Introduction

Wildlife monitoring programmes play a key role in understanding ecological systems and constitute the basis for management decisions and actions (Yoccoz, Nichols & Boulinier 2001). As well as tracking a species’ status in the wild, long-term population studies provide an insight into a population's usual abundance and variability prior to anthropogenic activity and can be used to determine whether a decline is part of a natural cycle or the result of external pressure (Willis et al. 2007). Monitoring population declines is an especially important step in tackling biodiversity loss, as severe population reductions act as a prelude to species extinction (Ceballos & Ehrlich 2002; Collen et al. 2009). Yet, a recent study has raised concern over the number of monitoring programmes that have ‘monitored populations to extinction’, with no clear guidelines on how to act when declines are first diagnosed (Lindenmayer, Pigott & Wintle 2013). Here, we explore how differences in decline shape could represent useful ‘trigger points’ within monitoring programmes, highlighting when and where rapid management intervention is required.

Recent analyses across a set of a long-term mammalian population time series have revealed several key distinctions in wildlife population declines (Di Fonzo, Collen & Mace 2013), which provide a more nuanced understanding of extinction risk compared with simple measures of average population change or cumulative percentage loss (i.e. Criterion A of the Red List; IUCN 2001). Di Fonzo, Collen & Mace (2013) identified three principal decline-curve types of increasing severity: quadratic concave (i.e. recovering), exponential concave (i.e. decelerating) and quadratic convex (i.e. accelerating) decline curves (Fig. 1a,b; Table S1 within Appendix S1 in Supporting Information), that carry information regarding the potential urgency of conservation interventions. Moreover, there is some evidence that dominant threats can be diagnosed from distinctions in long-term abundance trend data using flexible Bayesian frameworks (Shoemaker & Akçakaya 2015).

In order to evaluate whether these decline types represent useful triggers for conservation action, we must first determine whether their presence is associated with any particular biological, environmental or anthropogenic conditions. While it is well known that certain biological and ecological attributes predispose species to higher extinction risk (e.g. large body size, low population density, small geographic range, specific dietary requirements; Purvis et al. 2000), recent studies have revealed limited associations between intrinsic traits and population level trends (Cowlishaw, Pettifor & Issac 2009; Mace et al. 2010; Collen et al. 2011). Instead, they find site-specific characteristics (such as anthropogenic pressure intensity or certain environmental conditions) to be better determinants of population decline. Although these studies do not identify any link between negative population trends and species with an intrinsically higher risk of extinction, this does not preclude associations with specific decline types. For instance, species at a greater risk of extinction could exhibit a higher frequency of severe quadratic convex declines, which might have been missed in previous population trend analyses (i.e. based on simple linear regression). A higher frequency of switching between accelerating quadratic convex declines and recovering concave trajectories (i.e. exhibited during the process of ‘critical slowing down’ prior to catastrophic declines; Scheffer et al. 2009; Drake & Griffen 2010) is a further potential indicator of proximate extinction, which could be built within monitoring programmes to rapidly tackle at-risk populations before it is too late.

In this study, we investigate the association between the decline-curve dynamics of a sample of mammalian population data from a data base of vertebrate abundance time series and a range of characteristics (defined in Table 1 and Table S2 within Appendix S1). Specifically, we use generalized linear mixed modelling to investigate whether the presence of different decline-curve types is dependent on particular species-specific traits, levels of local, anthropogenic pressure or based on particular attributes of the time series itself. Through this study, we hope to provide insight into the conditions contributing to different decline curves, which could be used to pre-empt the application of particular conservation management strategies. We chose to focus our study on mammals as they are a very well-monitored and extensively studied vertebrate group, with a quarter of species recorded to be threatened with extinction (Schipper et al. 2008).

Materials and methods

Data Manipulation

We first selected 75 high-quality mammalian population time series for our analysis, representing 33 species, spanning six orders, which we drew from a vertebrate population abundance data base (Living Planet Database (LPD); Collen et al. 2009) based on criteria from Di Fonzo, Collen & Mace (2013). The minimum requirement was that time series had population counts which spanned more than 5 years, had less than 8 year gap between data points, were collected from 1900 onwards, exhibited stable or declining abundance trends (using linear regression), exhibited low environmental stochasticity (based on whether the total reduction in population abundance was less than the difference between the upper and lower 95% confidence interval around the linear model fit) and were reported as being affected by one or more threats. All the data manipulation and analyses here after were carried out using the statistical programming software, R 3.1.0 (R Development Core Team 2015).

We then applied the Kalman filter (as used by Knape & de Valpine 2012) to the time-series data to account for potential sources of uncertainty arising from count errors using the ‘dlmMLE’ function (‘dlm’ package; Petris 2015). Next, we identified the different decline curves using the following steps, described in Di Fonzo, Collen & Mace (2013), and summarized as follows: (i) we smoothed the Kalman-filtered time series using a generalized additive modelling framework (Wood 2006); (ii) detected statistically significant switches in dynamics (which we term ‘switch points’) based on changes in the trend's second derivative sign (fully described in Di Fonzo, Collen & Mace 2013); and (iii) fit a range of algebraic functions between switch points to determine the different curve types based on lowest Akaike Information Criterion corrected for small sample size (AICc; Sugiura 1978), using a threshold of four AICc (Burnham & Anderson 2004). We excluded eight linear declines from the data set as these decline types could not be robustly associated with a specific pressure type (Di Fonzo, Collen & Mace 2013). This left us with 85 decline-curve types (35 quadratic convex, 29 quadratic concave and 21 exponential concave) from 60 high-quality mammalian population time series, distributed around the globe (Fig. 2; Table S3). The above analyses were carried out using the following R packages: ‘AICcmodavg’ (Mazerolle 2015), ‘mgcv’ (Wood 2006), ‘timeSeries’ (Rmetrics Core Team 2015) and ‘msm’ (Jackson 2015). The decline-curve data sets are uploaded in Dryad (Entry DOI:10.5061/dryad.88nd5), and the R scripts for this study are uploaded in Appendix S2 in Supporting Information.

We subsequently retrieved information for each population on several anthropogenic pressures, species-specific and time-series-related characteristics (see Table 1), which we hypothesized might influence decline-curve dynamics and removed all decline curves with incomplete information. Home range data and population density were retrieved from the PanTHERIA data base of mammalian life-history traits (Jones et al. 2009). Generation length was retrieved from Pacifici et al. (2013). We obtained information on the number of distinct threats affecting each population from the Living Planet Index data base (Collen et al. 2009). Maximum finite rate of population growth (r_max) was calculated based on Equation 1 from Slade, Gomulkiewicz & Alexander (1998), using data on litter size, maximum longevity and age at sexual maturity from Jones et al. (2009). We solved this equation using the ‘uniroot.all’ function in the ‘rootSolve’ package (Keith et al. 2015) and selected the highest r_max value in cases where there was more than one solution. To obtain values for the two spatial variables (‘Human Appropriation of Net Primary Productivity as a percentage of Net Primary Productivity’ and the ‘Human Influence Index’), we created buffer polygons with a radius of 1, 10 and 60 km around each population's point location using the ‘Analysis Tool’ available in ArcMap v. 9.3 (ESRI 2008) and calculated the mean value across the buffers using the ‘Zonal Statistics’ tool. We chose this particular range of buffer sizes as they correspond to the 25th, 50th and 75th percentile (rounded-up to the nearest tenth) of home range sizes in km² across the species with available home range data in the filtered data set (n = 21).

A Kruskal–Wallis test (Kruskal & Wallis 1952) confirmed there were no statistically significant differences between the mean values at different buffer sizes; therefore, we continued the analysis at the 10 km level (see Table S4). Spearman-rank tests (Spearman 1904) were carried out to test for co-linearity between continuous explanatory variables (Table S5), and those with a correlation coefficient of greater than 0·7 were removed from the analysis, based on the most commonly used threshold (Dormann et al. 2013).

