Testing for changes in population trends from low-cost ecological count data

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Skip to main content Home About Submit ALERTS / RSS Search for this keyword Advanced Search New Results Testing for changes in population trends from low-cost ecological count data View ORCID Profile Liam Singer , View ORCID Profile Madleina Caduff , View ORCID Profile Thierry Aebischer , Primaël Tabiti , View ORCID Profile Aimée Freiberg , View ORCID Profile Jens Ingensand , View ORCID Profile Christoph Leuenberger , View ORCID Profile Daniel Wegmann doi: https://doi.org/10.1101/2025.01.08.631844 Liam Singer a Department of Biology, Université de Fribourg , Chemin du Musée 10, CH-1700 Fribourg, Switzerland c Swiss Institute of Bioinformatics , 1700 Fribourg, Switzerland Find this author on Google Scholar Find this author on PubMed Search for this author on this site ORCID record for Liam Singer Madleina Caduff a Department of Biology, Université de Fribourg , Chemin du Musée 10, CH-1700 Fribourg, Switzerland c Swiss Institute of Bioinformatics , 1700 Fribourg, Switzerland Find this author on Google Scholar Find this author on PubMed Search for this author on this site ORCID record for Madleina Caduff Thierry Aebischer d Aire de Conservation de Chinko, African Parks RCA , Bangui, Central African Republic Find this author on Google Scholar Find this author on PubMed Search for this author on this site ORCID record for Thierry Aebischer Primaël Tabiti d Aire de Conservation de Chinko, African Parks RCA , Bangui, Central African Republic Find this author on Google Scholar Find this author on PubMed Search for this author on this site Aimée Freiberg a Department of Biology, Université de Fribourg , Chemin du Musée 10, CH-1700 Fribourg, Switzerland c Swiss Institute of Bioinformatics , 1700 Fribourg, Switzerland Find this author on Google Scholar Find this author on PubMed Search for this author on this site ORCID record for Aimée Freiberg Jens Ingensand e School of Engineering and Management Vaud, HES-SO, University of Applied Sciences and Arts Western Switzerland Find this author on Google Scholar Find this author on PubMed Search for this author on this site ORCID record for Jens Ingensand Christoph Leuenberger b Department of Mathematics, Université de Fribourg , Chemin du Musée 23, CH-1700 Fribourg, Switzerland Find this author on Google Scholar Find this author on PubMed Search for this author on this site ORCID record for Christoph Leuenberger Daniel Wegmann a Department of Biology, Université de Fribourg , Chemin du Musée 10, CH-1700 Fribourg, Switzerland c Swiss Institute of Bioinformatics , 1700 Fribourg, Switzerland Find this author on Google Scholar Find this author on PubMed Search for this author on this site ORCID record for Daniel Wegmann For correspondence: daniel.wegmann{at}unifr.ch Abstract Full Text Info/History Metrics Supplementary material Preview PDF Abstract Accurate and up-to-date knowledge of population trends is essential for effective biodiversity conservation, as is assessing the impact of conservation measures designed to alter these trends. Estimating population trends is challenging, however, either due to altogether insufficient data or due to so-called noisy data that do not readily allow for standard statistical analyses. In addition, many existing methods require monitoring data over long periods of time, which is in contrast to the quick interventions needed by conservation projects, especially when endangered species are involved. To address these issues, we here present birp , a novel Bayesian tool that maximizes the power to test for population trends and changes in trends under arbitrary designs, including the canonical before-after (BA), control-intervention (CI) and before-after-control-intervention (BACI) designs often used to assess conservation impact. Our model builds on classic Poisson and negative binomial models for ecological count data and infers changes in population trends jointly from data obtained with multiple survey methods such as track counts, camera trap surveys, or distance sampling, and also from limited and noisy data not necessarily collected in standardized ecological surveys. By focusing on the change itself, our method side-steps common challenges of estimating population trends and does not need to know about absolute population densities or detection probabilities. birp is open-source and available as both a standalone command line tool as well as an R-package for fast and easy use. We illustrate the power of our tool through extensive simulations and show that changes in trends are accurately estimated under various designs, even when data are noisy and sparse, and thereby enables biodiversity research also in regions that are remote and difficult to access. Using birp , we further test for changes in population trends of Tasmanian devils in Australia and of apex predators and their main prey in the Central African Republic. Based on these results we give general guidelines on survey designs that maximize the power to detect trends. Introduction Accurate knowledge about population trends is crucial for biodiversity monitoring and effective conservation practice. The IUCN red list, for instance, uses population trends to classify their species and help inform conservation actions ( Rodrigues et al., 2006 ; Maes et al., 2015 ). Similarly, the decline of native populations is regarded as a key criteria in assessing the impact of invasive species ( Bacher et al., 2018 ; Pyšek et al., 2020 ). Since trends are considered crucial predictors on the state of biodiversity, management interventions are expected to slow, halt or even reverse negative trends. Similarly, a new threat or an increase of pressure on biodiversity is expected to lead to more negative trends. Hence, the effectiveness of management interventions and the impact of biodiversity threats is best evaluated through counterfactual inference that compares realized trends to those expected in the absence of the intervention or the previous pressure regime ( Butchart et al., 2010 ; Wauchope et al., 2021 ; Volery et al., 2023 ). Indeed, many ecological surveys are conducted primarily to answer the question whether a population trend changed following a specific intervention (e.g. Geldmann et al., 2013 ; Terraube et al., 2020 ; Pellissier et al., 2020 ; Wauchope et al., 2022 ; Barnes et al., 2023 ; Langhammer et al., 2024 ). An estimate of the expected trend in the absence of an intervention, the so-called counterfactual , is difficult to obtain experimentally as natural conditions can rarely be truly replicated ( Ferraro, 2009 ). As a consequence, in ecology, the counterfactual is commonly estimated using one of three quasi-experimental approaches ( Butsic et al., 2017 ; Wauchope et al., 2021 ; Volery et al., 2023 ): If data is available from control locations without intervention, they may be used to estimate the counterfactual, an approach termed control-intervention ( CI ). In many cases, however, such locations are not available, in which case the counterfactual may be predicted from time-series data obtained prior to the intervention, an approach termed before-after ( BA ). Finally, in some cases, the counterfactual may be estimated from control locations and data prior to the intervention jointly, an approach termed before-after-control-intervention ( BACI ). Not surprisingly, the BACI design was found to be the most powerful, if such data was available (e.g. Wauchope et al., 2022 ). A common approach to infer population trends is to first estimate and then compare absolute population densities at different points in time ( Collen et al., 2009 ). However, inferring population densities requires to deal with the stochastic nature of survey data as well as extensive, localized knowledge on how to translate the density of observations into that of populations (e.g. Buckland et al., 2001 ; Webster et al., 2010 ; Chelintsev, 1995 ). The reason for this is two fold: First, observations are imperfect such that true presences may be missed. Estimating population densities thus requires knowledge on the probability of detection, or means to estimate it from the data. Second, many surveys gather indirect observations (e.g. tracks or nests) that are detectable for an extended amount of time. Yet, translating such indirect observations into population densities requires knowledge on the rate at which animals leave observable traces as well as the rate at which they decay (e.g. Mathewson et al., 2008 ; Kühl et al., 2008 ; Keeping and Pelletier, 2014 ). While for some species, calibrated protocols have been established (e.g. Midlane et al., 2015 ; Russell et al., 2012 ; Funston et al., 2010 ), such protocols are extremely laborious to obtain and they are not easily transferable between regions. If the interest lies specifically in assessing population trends rather than absolute densities, however, (non-)linear models may be fitted directly to time-series data of direct or indirect observations ( Collen et al., 2009 ; Farmer et al., 2007 ; Fewster et al., 2000 ; Wauchope et al., 2019 ). From such models, important change points, i.e. times at which the trend changes, can then be inferred post-hoc , for instance through second derivatives ( Fewster et al., 2000 ). However, observational data results from stochastic processes modulated by local conditions, survey methods and survey efforts. To draw meaningful inference, these must thus either be perfectly standardized across surveys, or accounted for in the statistical analysis. Due to logistical challenges, ecological surveys in difficult-to-access and remote regions are often not conducted in a perfectly standardized manner. As a consequence, the majority of published biodiversity surveys were conducted in few, highly accessible areas where socio-economic conditions are favorable for research, and mostly focus on a few well-studied taxa only (e.g. Boakes et al., 2010 ; Ahrends et al., 2011 ; Meyer et al., 2015 , 2016; Di Marco et al., 2017 ; Hickisch et al., 2019 ; Lenoir et al., 2020 ; Wauchope et al., 2022 ). While technological advances like camera traps ( Sollmann et al., 2015 ) may reduce the logistical burden of conducting surveys in understudied regions, the data obtained are likely to remain scarce and noisy. Yet, filling the remaining spatial biodiversity knowledge gaps is crucial, considering that conservation priorities are usually identified based on species richness, endemism and threat, estimates of which are strongly correlated with the research effort spent in a given area ( Ahrends et al., 2011 ; Hickisch et al., 2019 ). There is thus a need for statistical methods that can reliably assess how biodiversity is changing also from such limited and noisy data ( Bowler et al., 2024 ). While it is possible to include the survey effort as a covariate in linear models of population trends (e.g. Wauchope et al., 2019 ), statistical power is gained when using approaches that model the survey process explicitly. Two types of such models have been proposed: First, deterministic models, which usually assume an exponential change in the form of N t = N 0 exp(− γt ) and aim at inferring the rate of change γ using a wide range of observation models (e.g. Link and Sauer, 1997a ; Williams et al., 2016 ; Moore and Barlow, 2011 ; Kéry et al., 2009 ). Second, stochastic models, which generally assume that the population trajectory is Markovian such that N t | N t −1 ∼ f ( N t −1 , θ ) with some transition probability function f between two consecutive states and relevant ecological parameters θ (often referred to as state-space models in ecology, e.g. Wang, 2007 ; Pedersen et al., 2011 ; Boyd et al., 2018 ; Sollmann et al., 