase of the population) is detected. The first bar on the left of
each graph is a decrease of 50\% over the time period, the rightmost a
decrease of 10\%\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image16/image16}

\end{center}
\end{figure}

\textbf{Fig.5} Statistical power for sample trends in detecting the
direction (left) and the magnitude (right) of a significant complete
trend in census data in GPNP over the last 65 years\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image4/image4}

\end{center}
\end{figure}

\textbf{Fig.6} Percentage of false, missed and opposing trends in the
retrospective power analysis given the number of sectors sampled

\selectlanguage{english}
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\end{document}