Results

Although quadratic convex and combined concave decline curves exhibited a higher degree of phylogenetic non-randomness compared to the other decline-curve types (Table S6; their D statistics were approximately 0), we did not account for phylogenetic structure in subsequent analyses as this was neither a strong phylogenetic signal, nor supported by a large sample size (n = 35 and 58 respectively). Spatial autocorrelation was also absent across decline curves (based on Moran's I and Geary's C tests; all P-values > 0·1; Fig. 2). We identified 20 plausible models which described quadratic convex decline curves with less than two AICc difference, containing a total of nine parameters in a variety of additive interactions (Tables 2 and S7). We found that only one time-series characteristic consistently explained the presence of quadratic convex decline curves, with the remaining time-series descriptors, biological traits and anthropogenic measures being less important. Specifically, we found that quadratic convex declines were associated with a change in trajectory towards the start of the time series (inferred from the negative ‘proximity to end’ estimate; Table 2). The presence of quadratic concave declines was best explained by 23 models containing additive interactions between a total of nine parameters (Tables 3 and S8). ‘Proximity to end’ was the only parameter that was consistently associated with the presence of quadratic concave declines (with a positive model-averaged estimate), suggesting that this curve type is more likely to occur towards the end of a population time series. We found that the presence of exponential concave declines was best explained by 58 models, with a combination of 12 variables in additive interactions (Tables 4 and S9). None of these demonstrated a consistent effect on the presence of exponential concave declines. When we combined concave declines together in one category, their presence was best explained by 11 models, with a combination of seven variables in additive interactions, in addition to the null model (Table S10). None of these had a consistent effect on the presence of this particular decline category (Table 5).

Discussion

Our broad-scale study of the distribution of different wildlife population decline curves in relation to species-specific biological, anthropogenic and time-series characteristics predictors identifies a novel association with useful management implications. We detect a key time-series characteristic associated with the presence of quadratic convex and quadratic concave declines, which can be explained in the context of different threatening processes. Quadratic convex declines were more likely to occur towards the start of the time series: a phenomenon that could coincide with the appearance of a novel threat (Mace et al. 2010). Quadratic concave declines, instead, were more likely at the end of the analysed time series. When a new pressure appears, there are four possible responses of a population: it may (i) decline and then recover, (ii) decline to extinction, (iii) decline and stabilize or (iv) be unaffected. Early in this process, quadratic convex declines will be more likely, especially under severe or increasing pressures (e.g. an accelerating rate of habitat conversion). Over the course of time, as the pressure stabilizes or reduces, a population that has not declined to extinction must either recover or stabilize, resulting in a concave (or decelerating) curve. The quadratic convex decline-curve type is also the least prevalent across our time-series data set (compared to concave declines), which may be a result of a ‘filter effect’ (Balmford 1996) causing an inherent bias against the identification of quadratic convex declines. Populations that declined in a quadratic convex manner in the past may have done so too quickly for monitoring to take place (exemplified in the Indian Gyps vultures decline; Prakash et al. 2012), or population monitoring may have only started following a severe decline, thereby missing earlier quadratic convex sections.

The tendency for quadratic convex declines to occur in the beginning of population time series means that its identification could act as a signal for a novel threat, following which increased monitoring in combination with rapid conservation effort focussing on threat abatement should be put in place. Quadratic convex declines have also been diagnosed with high levels of accuracy under simulated deteriorations in data quality (Di Fonzo, Collen & Mace 2013), providing further support for its efficacy as a trigger for interventions. Instead, if concave declines were detected, it could be inferred that the population has already undergone a period of severe decline, from which its rate of decline is slowing or even recovering. The presence of this decline type suggests a different type of management strategy might be required, focussed on supporting recovery (e.g. through improving the species’ habitat). The identification of different decline curves therefore represents a novel trigger for adaptive management practices (sensu Walters 1986), which should be considered in addition to recently proposed measures of ‘percentage decline in population size, duration of population decline, loss of numbers of subpopulations or reduction in the distribution of a species’ (Lindenmayer, Pigott & Wintle 2013).

Our analyses do not pick up any relationship between body mass and severe quadratic convex declines, which we had hypothesized as more likely in larger species with an intrinsically higher risk of extinction. Similarly, we found no relation between species with lower r_max (i.e. of slower life-history speed, with greater risk of extinction) and a higher likelihood of quadratic populations declines. Such lack of associations may be due to the low number of quadratic convex declines in this data set and warrants further investigation. A further characteristic of this data set which may preclude these associations is that the species included are generally on the slower scale of the mammalian life-history speed continuum, thus reducing the statistical power of our analyses. It would be valuable to follow up these findings by collecting information on the intraspecific variation in life-history characteristics across the various populations (as identified in Gonzláez-Suárez & Revilla 2013) in order to obtain more precise estimates for such decline-curve associations.