2015 ; Dennis et al., 2006 ; Hostetler and Chandler, 2015 ; Buckland et al., 2004 ; Dennis and Ponciano, 2014 ; De Valpine and Hastings, 2002 ). While these approaches proved powerful, many implementations require substantial amounts of data and are generally tailored to specific data sets with no widely applicable tool available. We here develop novel methods to infer population trends, which require minimal standardization and parametrization, and directly build on those of Link and Sauer (1997a) , who first realized that detection probabilities may not need to be known nor learned if data can be stratified such that detection probabilities are time-invariant within each stratum. In their work, they used bird observers as examples: Consider two observers with very different detection abilities that both survey the same location. They may end up reporting vastly different observations. If the surveyed species doubled in numbers, however, both are expected to detect twice as many birds. Here we build on this idea and extend it to potentially stochastic trends under arbitrary models of change, including the canonical BA, CI and BACI designs, but also more complicated designs with multiple times of change or multiple groups of control or intervention sites. We further propose appropriate priors, relax the assumption of time-invariant detection probabilities through covariates, conceive a fast approach for Bayesian estimation, and develop birp , a user-friendly implementation of this family of methods. Materials and Methods Modeling population trends Our Bayesian methods presented here infer population trends from count data obtained with (potentially) multiple survey methods such as track counts, camera trap images, distance sampling or metabarcoding sequences. The methods build on classic Poisson and negative binomial models for ecological count data (in particular Link and Sauer, 1997a ; Kéry et al., 2009 ), for which we develop the R-package birp for fast and easy use. We introduce both a deterministic and a stochastic variant that allow to directly test whether population trends changed at specific times, while integrating over the uncertainty associated with absolute densities and detection probabilities. Deterministic approach We assume that the data were obtained from J locations, which can be grouped into G groups such that all locations within group g share the same trend. Following others (e.g. Williams et al., 2016 ; Moore and Barlow, 2011 ; Weir et al., 2009 ; Kéry et al., 2009 ; Wauchope et al., 2022 ), we model these trends in population sizes exponentially with rates γ l , where a positive and negative value indicates an increase or decrease in abundance, respectively. Similar to e.g. Wauchope et al. (2022) , we allow for structural changes to happen at fixed times T m , m = 1, … M − 1 at which the rate parameters may change. The trends are thus divided into M epochs, of which epoch m runs from T m −1 to T m . Let γ = { γ 1 , …, γ L } denote the set of all rates and γ ( g, m ) ∈ γ denote the relevant rate for locations of group g during epoch m . Figure 1 shows visualizations of this model for several cases, including a canonical before-after (BA) design, in which there is a single group and two epochs with two potentially different rates γ (1, 1) and γ (1, 2) ( Figure 1A ) and a canonical before-after-control-intervention (BACI) design, in which there are two epochs and two groups, of which the intervention group has two potentially different rates γ (1, 1) and γ (1, 2) for the two epochs, while the control group has the former rate in both epochs ( γ (2, 1) = γ (2, 2) = γ (1, 1), Figure 1B ). However, the approach described here allows for an arbitrary number and combination of groups, epochs and rates (e.g. Figure 1C ). Download figure Open in new tab Figure 1: Illustrations of birp models with γ = (−0.25, 0.1, −0.1) and K = 7 survey times under different designs. The deterministic population trend φ j ( t , γ ) is shown in black and possible stochastic realizations in color. (A) A canonical before-after design with M = 2 epochs and an intervention at T 1 . (B) A canonical before-after-control-intervention (BACI) design with M = 2 epochs and G = 2 groups: a control group (blue) and a group (pink) affected by an intervention at time T 1 . (C) A complex BACI design as in B but with an additional group (gold) affected by an additional intervention at T 2 . Our method then infers the rates γ from observations obtained during multiple surveys k = 1, …, K conducted at known times t k . The data does not contain information about changes in the rates before the first ( t 1 ) or after the last ( t K ) survey. We therefore assume t 1 < T m < t K for m = 1, …, M − 1 and define T 0 = t 1 and T M = t K ( Figure 1 ). Let us denote by the abundance at t k relative to that at the first time point t 1 = T 0 for location j = 1, …, J . For example, φ j ( t k , γ ) = 1 implies the same abundance as at time t 1 and φ j ( t k , γ ) = 0.1 a reduction by 90%. Under the deterministic exponential model proposed above, for any location j in group g and for any t k in the first epoch T 0 ≤ t k ≤ T 1 . At T 1 , rate γ ( g , 2) will take over. For a survey time for any location j in group g . More generally, and denoting by g ( j ) the group to which location j belongs, where we introduced the factors which indicate whether the sampling time t k lies after, within or before epoch m , in which case the rate γ ( g ( j ), m ) was active for the full length of the epoch, the time passed since the start of the epoch, or not at all, respectively. Stochastic approach Stochastic models allow to account for differences in realized population trajectories between survey locations that are part of the same group. Many existing stochastic trend models use integer abundances in discrete time (e.g. Wang, 2007 ; Pedersen et al., 2011 ; Boyd et al., 2018 ; Sollmann et al., 2015 ; Dennis et al., 2006 ; Hostetler and Chandler, 2015 ; Buckland et al., 2004 ; De Valpine and Hastings, 2002 ), often with considerable realism such as density dependence or migration. To maximize the statistical power to infer trends from small data sets, we here follow Dennis and Ponciano (2014) ; De Valpine and Hastings (2002) and take a continuous approach. A natural stochastic variant of the deterministic model described above is geometric Brownian motion (see Karlin and Taylor, 1981 , Ch. 15), which describes the stochastic trend specific to location j using the stochastic differential equation: where the volatility σ is the only additional parameter, is the rate γ ( g ( j ), m ) relevant at time t , and W ( t ) ∼ 𝒩 (0, t ) is standard Brownian motion. Thus, the change in the relative population abundance is modeled using a deterministic term governed by γ g ( j ) ( t ), which is analogous to the deterministic case, modulated by a stochastic (unpredictable) term governed by σ . The expectation of the individual population trends follows the trend of the deterministic model above: The probability density has the closed form ( Karlin and Taylor, 1981 ) where we introduced the factors analogous to eq. 2 : These factors indicate whether the interval between the sampling times t k −1 and t k covers the epoch m fully, partially in the beginning, partially in the end, partially within, or not at all. Accordingly, the rate γ ( g ( j ), m ) was active for the full length of the epoch m , from the start of the epoch until t k , from t k −1 until the end of the epoch, from t k −1 until t k , or not at all, respectively. Note that this model could easily be extended to a correlated Brownian process to account for correlations between locations. Modeling observations Let n ij = ( n ij 1 , …, n ijK ) denote the observed counts obtained with survey method i at location j during survey times k = 1, …, K at known times t k . We assume these counts are proportional to known efforts s ijk and unknown rates λ ij ( t k ). The efforts s ijk may represent e.g. the length of a particular transect or camera trap survey, but can also be modeled using covariates (see below). The rates λ ij ( t k ) describe the rates at which observations are recorded per unit of effort (e.g. two captures per week of camera trapping). They are modeled as a function of both the population abundance at time t k and the method- and site-specific detection probability δ ij as where is the number of individuals at time T 0 . The relative population abundance φ j ( t k , γ ) is given by eq. 1 for the deterministic case and eq. 3 for the stochastic case. Since population abundance and detection probability are confounded, they can not be estimated individually. However, as we show below, by conditioning on the total number of observations, the potentially complex process of detection ( Hofmeester et al., 2019 ) does not need to be modeled. Similar to Link and Sauer (1997a) , we model observed counts as following either a Poisson or a negative binomial distribution. Under the Poisson assumption of mean equals variance, the model only needs to learn a single parameter and therefore likely has more statistical power to detect a trend from limited data. However, ecological survey data are often overdispersed and violate this assumption. Negative binomial models account for this overdispersion by learning an additional parameter reflective of the variance in the data set. We implemented both approaches and outline their details below. Poisson case Here, we assume that the counts n ijk are Poisson distributed as with rates λ ij ( t k ) given by eq. 4 . To make inferences about the rate parameters γ , we condition on the total number of observations across all survey times υ ij = ∑ k n ijk (see Link and Sauer, 1997a ). The conditional distribution of n ij given υ ij is multinomial ( Johnson et al., 1997 ) with probabilities Thanks to conditioning, the nuisance parameters and δ ij cancel out from the equation since neither depends on the specific survey times k . We relax and discuss our assumption of time-invariant detection probabilities below. The multinomial likelihood of the full observation vector n = ( n 11 , …, n IJ ), conditional on υ = ( υ 11 , …, υ IJ ), is We use the non-informative Jeffrey’s prior for γ , which has a closed-form solution for the deterministic case (see SI). For the stochastic model, Jeffrey’s prior cannot be determined analytically. But since , the deterministic solution will serve as a good approximation. Negative Binomial (NB) case Often, ecological count data are overdispersed and modeled using the negative binomial (NB) distribution (e.g. Link and Sauer, 1997a ; Lindén and Mantyniemi, 2011). The NB distribution arises naturally when treating the Poisson rates as latent variables that follow a conjugate Gamma distribution. Following Link and Sauer (1997a) , we choose such that with a method-specific, constant variance/mean ratio of which corresponds to the Poisson case in the limit a i → 0. The conditional distribution of n ij given υ ij is Dirichlet-Multinomial (DM, Johnson et al., 1997 ) with parameters In contrast to the Poisson case, the nuisance parameters and δ ij do not cancel out and have to be inferred along γ and a i . However, they have little influence on the data as the mean is independent of them: with probabilities π ij = ( π ij 1 , …, π ijK ) ′ identical to the Poisson case ( eq. 5 ): The Dirichlet-multinomial likelihood of the full observation vector n = ( n 11 , …, n IJ ), conditional on υ = ( υ 11 , …, υ IJ ), is where α ij 0 = ∑ k α ijk . Note that , δ ij and a i are confounded with multiple combinations of values leading to the same Dirichlet-Multinomial likelihood. To avoid the resulting non-identifiability issues, we define and infer the vectors µ = ( µ 11 , …, µ IJ ) and b = ( b 1 , …, b I ) with the additional constraint ∑ j µ ij = 1. The prior is of the form ℙ ( µ, b, γ ) = ℙ ( µ ) ℙ ( b | µ ) ℙ ( γ | µ, b ), for which Jeffrey’s prior can not be determined analytically. Similar to Link and Sauer (1997a) , however, we can use a quasi-likelihood approximation ( Wedderburn, 1974 ), for which Jeffrey’s prior ℙ ( µ, b, γ ) ≈ ℙ ( µ ) ℙ ( b ) ℙ ( γ ) with ℙ ( γ ) identical to the Poisson case. We further choose the improper prior ℙ ( µ ) ∝ 1 for all µ that adhere to the constraint ∑ j µ ij = 1 and ℙ ( µ ) = 0 otherwise, and the exponential prior b i ∼ Exp( λ b ), ℙ ( b ) = Π i ℙ( b i ) with default rate λ b = 0.1. Modeling survey efforts using covariates Accounting for covariates is straightforward under the models proposed above. Following others (e.g. Link and Sauer, 1997a , 1998 ; Kéry et al., 2009 ; Fewster et al., 2000 ), we model the effective efforts as a function of C i user-provided and method-specific covariates such as, for instance, the number of kilometers walked on a transect. Let denote covariate c = 1, …, C i for method i , location j and survey time k , and let denote the corresponding coefficient. The effort is then modeled as where, to avoid non-identifiability issues, we standardize the covariates per method such that and impose the constraint that . In case the user supplies a single covariate ( C i = 1), we set and do not infer the regression coefficient . Note that the covariates must be positive and that we do not include an intercept since an actual effort of zero (e.g. walking zero kilometers) will always result in zero counts. Relaxing the assumption of time-invariant detection rates Count data does not allow to distinguish between changes in population abundance and changes in detection rates. In the models introduced above, we assumed that detection rates may vary between location and methods, but are constant through time. As we argue in more detail in the discussion, this assumption is likely appropriate for many ecological surveys. However, it can be relaxed through user-provided covariates, albeit at the cost of lower statistical power due to a higher number of parameters that need to be inferred. Let δ ijk denote the detection probability of method i at location j and survey time k . We may model it through D i method-specific covariates such as, for instance, the experience of the observer, the time of the day or habitat structure. Let denote covariate d = 1, …, D i , for method i , location j and survey time k . The detection rate is then modeled as where β i 0 and are method-specific intercepts and coefficients (see SI for implementation details). Note that some covariates may be negative (e.g. temperature). Bayesian inference Given observed counts n , we use Markov chain Monte Carlo (MCMC) methods with Metropolis-Hastings updates and symmetric proposal kernels to obtain posterior samples under the models described above. In all models, the inferred parameters include γ . For NB models, we further infer b and µ , and for stochastic models additionally σ and the vector of for all locations and timepoints. In case covariates are used, we also infer the vector of relevant coefficients, here denoted by β . For fast convergence of the MCMC, we obtain initial estimates of the γ parameters using ordinary least squares (see SI). With the inferred γ , we can address highly relevant questions for species monitoring and conservation: (i) Is a population increasing or decreasing during epoch m ? This question is answered by the posterior probabilities ℙ ( γ ( g, m ) > 0| n ) or ℙ ( γ ( g, m ) < 0| n ) for the population represented by group g . (ii) Did a population trend change at a specific time T m ? This question, which arises in BA designs, has so far only been addressed indirectly and under parameter rich models (e.g. Fewster et al., 2000 ). It can readily be answered by determining the posterior probabilities ℙ ( γ ( g, m ) > γ ( g, m + 1)| n ) or ℙ ( γ ( g, m ) < γ ( g, m + 1)| n ). iii) Does the trend differ between two populations during epoch m ? This question, which arises in CI designs, can be answered by determining the posterior probabilities ℙ ( γ ( g, m ) > γ ( g ′ , m )| n ) or ℙ ( γ ( g, m ) < γ ( g ′ , m )| n ) for the populations represented by groups g and g ′ , respectively. (iv) Finally, the model is flexible regarding the experimental design and arbitrary rates may be compared. In a BACI design, for instance, the control populations represented by group g share the same rate in both epochs ( γ ( g, m ) = γ ( g, m + 1) = γ 1 ), while the populations with an intervention represented by group g ′ share that rate in the first ( γ ( g ′ , m ) = γ 1 ) but have a potentially different rate in the second epoch ( γ ( g ′ , m + 1) = γ 2 ). In such a case, the posterior probabilities of interest are ℙ ( γ 1 > γ 2 | n ) or ℙ ( γ 1 < γ 2 | n ). Distinguishing between Poisson and negative binomial models The use of MCMC methods does not lend itself easily to determine whether a Poisson or NB model should be used via Bayesian model selection. To overcome this limitation, we implemented the following simulation-based approach to check whether the inferred overdispersion is outside the range of values expected under a Poisson model: For a given data set n , we first infer all parameters under the NB model. Let θ nb denote that set of inferred parameters, of which we denote the inferred overdispersion parameter b as b * . We then perform R simulations under the conditional Poisson model (i.e. the multinomial distribution conditioned on the observed counts υ ij ) using the subset of inferred parameters θ p ⊂ θ nb relevant for the Poisson model. This includes γ , but also β in case of covariates as well as σ and in case of a stochastic model. For each simulation r = 1, …, R , we infer parameters under the NB model and store the posterior mean estimates of b as b r . We then compare the value b * obtained with the real data against the values b 1 , …, b R obtained under Poisson simulated data and compute the method-specific p-value as the fraction of Poisson simulations for which the overdispersion parameter was inferred to be larger than that inferred on the real data. We recommend to use the NB model whenever p i is below a chosen threshold (e.g. 0.05) and hence the Poisson model is rejected for at least one method i . Performance against simulations To assess the performance of birp , we generated simulations under various conditions. First, we examined the effect of the number of time points, the number of available observations, and the number of locations on the power to detect a trend in a single group in a single epoch with γ (1, 1) = γ under the deterministic Poisson model. Second, we explored the effect of increased variation under the NB model (varying a ) and the stochastic model (varying σ ) under the same one epoch model with γ (1, 1) = γ . We further used these simulations to explore the effect of incorrectly assuming a Poisson distribution or a deterministic model. Third, we assessed the impact on power when using the simulation-based approach for distinguishing between Poisson and NB models. Fourth, we explored the power under three canonical designs: i) BA with two epochs, one group and γ (1, 1) = γ 1 and γ (1, 2) = γ 2 , ii) CI with one epoch, two groups and γ (1, 1) = γ 1 and γ (2, 1) = γ 2 , and iii) BACI with two epochs, two groups and γ (1, 1) = γ (1, 2) = γ (2, 1) = γ 1 and γ (2, 2) = γ 2 . For each design, we varied the number of time points and rates for both epochs as well as the number of available observations. Fifth, we explored the power of birp to detect changes under BA designs in the case of local extinctions and in case an intervention affects a population with a time lag. To model the number of available observations, we varied the expected number of observations at the first time point . Since only the product of and s ij 1 is relevant, we set the detection probabilities δ ij = 1 in all cases, drew the efforts s ijk ∼ Γ(1, 2) from a Gamma distribution and scaled these such that the average effort was 1.0, and set to either 10 or 50. We conducted 1,000 replicate simulations in all cases and repeated the few simulations that resulted in zero counts for all time points. We then inferred trends on these simulations using 10 4 iterations as burnin and 10 5 iterations of sampling. In case of single group, single epoch trends, we only simulated negative or constant trends and determined the power of birp to detect the directionality of a trend through the fraction of all replicates for which the posterior ℙ ( γ 0.95 if we simulated γ < 0 and the fraction for which ℙ ( γ 0.975 or ℙ ( γ > 0| n ) > 0.975 if we simulated γ = 0. In case of BA, CI and BACI designs, we used the fraction of replicates for which ℙ ( γ 1 0.95 if we simulated γ 1 γ 2 | n ) > 0.95 if we simulated γ 1 > γ 2 and the fraction for which P( γ 1 0.975 or ℙ ( γ 1 > γ 2 | n ) > 0.975 if we simulated γ 1 = γ 2 . Application to Tasmanian devil transect surveys We used Tasmanian devil ( Sarcophilus harrisii , hereafter devil ) count data comprising 35 years of surveys conducted by the Tasmanian State Government (Australia) between 1985 and 2019 ( Driessen and Hocking, 1992 ). The data covers between 128 and 172 spotlight road transects of 10 km length per year. We obtained the data from Cunningham et al. (2021) , and, following their analysis, aggregated the survey counts into six national biogeographic regions of Tasmania, with multiple transects (locations) being sampled within each region. For each region, we tested whether the devil population trends changed following the outbreak of devil facial tumour disease (DFTD) ( Hawkins et al., 2006 ; McCallum et al., 2007 ). To investigate the power of birp in identifying changes in trends, we reanalyzed temporal subsets of this data, using five years of data before and one to eight years of data after the outbreak in a region. We used the estimates of disease outbreak per location of Cunningham et al. (2021) and identified the first survey for which the disease was estimated to be present at half or more of all locations of a bioregion. We denote that survey time as year 1 and set the time of change to the middle between that and the previous survey ( T 1 = 0.5). To quantify whether our method accurately detects no change in trends if there was none, we also conducted a second analysis, again using data spanning thirteen years as described above, but shifted eight years earlier to end shortly before the outbreak of the disease. We inferred trends with two epochs with γ (1, 1) = γ 1 and γ (1, 2) = γ 2 for each region separately and treating each transect as its own location, determined the appropriate model (Poisson or NB) for each case and used 10 4 iterations as burnin and 10 5 iterations of sampling. Application to track count and camera trapping surveys We applied birp to track count and camera trapping data obtained between 2012 and 2024 in the Aire de Conservation de Chinko (hereafter Chinko ) in the eastern Central African Republic. This region is a typical example of a remote and inaccessible region for which only limited biodiversity data was available until recently. For more information about the study area, see Aebischer et al. (2020) . Ongoing surveys have identified several flagship species, here we focus on the four apex predators, African wild dog ( Lycaon pictus ), Northern lion ( Panthera leo ssp. leo ), spotted hyena ( Crocuta crocuta ) and African leopard ( Panthera pardus ssp. pardus ), for which data from the Chinko was available. We complemented this data set with data from six mammalian herbivores, namely defassa waterbuck ( Kobus ellipsiprymnus ssp. defassa ), Eastern giant eland ( Tragelaphus derbianus ssp. gigas ), lelwel hartebeest ( Alcelaphus buselaphus ssp. lelwel ), lowland bongo ( Tragelaphus eurycerus ssp. eurycerus ), Western roan antelope ( Hippotragus equinus ssp. koba ) and Western Central African buffalo ( Syncerus caffer ssp. nanus ). We reanalyzed track count and camera trap surveys from 2012 to 2020 of Aebischer et al. (2020) , complemented with more recent surveys conducted using the same protocol. Since the main observer for track counts changed as of 2022, we treated the surveys from 2022, 2023 and 2024 as a different method in our analysis. The road substrate was variable, which likely had an impact on the detection of tracks ( Funston et al., 2010 ). To make the track counts comparable, we constructed a road network from GPS-tracks and divided the Chinko roads into separate segments of up to 500 meters such that segments never included intersections and were always surveyed in full. For each segment, we then calculated the effort as the number of times it was surveyed in a season (see SI for details). We combined the track count data with data from camera trapping surveys collected from 2012 to 2019 from Aebischer et al. (2020) . Detection probabilities most likely varied between locations and seasons ( Ait Kaci Azzou et al., 2021 ). To make observations comparable, we only compared data from camera traps that were installed at the same location (within 40 meters) across different years and considered images taken during the dry (November - April) and wet (May - October) seasons as separate methods. We included consecutive observations of the same species if they were obtained more than 10 minutes apart. The efforts represent the time span during which the camera trap was active, after cutting off 30 minutes at the beginning and at the end of each trapping session to minimise the influence of human presence in the data. While the entire eastern Central African Republic wilderness is formally protected as national parks, nature reserves and hunting zones, it was not managed until recently ( Blom et al., 2004 ). As of 2016, the African Parks Network enforces a strict reserve with a constantly enlarged core zone from about 2,300 km 2 in 2016 to 30,900 km 2 in 2023. To test for changes in population trends following the implementation of active law enforcement, we ran the deterministic, NB model of birp with a single group, two epochs with rates γ (1, 1) = γ 1 and γ (1, 2) = γ 2 and a time of change at T 1 = 2017, using 10 4 iterations as burnin and 10 5 iterations of sampling. For each species, we then used the simulation-based approach described above to test whether the inferred overdispersion parameters were significantly different (p-value < 0.05) from the values expected under a Poisson model. For species where the Poisson model could not be rejected, we re-analyzed the data under a Poisson model. For each species, we then obtained the probabilities ℙ ( γ 1 < γ 2 | n ) and visualized two-dimensional posteriors ℙ ( γ 1 , γ 2 | n ) with the function plot epoch pair of the birp R-package. Data availability and implementation All methods were implemented in the C++ program birp , available through a git repository at bitbucket.org/wegmannlab/birp_cpp/wiki/Home, together with a wiki detailing its usage. The accompanying package birp for R ( R Core Team, 2024 ) can be downloaded as described here: bitbucket.org/wegmannlab/birpr/wiki/Home. As input, birp requires one file per survey method (e.g camera trapping) with information about the location, time point, number of observations and the survey effort or effort covariates. The input files generated for the data analyzed here are available at (link will be revealed upon acceptance), along with the final road network for the Chinko. Results Performance against simulations Power to detect trends in a single epoch We first investigated the power of birp to detect trends from data of a single epoch and a single group under a deterministic Poisson model. As expected, the power of birp was higher if the trend was stronger (i.e. γ was more negative), if more time points were surveyed, or if more observations were obtained per survey ( Figure 2A ). However, power was generally high even for limited data. For J = 5 locations and K = 2 survey times, for instance, the deterministic Poisson model correctly estimated the simulated trends in >95% of the cases if γ ≤ −0.5 and if an expected number of observations were simulated for the first time point ( Figure 2A ). When using K = 5 survey times, the same > 95% power was reached with γ ≤ −0.2, already with . These simulations further show that the posterior probabilities reported by birp are well calibrated: if no trend was simulated (i.e. γ = 0), no trend was detected in 94.7% of all simulations across all cases, close to the 95% expected under the chosen posterior threshold. Download figure Open in new tab Figure 2: Simulation results for a single epoch and group. Shown are the fraction of 1000 replicates for which the model correctly inferred the directionality of the negative trend with > 0.95 posterior support. For the case of γ = 0, the power was calculated as the fraction of replicates for which any change was inferred with that posterior support. (A) A deterministic Poisson model with J = 5 locations for different values of γ, K = 2 or K = 5 time points and or expected observations at the first time point. (B) A deterministic Poisson model as in A but for a fixed γ = − 0.2 and different number of locations. (C) A deterministic negative binomial (NB) model with J = 5 locations for different values of γ and overdispersion parameter a . The pink and blue lines correspond to inference under the NB and Poisson model, respectively, on the same simulations. (D) A stochastic Poisson model with J = 5 locations for different values of γ and volatility parameter σ . The light blue and green lines correspond to inference under a stochastic and deterministic Poisson model, respectively, on the same simulations. We next investigated the effect of surveying additional locations in the above setting while fixing γ = −0.2. As expected, the power to detect a negative trend increased sharply with the number of locations surveyed ( Figure 2B ). With J = 10 locations or more, maximum power was reached if surveys were conducted with K = 5 and . We next investigated the power under negative binomial (NB) models with J = 5, K = 5 and . While NB models account for overdispersion, more parameters need to be learned and the data is generally less informative than under the Poisson case. Consequently, the power to detect trends decreased with increasing dispersion parameter a ( Figure 2C ). To investigate the impact of the Poisson assumption, we also re-estimated the parameters under the Poisson model (blue lines in Figure 2C ), which led to increased power, at the cost of considerable false-positives at γ = 0 in case of large overdispersion ( a ≥ 1). The simulations above were conducted under the deterministic model. We next explored the power under stochastic models using J = 5, K = 5 and . As expected, power decreased with increasing volatility σ ( Figure 2D ). For a small volatility of σ = 0.1 and using γ = −0.2, for instance, birp correctly estimated the simulated trends in 79% of the cases. For a larger volatility of σ = 0.5 and σ = 1, the power dropped below 74% and 40%, respectively, even for γ = −1. To check for the importance of accounting for variation in local trends, we inferred trends under the deterministic model on the data simulated under the stochastic model. As shown in Figure 2D , a violation of the assumption of a deterministic trend leads to generally higher power, at the cots of many false-positives at γ = 0, particularly if volatility was high (e.g. 78% in case of σ = 1). If volatility was small (e.g. σ = 0.1), however, the benefit of inferring fewer parameters seemed to outweigh the violation of the deterministic assumption as we observed a gain in power without an elevation of false-positives. Testing for the Poisson assumption The above simulations show that while using Poisson models on overdispersed data leads to increased false positives, they have higher power in case the Poisson assumption is justified. To guide the choice of the appropriate model, we propose a simulation-based approach that evaluates whether the inferred overdispersion parameter differs significantly from what would be expected under a Poisson model ( Figure 3A ). To assess the power of this approach, we generated simulations under both deterministic Poisson and NB models with varying overdispersion for a single epoch and a single group, using J = 5 locations, K = 5 time points and . For each simulation, we then performed inference under i) the Poisson model, ii) the NB model, and iii) the model identified as appropriate based on the simulation-based test. These simulations confirmed the elevated false-positive rates when inferring under a Poisson model from overdispersed data simulated with no trend ( γ = 0, Figure 3B ). However, the inference was well calibrated when inferring under a NB model, albeit at the cost of lower statistical power ( Figure 3C ). When deciding on the appropriate model individually for each simulation ( Figure 3D ), power could be increased (particularly when overdispersion was small) without any excess in false-positives. Of all simulations conducted with a = 0.1, for instance, the Poisson model was deemed appropriate in 93% cases, which resulted in an increase in power to detect a negative trend from 37.6% to 60.5% for γ = −0.1. For simulations conducted with a = 1, a = 10 or a = 100, in contrast, the Poisson model was deemed appropriate for only 50%, 2% and 1% cases, respectively. Download figure Open in new tab Figure 3: Simulation results for distinguishing between Poisson and negative binomial (NB) models. (A) Scheme illustrating the simulation-based approach used to assess whether the inferred overdispersion b * (blue) is outside the range of values b expected under a Poisson model (orange). If the p-value is smaller than some threshold (e.g. 0.05), the Poisson model is rejected and the data should be analyzed using a NB model. (B,C,D) Simulations for J = 5 locations, different values of γ (x-axis) and under deterministic Poisson models (black diamonds) or deterministic NB models with different values of the overdispersion parameter a (colors). Power was calculated as for Figure 2 . (B) Inference under a Poisson model. (C) Inference under a NB model. (D) Inference under the model deemed appropriate by the simulation-based approach. Power to detect changes in trends We next investigated the ability of birp to detect changes in population trends by simulating data under canonical before-after (BA), control-intervention (CI) and before-after-control-intervention (BACI) designs. We focused on deterministic Poisson models in all cases. Under a BA design, power was generally higher if more time points were available, if more observations were available, and if the rates γ (1, 1) = γ 1 and γ (1, 2) = γ 2 of the two epochs differed more substantially ( Figure 4 , first row). In case of five time points in each epoch (5/5 case) and γ 1 = 0 and γ 2 = −1, for instance, a decrease in trend was detected with ℙ ( γ 2 0.95 in 94% of the simulated replicates. If γ 1 = −0.2, however, that power dropped to 49%. The simulations also show the importance of having multiple time points per epoch. The power was, for instance, substantially higher if two time points were surveyed in each epoch (2/2 case), rather than two before and only one after the time of change T 1 (2/1 case), and notably also higher than if five points before and one after were surveyed (5/1 case), despite the higher total