Statistical analysis of the distribution of decline-curve types across IUCN Red List extinction risk categories did not reveal any clear patterns. Contrary to expectation, more severe, quadratic convex declines were no more likely to occur in populations of more threatened species than species with lower extinction risk, despite IUCN Criteria A being based on ‘high population decline rate’. The inconsistency between decline-curve severity and species extinction risk may be explained by the fact that percentage loss in population size and population decline-curve type measure two different processes. The first metric considers the species’ risk of extinction based on the total reduction in population abundance, whereas the latter provides information on how the rate of population decline changes over time. This result highlights how determining populations’ decline-curve type could be a valuable refinement to IUCN Criterion A, indicating a higher risk of extinction for species with populations declining at an accelerating rate (Di Fonzo, Collen & Mace 2013).

Some potential limitations of the information used here may have affected our findings. First, there was a temporal mismatch between individual population time series and the predictors of anthropogenic pressure (i.e. HII and HANNP) that will have reduced their explanatory power. The population time series used in this analysis were recorded between 1950 and 2010, ranging across all decades, whereas the human pressure data sets provided single ‘snapshots’ of the populations’ local conditions that may not represent the pressures affecting the population during the time it was monitored. Secondly, population abundance estimates were compiled from numerous sources, using a variety of monitoring methods and sampling efforts, including potential observation errors that may have obscured our ability to correctly categorize decline curves (as described in Solow 1998; Knape & de Valpine 2012). Our analyses were also limited to the time frame over which the data were collected, preventing us from assessing the declines with respect to a historical population baseline (recommended by Porszt et al. 2012; d'Eon-Eggertson, Dulvy & Peterman 2015) or over a period of three generation lengths (recommended by the IUCN Standards and Petitions Subcommittee 2010). This may have reduced our ability to diagnose declining populations to begin with, however would not have subsequently affected the analysis. Additionally, our study did not take into account the effect of possible conservation actions on population dynamics, which could have potentially biased our decline-curve results. The paucity of information on the presence and timing of conservation actions across the data set prevented us from using it within a comparative analysis; however, its possible influence on population dynamics merits future investigation through more detailed case studies. Finally, we included a high number of explanatory variables in our models (with respect to sample size), which may have resulted in potential model overfitting. To reduce the chances of this occurring, we recommend augmenting the data set through an updated search for suitable population time series, expanding the study to include data from more animal taxa. A larger data set would enable division of the time series into training and testing sets, maximizing performance on the training set and testing its efficacy on the unseen portion of data.

In addition to acting as a novel type of conservation trigger, dividing observed mammal population decline-curve types into categories of increasing decline severity has the potential to provide important insights for wildlife management. Although not a prioritization mechanism on its own, differences in decline type could be incorporated within Criterion A of the IUCN Red List as a more continuous indicator of the urgency with which species’ population declines need addressing (while also considering the potential costs and probability of success associated with their recovery). For example, species with a greater number of severe declines could be allocated a higher risk of extinction within the same Red List category, despite not having met the criteria for moving up a category at the species level. Assessing differences in population decline-curve dynamics could also be useful for prioritizing Red List (re)assessments, where an elevated presence of quadratic convex population declines could instigate a closer evaluation of a species’ extinction risk. We acknowledge that such changes in the IUCN extinction risk assessment protocol might be difficult to implement unanimously due to differences in the availability of population time series data for highly threatened species and budget limitations to monitoring additional populations. Extinction risk categorizations that include information on population decline-curve type may therefore be of greater use to local/regional assessments, where the data are generally of higher resolution.

Our study illustrates how the identification of different decline-curve types can provide a signal for a change in management action: quadratic convex declines could be used to trigger rapid threat abatement efforts to prevent potential extinctions, while the identification of quadratic concave dynamics could promote increased conservation effort towards the recovery of dwindling populations. In order to confirm the efficacy of these decline-curve types as signals for management change, we recommend performing this analysis over a much wider sample of population time series, both already collected and in the process of being monitored. Further testing on the power to detect differences in decline dynamics in wild populations will also be required (sensu Nichols & Williams 2006).