number of time points. That said, our simulations show that strong changes in trends can be detected with rather few time points: In the 2/1 case, the power to detect a change from γ 1 = 0 to γ 2 = −1 was still 27% for . Since these simulations were conducted for a single location, we expect much higher power for real world scenarios, as shown for the single epoch case ( Figure 2B ) above. Finally, the simulations again show that birp posterior probabilities are well calibrated: Across all cases in which we simulated no change in trend, a change was detected in exactly the expected 5% of all simulations. Download figure Open in new tab Figure 4: Simulation results for canonical before-after (BA, first row), control-intervention (CI, second row) and before-after-control-intervention designs (BACI, third row) for different values of γ 2 in the second epoch or intervention group (x-axis) and for different numbers of survey times (colors). Columns differ in the expected number of observations at the first time point and the value of γ 1 . For each design, the first column shows an illustration of the population trend φ j ( t , γ ) for four specific simulation settings, indicated by the matching color and plotting symbol. In case of CI and BACI designs, the dotted lines represent the trend of the control group. The reported power refers to the fraction of 1000 replicates for which the model correctly inferred the directionality of the change with >0.95 posterior support. For the cases of γ 1 = γ 2 , we report the fraction of replicates for which any change was inferred with that posterior support. To simulate CI designs, we focused on cases with two or five time points in a single epoch, surveyed for one control and one intervention location with rates γ (1, 1) = γ 1 and γ (2, 1) = γ 2 ( Figure 4 , second row). In case of γ 1 = 0, power was very comparable but slightly (about 5%) lower than under the BA cases with equal number of surveyed time points. For and γ 2 = −1, for instance, birp detected a change in 94% of simulated BA cases with five time points per epoch (5/5 case), but only in 90% of simulated CI cases with five time points per group (-/5 case). From a statistical point of view, these BA and CI designs are indeed very similar, since in both cases two rates have to be inferred, but the BA case can take advantage of an additional comparison, namely the one across the time of change. In case of γ 1 = −0.2, however, the power was generally higher under comparable CI than BA designs in our simulations. This is because in the CI design as simulated here, both control and intervention locations started with the same expected number of observations at the first time point, while in the BA design the expected number of observations is already reduced at the beginning of the second epoch. We finally explored the power of BACI designs with two locations, under which the control location had the same rate in both epochs ( γ (1, 1) = γ (1, 2) = γ 1 ), while the intervention location had a potentially different rate in the second epoch ( γ (2, 1) = γ 1 , γ (2, 2) = γ 2 , Figure 4 , third row). Under BACI designs, the power to detect changes in trends was higher than under either BA or CI designs due to the additional available data. For the 2/2 case with γ 1 = 0, γ 2 = −1 and , for instance, the power doubled from 23% under a BA design to 46% under a BACI design. For the same parameters but for the 2/1 case, the power increase was from 12% to 18%. Interestingly, the picture was less clear when both design had the same number of surveys. The 2/2 BA case, for instance, had a higher power (23%) than the 2/1 BACI case (18%). This pattern was observed for all but one combinations of γ 1 and γ 2 tested and suggests that under the conditions explored here, additional information for γ 2 , as provided under BA designs, may be more relevant than additional information for γ 1 , as provided under BACI designs. All-zeroes and time lags Wauchope et al. (2021) recently identified two cases under which the detection of changes in trends are complicated. The first case, termed all-zeroes , describes a situation under which the species went locally extinct and hence only zero counts were obtained for the second epoch. Such cases are detrimental to approaches that compare trends inferred independently for each epoch, as all zero counts do not allow for trend estimation or are interpreted as a stable trend. We here explored the power to detect such changes under the birp model, which also exploits the comparison across epochs. As shown in Figure 5AB , birp has no issues with all zeroes cases and retains high power. Importantly, however, the power strongly depends on the trend during the first epoch, since strongly negative trends lead to populations that are already close to extinction before the time of change. Since all our simulations start with , that is particularly visible in the the 5/1 and 5/5 cases, for which the expected number of counts at the time of change T 1 is 0.11 in case of γ 1 = −1, and hence already very close to 0.0. Download figure Open in new tab Figure 5: Simulation results for cases of local extinctions (A,B) and time lags (C,D). Panels A and C show the population trend φ j ( t , γ ) for four specific simulation settings, indicated by the matching color and plotting symbol. Power was calculated as for Figure 4 . (A, B) All-zeroes scenario where a constant or negative population trend in the first epoch ( γ 1 ) is followed by a local extinction event resulting in all-zero counts in the second epoch. We varied the rates of the first epoch γ 1 (x-axis in B) as well as the number of survey times (colors). (C, D) Time lag scenario where the impact of an intervention manifests after some time only. We varied the rates of the second epoch γ 2 (x-axis in D) as well as the magnitude of the time lag (colors). The second case with complications raised by Wauchope et al. (2021) was a situation under which the impact of an intervention manifested after some time lag only. While birp could be readily extended to also infer the most likely time of change, we here explored the power under time lag scenarios under the 5/5 case with γ 1 = 0. As shown in Figure 5CD , and as expected, long time lags diminish the power to detect changes in trends much more than short time lags. For the case of γ 2 = −1, for instance, the power dropped from 94% in case of no time lag to 91%, 76% and 15% in case the effect lagged by one, two or four survey seasons, respectively. Importantly, and across all values of γ 2 tested, the model appears robust to misspecifications of one survey season. Application to Tasmanian devil transect surveys Tasmanian devils were previously reported to have declined across almost their entire geographical range in the last three decades after the emergence of the highly contagious devil facial tumour disease (DFTD, Lazenby et al., 2018 ; Cunningham et al., 2021 ; Volery et al., 2023 ). Using birp on transect count data, we tested whether the emergence of DFTD resulted in more negative population trends in each Tasmanian bioregion. As shown in Figure 6A , annual counts varied greatly over time and between bioregions. Consequently, and while our model detected a change to a more negative population trend for five out of six bioregions, results varied in the certainty depending on how many surveys were used ( Figure 6C ). We found the strongest support for a change to a more negative trend for the bioregion Ben Lomond, for which the posterior probability ℙ ( γ 1 > γ 2 ) > 0.99 whenever we used three or more surveys after the time of change T 1 . For the bioregion Northern slopes, we also found support for such a change with ℙ ( γ 1 > γ 2 ) > 0.95 if tested using three or four surveys after T 1 , but with greater uncertainty if more surveys were used. For the bioregions Central Highlands and Southern Ranges, in contrast, we found considerable support for a more negative trend only if five or six surveys after T 1 were used, respectively. For the bioregion South East, we identified strong support for a more negative trend with P( γ 1 > γ 2 ) > 0.99 right after T 1 , but not when using additional surveys. For the bioregion King, finally, we estimated a lower population density before T 1 . Download figure Open in new tab Figure 6: Analysis of Tasmanian devil transect count data. (A) The average counts for each region for 21 years, aligned at the time at which the disease was estimated to be present at half or more of all locations in a bioregion, denoted by year 1. (B,C) Posterior probabilities that γ 1 (first epoch) was larger than γ 2 (second epoch) on a logarithmic scale for two potential times of change marked with dashed vertical lines: The year 0.5 (C) and eight years earlier (B). In both cases, every analysis used five years before the time of change and results are shown for temporal subsets of one to eight years of data after the time of change, as indicated with the arrows. The horizontal lines indicate the 95% certainty threshold value. This variation in results is both compatible with a visual inspection of the data as well as previous analyses. For the bioregion King, for instance, reported counts were monotonously increasing for four surveys starting from the one just before T 1 and culminating in the third survey after T 1 , for which the highest counts across the entire time series was reported. There is thus little evidence for a negative effect of DFTD, presumably because the disease did not spread into the entire region ( Lazenby et al., 2018 ). In case of the Southern Ranges, the highest counts of the time series were also reported two years after T 1 , and the decline consequently only manifested with some delay. In contrast, the counts for South East showed high variability already prior to T 1 , then a major drop right after T 1 , followed by an apparent recovery. Consequently, a more negative trend was only inferred when using a single survey after T 1 . It thus appears as if the devils were differently affected by DFTD in these bioregions, and their trends also likely affected by additional, more local factors. The region with the clearest signal of an effect was Ben Lomond, for which we inferred strongly different trends before and after T 1 . Volery et al. (2023) previously reported strong impact of DFTD also for the bioregion Northern Slopes from survey four after T 1 until the end of the series. Here, we found strong support when using three or four surveys after T 1 , but not when using longer time-series. There is, however, a fundamental difference in the quantities that were estimated: birp estimates population trends, whereas Volery et al. (2023) estimated impact as the per year population size reduction compared to the expectation in the absence of the disease, a signal that may remain detectable even after a trend stabilized or reversed. We argue that both serve important purposes: while the latter may be used to showcase the impact of a past event, trends are informative about the trajectory of a population and hence the need of further interventions. To investigate the robustness of birp , we also tested for a change to a more negative trend eight years prior to the emergence of DFTD, a time for which no ecologically relevant change is known ( Figure 6B ). Reassuringly, we did not infer a change in rates for any regions and regardless of the number of time points considered after the time of change (one through eight). Application to track count and camera trapping surveys We inferred population trends for four apex predators and six of their associated prey species in the Chinko before and after the implementation of active law enforcement. We used the NB model for all but three species (African wild dog, African leopard and Northern lion), for which the Poisson model could not be rejected. For most species, our inference indicated a currently positive growth rate and also a positive change in trend following law-enforcement, albeit with variable posterior support ( Figure 7 ). For three out of the six prey species, namely African buffaloes (1,300 observations across all methods and years), waterbucks (961 observations) and giant elands (375 observations), posterior support for a positive change was ℙ ( γ 1 < γ 2 | n ) = 1, and only slightly lower for hartebeests (0.99, 188 observations) and roan antelopes (0.93, 698 observations). Of those, we inferred that the large ungulates giant eland and hartebeest are currently growing at the fastest rates with posterior mean estimates for γ 2 at 0.46 and 0.41, respectively. In contrast, waterbucks, African buffaloes and roan antelopes were estimated to grow with rates 0.25, 0.21 and 0.16, respectively. No change in trend was inferred for bongos (233 observations), for which Aebischer et al. (2020) speculated that due to their use of close-canopy forests and their elusive behavior, they were barely affected by direct poaching by herders. However, we inferred a currently growing population with ℙ ( γ 2 > 0| n ) = 0.88 at a rate of 0.23 also for that species. Download figure Open in new tab Figure 7: Analysis of survey data from the Chinko. Shown are two dimensional posterior densities for the trends before ( γ 1 , x-axis) and since active law enforcement was enacted ( γ 2 , y-axis) for four apex predator species (right, brown) and six of their potential prey species (left, blue). The total number of observations is shown in the top-left corner; the posterior support for a more positive trend since law enforcement was enacted in the lower-right corner. We also found posterior support with ℙ ( γ 1 0.9 for a positive change for three out of the four predator species, namely for African leopard (0.96, 288 observations), African wild dog (0.92, 36 observations) and Northern lion (0.91, 38 observations), albeit with a larger uncertainty regarding the exact growth rates for wild dogs and lions due to their small number of observations. All three species were inferred to currently grow with posterior mean estimates for γ 2 at 0.20, 0.17 and 0.27, respectively. For hyenas, which had the largest number of observations (540), we only found moderate support for both a currently growing population (ℙ ( γ 2 > 0| n ) = 0.78) as well as a positive change in trend since 2017 (ℙ ( γ 1 < γ 2 | n ) = 0.73). This may reflect that in contrast to lions, hyenas were not directly targeted by poaching herders. Our results are in good agreement with those of Aebischer et al. (2020) , despite methodological differences: First, we analysed the data under the negative binomial model rather than the Poisson model for most species to account for the overdispersion present in the data. Second, we treated each road segment as an individual location and camera trap observations from dry and wet seasons as different methods, choices which better accounts for spatial and temporal variation in detectability. Third, we directly tested for a change in trends, rather than inferring trends individually for the different epochs. The somewhat lower current growth rates observed for some predators (especially lions and leopards) may be due to our use of much longer time series in the second epoch, over which trends may be more moderate than what was observed right after law enforcement was enacted. Discussion Assessing population trends is an important goal of ecological monitoring. In the context of biodiversity conservation, trends can act as early warning tools in case of a novel or increased threat (e.g. intensified poaching or the arrival of a new disease) as well as direct ways to assess the impact of an intervention or policy change by testing if a negative trend was slowed or reversed. In both cases, it is particularly important to be able to identify such changes in trends quickly, as biodiversity once lost can not be restored. A common challenge in inferring trends is that a change in observed counts may be due to an actual change in abundance, but may also reflect a change in survey efforts or detection probabilities. To minimize variation in the latter two, surveys may be highly standardized, in which case trends can be inferred using standard regression techniques (e.g. Collen et al., 2009 ; Farmer et al., 2007 ; Fewster et al., 2000 ; Wauchope et al., 2019 ). However, monitoring projects often face logistical challenges and financial constraints, resulting in sparse and noisy data that can not be analysed using these standard techniques without violating their underlying assumptions, and in particular the assumption of a common error distribution for all data points. Alternatively, variation in survey efforts and detection probabilities may be modelled and learned. While such models have been used successfully (e.g. Wauchope et al., 2019 , 2022 ), they require more data as additional parameters have to be inferred (e.g. location-specific intercepts). To address this issue, we here present birp , a collection of novel Bayesian methods that build on those of Link and Sauer (1997a) , who proposed to stratify data such that time-invariant detection probabilities can be assumed within each stratum. In the deterministic Poisson case, that stratification of data is highly efficient as no additional parameter needs to be learned, yet allows the joint analysis of data obtained with multiple survey method and renders the inference robust to spatial variation in detection probabilities, while fully accounting for variable efforts. While the negative binomial and stochastic models do require additional parameters, the expected trends are independent of those and, with the exception of the overdispersion a and volatility σ , they are nuisance parameters with little effect on the data. Here we developed a fast and user-friendly Bayesian inference tool that uses Jeffrey’s prior, relaxes the assumption of time-invariant detection probabilities through covariates, supports arbitrarily complex experimental designs, including canonical BA, CI, or BACI designs with potentially multiple control and intervention groups, and thus enables rapid trend inference and intervention analysis for a wide range of ecological systems. Time-invariant detection probabilities and survey design The assumption of time-invariant detection probabilities within strata has implications for efficient survey designs. For track-count surveys of apex predators, for instance, detection probabilities were reported to differ between soil types (e.g. Funston et al., 2010 ). Similarly, detection probabilities of camera traps are known to vary greatly with habitat structure (e.g. Ait Kaci Azzou et al., 2021 ; Hofmeester et al., 2019 ). However, such variation is readily accounted for if the same locations are repeatedly surveyed. In our analysis of track count data, for instance, we stratified the data by segments of the surveyed road network. While there may also be variation in detection probabilities within a segment, that variation is of no concern if the segments are defined such that they are always surveyed in full. For the camera trapping data, we similarly only considered locations at which camera traps were mounted at multiple survey times, if possible at the very same tree. Detection probabilities likely also vary across methods, observers or even across devices. Ideally, data is thus collected such that the very same method, observer and even device is employed at a specific location across surveys, as illustrated in Figure 8 . Download figure Open in new tab Figure 8: Different experimental designs that allow (panels A, D and E), partially allow (panel F) or do not allow (panels B and C) for stratification. Two different observers (or survey methods) are indicated with a black square and a white triangle, respectively. Lines connecting the observers across survey times indicate surveys that are comparable in terms of detection probabilities and thus form a stratum. The horizontal dashed line indicates a potential time of change (TOC). Detection probabilities may further vary due to other factors such as the weather condition, the season or the time of the day. Some of these factors can again be readily stratified. If surveys were conducted at night and day, for instance, they should be considered as different “methods”, even if conducted by the same observer, and analogously for different seasons or weather conditions. As the number of such factors to consider grows, however, so will the resources required for planning and logistics to ensure enough comparisons are available within each stratum. It may thus be necessary to merge similar strata, such as, for instance, observers with similar skills. Alternatively, birp also allows to model variation in detection rates through covariates, although additional parameters then need to be estimated. Importantly, the covariates do not need to be sufficient for absolute but for relative detection probabilities only. More difficult are cases in which detection rates vary with population abundance. There is, however, little evidence that this is a general issue for count data but rather for specific analyses. Under occupancy models, for instance, only the presence of a species is recorded, and as expected, several studies found that it becomes easier to detect a species as its abundance increases (e.g. Royle and Nichols, 2003 ; McCarthy et al., 2013 ; Bornand et al., 2014 ). The major factors affecting individual counts, however, do not include abundances but rather the target species, the survey location, the observer or the weather conditions (e.g. Thompson and La Sorte, 2008 ; Diefenbach et al., 2003 ) - factors that are all readily accounted for via stratification. Furthermore, in case detection probabilities for individual count data were indeed increasing with abundance, the methods proposed here would nevertheless correctly infer the directionality of a trend, but may overestimate its strength. The Poisson assumption Ecological data is often overdispered in that it violates the Poisson assumption of mean equal variance. To maximize statistical power, we here implemented both Poisson and negative binomial models, as well as a simulation-based test to guide users towards the appropriate model for their data. However, we note that zero-inflation, a particular case of overdispersion that results from a sequence of non-observations due to the absence of a species at a specific location ( Blasco-Moreno et al., 2019 ), does not affect the models presented here: If a species is truly absent from a location, no counts were obtained across all years surveyed. Such locations bear no information regarding trends and, to avoid unnecessary calculations, are omitted. Population trends vs. population trajectories The terms population trend and population trajectory are sometimes used synonymously, although their meaning differs with potential implications for conservation management. We here refer to Link and Sauer (1997b) , who briefly discussed this terminology in their commentary and proposed to use the term trajectory to “describe the pattern of population fluctuation” and the term trend to “represent an average rate of change over a specific time interval”. Thus, a trend, according to this definition, is a smoothed trajectory ( Link and Sauer, 1997b ). Following this, it is easy to recognize that a population may show fluctuating trajectories while overall exhibiting a stable trend. Analogously, a declining population could temporarily show a positive trajectory as a result of random fluctuations. Hence, being able to distinguish longer-term trends from shorter-term variation in trajectories is important, yet these definitions are highly context and species dependent and there is no universally valid definition for when a trajectory turns into a trend. For these reasons, there is no consensus on the time span needed to reliably estimate population trends, but several authors recommend several dozen visits per year over several years (e.g. Brashares and Sam, 2005 ; Maclean et al., 2013 ; Embling et al., 2015 ). As shown through simulations and real data applications, birp may reliably identify changes in abundance from much fewer data points. Whether those inferred changes reflect random fluctuations or longer-term trends will be up to the experimenter to judge. The risk of misinterpreting random fluctuations as trends is, of course, greatly reduced when surveys were conducted over longer times, but also if many locations were surveyed such that random fluctuations effectively cancel out. We argue that from a biodiversity conservation perspective, it may be crucial to act even if a sharp change in trends was inferred from short time series only. Author Contributions LS, MC, DW and CL conceived the ideas and designed methodology; TA and PT collected the data; MC and LS implemented the methods, LS, MC, AF and JI analyzed the data; LS, MC and DW led the writing of the manuscript. All authors contributed critically to the drafts and gave final approval for publication. Inclusion statement Our study brings together authors from two different countries, including scientists based in the country where data collection for this study was carried out. All authors were engaged early on with the research and study design to ensure that the diverse sets of perspectives they represent was considered from the onset. Whenever relevant, literature published by scientists from the region was cited; efforts were made to consider relevant work published in the local language. Conflicts of interest The authors declare no conflict of interest. Acknowledgements We would like to thank the African Parks Network and its hard-working staff. Special thanks go to the ecological research team in the Chinko: Modeste Bambalo, Narcisse Bangali, Alain Gbando, Emmanuel Lango, Fernand Magraba, Vivien Pandi and Junior Wakassa. References ↵ Aebischer , T. , Ibrahim , T. , Hickisch , R. , Furrer , R.D. , Leuenberger , C. , Wegmann , D. , 2020 . Apex predators decline after an influx of pastoralists in former central african republic hunting zones . Biological Conservation 241 , 108326 . OpenUrl CrossRef ↵ Ahrends , A. , Burgess , N.D. , Gereau , R.E. , Marchant , R. , Bulling , M.T. , Lovett , J.C. , Platts , P.J. , Wilkins Kindemba , V. , Owen , N. , Fanning , E. , et al. , 2011 . Funding begets biodiversity . Diversity and Distributions 17 , 191 – 200 . OpenUrl CrossRef ↵ Ait Kaci Azzou , S. , Singer , L. , Aebischer , T. , Caduff , M. , Wolf , B. , Wegmann , D. , 2021 . A sparse observation model to quantify species distributions and their overlap in space and time . Ecography 44 , 928 – 940 . OpenUrl CrossRef ↵ Bacher , S. , Blackburn , T.M. , Essl , F. , Genovesi , P. , Heikkilä , J. , Jeschke , J.M. , Jones , G. , Keller , R. , Kenis , M. , Kueffer , C. , Martinou , A.F. , Nentwig , W. , Pergl , J. , Pyšek , P. , Rabitsch , W. , Richardson , D.M. , Roy , H.E. , Saul , W.C. , Scalera , R. , Vilà , M. , Wilson , J.R. , Kumschick , S. , 2018 . Socio-economic impact classification of alien taxa (SEICAT) . Methods in Ecology and Evolution 9 , 159 – 168 . doi: 10.1111/2041-210X.12844 . OpenUrl CrossRef ↵ Barnes , A. , Davies , J. , Martay , B. , Boersch-Supan , P. , Harris , S. , Noble , D. , Pearce-Higgins , J. , Robinson , R. , 2023 . Rare and declining bird species benefit most from designating protected areas for conservation in the uk . Nature ecology & evolution 7 , 92 – 101 . OpenUrl PubMed ↵ Blasco-Moreno , A. , Pérez-Casany , M. , Puig , P. , Morante , M. , Castells , E. , 2019 . What does a zero mean? Understanding false, random and structural zeros in ecology . Methods in Ecology and Evolution 10 , 949 – 959 . doi: 10.1111/2041-210X.13185 . OpenUrl CrossRef ↵ Blom , A. , Yamindou , J. , Prins , H.H. , 2004 . Status of the protected areas of the central african republic . Biological Conservation 118 , 479 – 487 . OpenUrl CrossRef ↵ Boakes , E.H. , McGowan , P.J. , Fuller , R.A. , Chang-qing , D. , Clark , N.E. , O’Connor , K. , Mace , G.M. , 2010 . Distorted views of biodiversity: spatial and temporal bias in species occurrence data . PLoS Biol 8 , e1000385 . OpenUrl CrossRef PubMed ↵ Bornand , C.N. , Kéry , M. , Bueche , L. , Fischer , M. , 2014 . Hide-and-seek in vegetation: time-to-detection is an efficient design for estimating detectability and occurrence . Methods in Ecology and Evolution 5 , 433 – 442 . OpenUrl CrossRef ↵ Bowler , D.E. , Boyd , R.J. , Callaghan , C.T. , Robinson , R.A. , Isaac , N.J. , Pocock , M.J. , 2024 . Treating gaps and biases in biodiversity data as a missing data problem . Biological Reviews . ↵ Boyd , C. , Barlow , J. , Becker , E.A. , Forney , K.A. , Gerrodette , T. , Moore , J.E. , Punt , A.E. , 2018 . Estimation of population size and trends for highly mobile species with dynamic spatial distributions . Diversity and Distributions 24 , 1 – 12 . doi: 10.1111/ddi.12663 . OpenUrl CrossRef ↵ Brashares , J.S. , Sam , M.K. , 2005 . How much is enough? estimating the minimum sampling required for effective monitoring of african reserves . Biodiversity & Conservation 14 , 2709 – 2722 . OpenUrl CrossRef ↵ Buckland , S.T. , Anderson , D.R. , Burnham , K.P. , Laake , J.L. , Borchers , D.L. , Thomas , L. , et al. , 2001 . Introduction to distance sampling: estimating abundance of biological populations . ↵ Buckland , S.T. , Newman , K.B. , Thomas , L. , Koesters , N.B. , 2004 . State-space models for the dynamics of wild animal populations . Ecological Modelling 171 , 157 – 175 . doi: 10.1016/j.ecolmodel.2003.08.002 . OpenUrl CrossRef Web of Science ↵ Butchart , S.H. , Walpole , M. , Collen , B. , Van Strien , A. , Scharlemann , J.P. , Almond , R.E. , Baillie , J.E. , Bomhard , B. , Brown , C. , Bruno , J. , et al. , 2010 . Global biodiversity: indicators of recent declines . Science 328 , 1164 – 1168 . OpenUrl Abstract / FREE Full Text ↵ Butsic , V. , Lewis , D.J. , Radeloff , V.C. , Baumann , M. , Kuemmerle , T. , 2017 . Quasiexperimental methods enable stronger inferences from observational data in ecology . Basic and Applied Ecology 19 , 1 – 10 . doi: 10.1016/j.baae.2017.01.005 . OpenUrl CrossRef ↵ Chelintsev , N. , 1995 . Mathematical principles of winter censuses of mammals . Byulleten Moskovskogo Obschestva Ispytatelei Prirody 100 , 3 – 19 . OpenUrl ↵ Collen , B. , Loh , J. , Whitmee , S. , McRae , L. , Amin , R. , Baillie , J.E.M. , 2009 . Monitoring change in vertebrate abundance: the living planet index . Conservation Biology 23 , 317 – 327 . URL: https://conbio.onlinelibrary.wiley.com/doi/abs/10.1111/j.1523-1739.2008.01117.x . OpenUrl CrossRef PubMed ↵ Cunningham , C.X. , Comte , S. , McCallum , H. , Hamilton , D.G. , Hamede , R. , Storfer , A. , Hollings , T. , Ruiz-Aravena , M. , Kerlin , D.H. , Brook , B.W. , et al. , 2021 . Quantifying 25 years of disease-caused declines in tasmanian devil populations: host density drives spatial pathogen spread . Ecology Letters 24 , 958 – 969 . OpenUrl CrossRef PubMed ↵ De Valpine , P. , Hastings , A. , 2002 . Fitting population models incorporating process noise and observation error . Ecological Monographs 72 , 57 – 76 . doi: 10.1890/0012-9615(2002)072 [0057:FPMIPN]2.0.CO;2. OpenUrl CrossRef Web of Science ↵ Dennis , B. , Ponciano , J. , 2014 . Density-dependent state-space model for populationabundance data with unequal time intervals . Ecology 95 , 2069 – 2076 . doi: 10.1890/13-1486.1 . OpenUrl CrossRef PubMed ↵ Dennis , B. , Ponciano , J.M. , Lele , S.R. , Taper , M.L. , Staples , D.F. , 2006 . Estimating density dependence, process noise, and observation error . Ecological Monographs 76 , 323 – 341 . doi: 10.1890/0012-9615(2006)76 [323:EDDPNA]2.0.CO;2. OpenUrl CrossRef Web of Science ↵ Di Marco , M. , Chapman , S. , Althor , G. , Kearney , S. , Besancon , C. , Butt , N. , Maina , J.M. , Possingham , H.P. , Rogalla von Bieberstein , K. , Venter , O. , Watson , J.E. , 2017 . Changing trends and persisting biases in three decades of conservation science . Global Ecology and Conservation 10 , 32 – 42 . URL: https://www.sciencedirect.com/science/article/pii/S2351989417300148 , doi: 10.1016/j.gecco.2017.01.008 . OpenUrl CrossRef ↵ Diefenbach , D.R. , Brauning , D.W. , Mattice , J.A. , 2003 . Variability in grassland bird counts related to observer differences and species detection rates . The Auk 120 , 1168 – 1179 . OpenUrl CrossRef ↵ Driessen , M.M. , Hocking , G.J. , 1992 . Review and Analysis of Spotlight Surveys in Tasmania, 1975-1990 . Department of Parks, Wildlife and Heritage Hobart, Tasmania, Australia . ↵ Embling , C. , Walters , A. , Dolman , S. , 2015 . How much effort is enough? the power of citizen science to monitor trends in coastal cetacean species . Global Ecology and Conservation 3 , 867 – 877 . OpenUrl CrossRef ↵ Farmer , C.J. , Hussell , D.J.T. , Mizrahi , D. , 2007 . Detecting Population Trends in Migratory Birds of Prey . The Auk 124 , 1047 – 1062 . doi: 10.1093/auk/124.3.1047 . OpenUrl CrossRef ↵ Ferraro , P.J. , 2009 . Counterfactual thinking and impact evaluation in environmental policy . New Directions for Evaluation 2009 , 75 – 84 . URL: https://onlinelibrary.wiley.com/doi/abs/10.1002/ev.297 , doi: 10.1002/ev.297 . OpenUrl CrossRef ↵ Fewster , R.M. , Buckland , S.T. , Siriwardena , G.M. , Baillie , S.R. , Wilson , J.D. , 2000 . Analysis of population trends for farmland birds using generalized additive models . Ecology 81 , 1970 – 1984 . doi: 10.1890/0012-9658(2000)081 [1970:AOPTFF]2.0.CO;2. OpenUrl CrossRef Web of Science ↵ Funston , P.J. , Frank , L. , Stephens , T. , Davidson , Z. , Loveridge , A. , Macdonald , D. , Durant , S. , Packer , C. , Mosser , A. , Ferreira , S.M. , 2010 . Substrate and species constraints on the use of track incidences to estimate african large carnivore abundance . Journal of Zoology 281 , 56 – 65 . OpenUrl CrossRef Web of Science ↵ Geldmann , J. , Barnes , M. , Coad , L. , Craigie , I.D. , Hockings , M. , Burgess , N.D. , 2013 . Effectiveness of terrestrial protected areas in reducing habitat loss and population declines . Biological Conservation 161 , 230 – 238 . URL: https://www.sciencedirect.com/science/article/pii/S0006320713000670 , doi: 10.1016/j.biocon.2013.02.018 . OpenUrl CrossRef Web of Science ↵ Hawkins , C.E. , Baars , C. , Hesterman , H. , Hocking , G. , Jones , M.E. , Lazenby , B. , Mann , D. , Mooney , N. , Pemberton , D. , Pyecroft , S. , et al. , 2006 . Emerging disease and population decline of an island endemic, the tasmanian devil sarcophilus harrisii . Biological Conservation 131 , 307 – 324 . OpenUrl CrossRef ↵ Hickisch , R. , Hodgetts , T. , Johnson , P.J. , Sillero-Zubiri , C. , Tockner , K. , Macdonald , D.W. , 2019 . Effects of publication bias on conservation planning . Conservation Biology 33 , 1151 – 1163 . OpenUrl CrossRef PubMed ↵ Hofmeester , T.R. , Cromsigt , J.P.G.M. , Odden , J. , Andrén , H. , Kindberg , J. , Linnell , J.D.C. , 2019 . Framing pictures: A conceptual framework to identify and correct for biases in detection probability of camera traps enabling multi-species comparison . Ecology and Evolution 9 , 2320 – 2336 . URL: https://onlinelibrary.wiley.com/doi/abs/10.1002/ece3.4878 , doi: 10.1002/ece3.4878 , arXiv: https://onlinelibrary.wiley.com/doi/pdf/10.1002/ece3.4878 . OpenUrl CrossRef ↵ Hostetler , J.A. , Chandler , R.B. , 2015 . Improved state-space models for inference about spatial and temporal variation in abundance from count data . Ecology 96 , 1713 – 1723 . doi: 10.1890/14-1487.1 . OpenUrl CrossRef ↵ Johnson , N. , Kotz , S. , Balakrishnan , N. , 1997 . Discrete Multivariate Distributions . Wiley Series in Probability and Statistics , Wiley . URL: https://books.google.ch/books?id=B4F9QgAACAAJ . ↵ Karlin , S. , Taylor , H.E. , 1981 . A second course in stochastic processes . Elsevier . ↵ Keeping , D. , Pelletier , R. , 2014 . Animal density and track counts: understanding the nature of observations based on animal movements . PLoS one 9 , e96598 . OpenUrl CrossRef PubMed ↵ Kéry , M. , Dorazio , R.M. , Soldaat , L. , Van Strien , A. , Zuiderwijk , A. , Royle , J.A. , 2009 . Trend estimation in populations with imperfect detection . Journal of Applied Ecology 46 , 1163 – 1172 . doi: 10.1111/j.1365-2664.2009.01724.x . OpenUrl CrossRef ↵ Kühl , H. , Maisels , F. , Ancrenaz , M. , Williamson , E. , 2008 . Best practice for surveys and monitoring of great ape population . Gland, IUCN SSC Primate Specialist Group . ↵ Langhammer , P.F. , Bull , J.W. , Bicknell , J.E. , Oakley , J.L. , Brown , M.H. , Bruford , M.W. , Butchart , S.H.M. , Carr , J.A. , Church , D. , Cooney , R. , Cutajar , S. , Foden , W. , Foster , M.N. , Gascon , C. , Geldmann , J. , Genovesi , P. , Hoffmann , M. , Howard-McCombe , J. , Lewis , T. , Macfarlane , N.B.W. , Melvin , Z.E. , Merizalde , R.S. , Morehouse , M.G. , Pagad , S. , Polidoro , B. , Sechrest , W. , Segelbacher , G. , Smith , K.G. , Steadman , J. , Strongin , K. , Williams , J. , Woodley , S. , Brooks , T.M. , 2024 . The positive impact of conservation action . Science 384 , 453 – 458 . URL: https://www.science.org/doi/abs/10.1126/science.adj6598 , doi: 10.1126/science.adj6598 . OpenUrl CrossRef PubMed ↵ Lazenby , B.T. , Tobler , M.W. , Brown , W.E. , Hawkins , C.E. , Hocking , G.J. , Hume , F. , Huxtable , S. , Iles , P. , Jones , M.E. , Lawrence , C. , et al. , 2018 . Density trends and demographic signals uncover the long-term impact of transmissible cancer in tasmanian devils . Journal of Applied Ecology 55 , 1368 – 1379 . OpenUrl CrossRef PubMed ↵ Lenoir , J. , Bertrand , R. , Comte , L. , Bourgeaud , L. , Hattab , T. , Murienne , J. , Grenouillet , G. , 2020 . Species better track climate warming in the oceans than on land . Nature ecology & evolution 4 , 1044 – 1059 . OpenUrl CrossRef PubMed Lindén , A. , Mäntyniemi , S. , 2011 . Using the negative binomial distribution to model overdispersion in ecological count data . Ecology 92 , 1414 – 1421 . doi: 10.1890/10-1831.1 . OpenUrl CrossRef PubMed Web of Science ↵ Link , W.A. , Sauer , J.R. , 1997a . Estimation of Population Trajectories from Count Data . Biometrics 53 , 488 – 497 . OpenUrl CrossRef ↵ Link , W.A. , Sauer , J.R. , 1997b . New approaches to the analysis of population trends in land birds: comment . Ecology 78 , 2632 – 2634 . OpenUrl CrossRef ↵ Link , W.A. , Sauer , J.R. , 1998 . Estimating Relative Abundance From Count Data . Austrian Journal of Statistics 27 , 83 – 97 . OpenUrl ↵ Maclean , I.M. , Rehfisch , M.M. , Skov , H. , Thaxter , C.B. , 2013 . Evaluating the statistical power of detecting changes in the abundance of seabirds at sea . Ibis 155 , 113 – 126 . OpenUrl CrossRef ↵ Maes , D. , Isaac , N.J. , Harrower , C.A. , Collen , B. , Van Strien , A.J. , Roy , D.B. , 2015 . The use of opportunistic data for iucn red list assessments . Biological Journal of the Linnean Society 115 , 690 – 706 . OpenUrl CrossRef ↵ Mathewson , P. , Spehar , S. , Meijaard , E. , Sasmirul , A. , Marshall , A.J. , 2008 . Evaluating orangutan census techniques using nest decay rates: implications for population estimates . Ecological Applications 18 , 208 – 221 . OpenUrl CrossRef PubMed ↵ McCallum , H. , Tompkins , D.M. , Jones , M. , Lachish , S. , Marvanek , S. , Lazenby , B. , Hocking , G. , Wiersma , J. , Hawkins , C.E. , 2007 . Distribution and impacts of tasmanian devil facial tumor disease . EcoHealth 4 , 318 – 325 . OpenUrl CrossRef Web of Science ↵ McCarthy , M.A. , Moore , J.L. , Morris , W.K. , Parris , K.M. , Garrard , G.E. , Vesk , P.A. , Rumpff , L. , Giljohann , K.M. , Camac , J.S. , Bau , S.S. , et al. , 2013 . The influence of abundance on detectability . Oikos 122 , 717 – 726 . OpenUrl CrossRef Web of Science Meyer , C. , Jetz , W. , Guralnick , R.P. , Fritz , S.A. , Kreft , H. , 2016 . Range geometry and socio-economics dominate species-level biases in occurrence information . Global Ecology and Biogeography 25 , 1181 – 1193 . OpenUrl CrossRef ↵ Meyer , C. , Kreft , H. , Guralnick , R. , Jetz , W. , 2015 . Global priorities for an effective information basis of biodiversity distributions . Nature communications 6 , 1 – 8 . OpenUrl ↵ Midlane , N. , O’riain , M.J. , Balme , G.A. , Hunter , L.T. , 2015 . To track or to call: comparing methods for estimating population abundance of african lions panthera leo in kafue national park . Biodiversity and Conservation 24 , 1311 – 1327 . OpenUrl CrossRef ↵ Moore , J.E. , Barlow , J. , 2011 . Bayesian state-space model of fin whale abundance trends from a 1991-2008 time series of line-transect surveys in the California Current . Journal of Applied Ecology 48 , 1195 – 1205 . doi: 10.1111/j.1365-2664.2011.02018.x . OpenUrl CrossRef ↵ Pedersen , M.W. , Berg , C.W. , Thygesen , U.H. , Nielsen , A. , Madsen , H. , 2011 . Estimation methods for nonlinear state-space models in ecology . Ecological Modelling 222 , 1394 – 1400 . , doi: 10.1016/j.ecolmodel.2011.01.007 . OpenUrl CrossRef PubMed Web of Science ↵ Pellissier , V. , Schmucki , R. , Pe’er , G. , Aunins , A. , Brereton , T. , Brotons , L. , Carnicer , J. , Chodkiewicz , T. , Chylarecki , P. , Del Moral , J. , et al. , 2020 . Effects of natura 2000 on nontarget bird and butterfly species based on citizen science data . Conservation Biology 34 , 666 – 676 . OpenUrl CrossRef PubMed ↵ Pyšek , P. , Hulme , P.E. , Simberloff , D. , Bacher , S. , Blackburn , T.M. , Carlton , J.T. , Dawson , W. , Essl , F. , Foxcroft , L.C. , Genovesi , P. , et al. , 2020 . Scientists’ warning on invasive alien species . Biological Reviews 95 , 1511 – 1534 . OpenUrl CrossRef ↵ R Core Team , 2024 . R: A Language and Environment for Statistical Computing . R Foundation for Statistical Computing . Vienna, Austria . URL: https://www.R-project.org/ . ↵ Rodrigues , A.S. , Pilgrim , J.D. , Lamoreux , J.F. , Hoffmann , M. , Brooks , T.M. , 2006 . The value of the IUCN Red List for conservation . Trends in Ecology and Evolution 21 , 71 – 76 . doi: 10.1016/j.tree.2005.10.010 . OpenUrl CrossRef PubMed Web of Science ↵ Royle , J.A. , Nichols , J.D. , 2003 . Estimating abundance from repeated presence–absence data or point counts . Ecology 84 , 777 – 790 . OpenUrl CrossRef Web of Science ↵ Russell , R.E. , Royle , J.A. , Desimone , R. , Schwartz , M.K. , Edwards , V.L. , Pilgrim , K.P. , Mckelvey , K.S. , 2012 . Estimating abundance of mountain lions from unstructured spatial sampling . The Journal of Wildlife Management 76 , 1551 – 1561 . OpenUrl ↵ Sollmann , R. , Gardner , B. , Chandler , R.B. , Royle , J.A. , Sillett , T.S. , 2015 . An openpopulation hierarchical distance sampling model . Ecology 96 , 325 – 331 . doi: 10.1890/14-1625.1 . OpenUrl CrossRef PubMed ↵ Terraube , J. , Van Doninck , J. , Helle , P. , Cabeza , M. , 2020 . Assessing the effectiveness of a national protected area network for carnivore conservation . Nature Communications 11 , 2957 . OpenUrl CrossRef PubMed ↵ Thompson , F.R. , La Sorte , F.A. , 2008 . Comparison of methods for estimating bird abundance and trends from historical count data . The Journal of Wildlife Management 72 , 1674 – 1682 . OpenUrl ↵ Volery , L. , Vaz Fernandez , M. , Wegmann , D. , Bacher , S. , 2023 . A general framework to quantify and compare ecological impacts under temporal dynamics . Ecology Letters 26 , 1726 – 1739 . doi: 10.1111/ele.14288 . OpenUrl CrossRef PubMed ↵ Wang , G. , 2007 . On the latent state estimation of nonlinear population dynamics using Bayesian and non-Bayesian state-space models . Ecological Modelling doi: 10.1016/j.ecolmodel.2006.09.004 . OpenUrl CrossRef ↵ Wauchope , H.S. , Amano , T. , Geldmann , J. , Johnston , A. , Simmons , B.I. , Sutherland , W.J. , Jones , J.P. , 2021 . Evaluating impact using time-series data . Trends in Ecology & Evolution 36 , 196 – 205 . doi: 10.1016/j.tree.2020.11.001 . OpenUrl CrossRef PubMed ↵ Wauchope , H.S. , Amano , T. , Sutherland , W.J. , Johnston , A. , 2019 . When can we trust population trends? A method for quantifying the effects of sampling interval and duration . Methods in Ecology and Evolution 2019 , 1 – 12 . doi: 10.1111/2041-210x.13302 . OpenUrl CrossRef ↵ Wauchope , H.S. , Jones , J.P. , Geldmann , J. , Simmons , B.I. , Amano , T. , Blanco , D.E. , Fuller , R.A. , Johnston , A. , Langendoen , T. , Mundkur , T. , et al. , 2022 . Protected areas have a mixed impact on waterbirds, but management helps . Nature 605 , 103 – 107 . OpenUrl CrossRef PubMed ↵ Webster , H. , McNutt , J.W. , McComb , K. , 2010 . Eavesdropping and risk assessment between lions, spotted hyenas and african wild dogs . Ethology 116 , 233 – 239 . OpenUrl CrossRef ↵ Wedderburn , R.W.M. , 1974 . Quasi-likelihood functions, generalized linear models, and the Gauss—Newton method . Biometrika 61 , 439 – 447 . , doi: 10.1093/biomet/61.3.439 , arXiv: https://academic.oup.com/biomet/article-pdf/61/3/439/690500/61-3-439.pdf . OpenUrl CrossRef Web of Science ↵ Weir , L. , Fiske , I.J. , Royle , J.A. , 2009 . Trends in anuran occupancy from northeastern States of the North American Amphibian Monitoring Program . Herpetological Conservation and Biology 4 , 389 – 402 . OpenUrl ↵ Williams , R. , Moore , J.E. , Gomez-Salazar , C. , Trujillo , F. , Burt , L. , 2016 . Searching for trends in river dolphin abundance: Designing surveys for looming threats, and evidence for opposing trends of two species in the Colombian Amazon . Biological Conservation 195 , 136 – 145 . , doi: 10.1016/j.biocon.2015.12.037 . OpenUrl CrossRef View the discussion thread. Back to top Previous Next Posted January 12, 2025. Download PDF Supplementary Material Email Thank you for your interest in spreading the word about bioRxiv. 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Share Testing for changes in population trends from low-cost ecological count data Liam Singer , Madleina Caduff , Thierry Aebischer , Primaël Tabiti , Aimée Freiberg , Jens Ingensand , Christoph Leuenberger , Daniel Wegmann bioRxiv 2025.01.08.631844; doi: https://doi.org/10.1101/2025.01.08.631844 Share This Article: Copy Citation Tools Testing for changes in population trends from low-cost ecological count data Liam Singer , Madleina Caduff , Thierry Aebischer , Primaël Tabiti , Aimée Freiberg , Jens Ingensand , Christoph Leuenberger , Daniel Wegmann bioRxiv 2025.01.08.631844; doi: https://doi.org/10.1101/2025.01.08.631844 Citation Manager Formats BibTeX Bookends EasyBib EndNote (tagged) EndNote 8 (xml) Medlars Mendeley Papers RefWorks Tagged Ref Manager RIS Zotero Tweet Widget Facebook Like Google Plus One Subject Area Ecology Subject Areas All Articles Animal Behavior and Cognition (7636) Biochemistry (17704) Bioengineering (13897) Bioinformatics (41963) Biophysics (21460) Cancer Biology (18598) Cell Biology (25525) Clinical Trials (138) Developmental Biology (13383) Ecology (19908) Epidemiology (2067) Evolutionary Biology (24325) Genetics (15613) Genomics (22512) Immunology (17738) Microbiology (40422) Molecular Biology (17190) Neuroscience (88634) Paleontology (667) Pathology (2835) Pharmacology and Toxicology (4825) Physiology (7645) Plant Biology (15158) Scientific Communication and Education (2046) Synthetic Biology (4302) Systems Biology (9825) Zoology (2271)