Compensation and additivity of anthropogenic mortality: life-history effects and review of methods


  • Guillaume Péron

    Corresponding author
    1. Patuxent Wildlife Research Center, U.S. Geological Survey, Laurel, MD, USA
    2. USGS Colorado Cooperative Fish and Wildlife Research Unit, Colorado State University, Fort Collins, CO, USA
    • Department of Wildland Resources, Utah State University, Logan, UT, USA
    Search for more papers by this author

Correspondence author. E-mail:


  1. Demographic compensation, the increase in average individual performance following a perturbation that reduces population size, and, its opposite, demographic overadditivity (or superadditivity) are central processes in both population ecology and wildlife management. A continuum of population responses to changes in cause-specific mortality exists, of which additivity and complete compensation constitute particular points. The position of a population on that continuum influences its ability to sustain exploitation and predation.
  2. Here I describe a method for quantifying where a population is on the continuum. Based on variance–covariance formulae, I describe a simple metric for the rate of compensation–additivity.
  3. I synthesize the results from 10 wildlife capture–recapture monitoring programmes from the literature and online databases, reviewing current statistical methods and the treatment of common sources of bias.
  4. These results are used to test hypotheses regarding the effects of life-history strategy, population density, average cause-specific mortality and age class on the rate of compensation–additivity. This comparative analysis highlights that long-lived species compensate less than short-lived species and that populations below their carrying capacity compensate less than those above.


The degree to which temporal and spatial changes in cause-specific mortality are reflected in dynamics at the population scale plays a key role in ecological processes, from predator–prey relationships (Sinclair & Pech 1996; Salo et al. 2010) to population response to environmental fluctuations (McCann, Botsford & Hasting 2003), and sustainable harvest, which is the topic of many studies, such as Burnham & Anderson (1984) and Boyce, Sinclair & White (1999). The non-trivial nature of the link between cause-specific mortality and total mortality has long been recognized. First, individuals that are removed by new or increasing sources of mortality may not be a random subset. For example, ducks shot by hunters in a study by Hepp et al. (1986) had lower body condition than the general population, thus were suspected to have lower natural survival probability than individuals that escaped hunters. By contrast, in ungulates, trophy hunters often remove those individuals that normally would have performed best (Mysterud & Bischof 2010). Secondly, population density can be locally modified by changes in cause-specific mortality, which can influence the fate of survivors. Survivors may benefit from compensatory density dependence (Bonenfant et al. 2009), as well as take the opportunity to colonize better habitat than originally available to them (Pulliam 1988; Turgeon & Kramer 2012). By contrast, changes in cause-specific mortality may induce reduction in the proportion of the range that is used (‘landscape of fear’; Kauffman, Brodie & Jules 2010), increase the risk of starvation because of stress and lost foraging opportunities (Arlettaz et al. 2007; Brøseth & Pedersen 2010), disrupt social structures (Williams, Lutz & Applegate 2004; Rutledge et al. 2010) and induce other Allee effects (Halliday 1980).

Because of these multiple mechanisms, there is a gradient of possible responses of populations to changes in cause-specific mortality (Nichols et al. 1984; Sandercock et al. 2011). Historically, two main hypotheses were considered: additivity and compensation. The former means that any individual that dies from the ‘additive’ cause would have survived if this cause was removed. The latter means that, if one reduces mortality from one cause, the spared individuals simply die from other causes. This was initially coined the ‘doomed surplus’ hypothesis (Errington 1956). These two hypotheses are, however, best viewed as particular points on a gradient of possible population responses to changes in mortality pattern (detailed below). The development of new statistical methods (Sedinger et al. 2010; Servanty et al. 2010) makes it timely to synthesize the information about compensation and additivity that is contained in long-term individual-based monitoring data (e.g. GameBirds database; Bird Banding Lab, USGS Patuxent Wildlife Research Center) and to compare populations to draw inference about the effect of life-history strategy and population density on compensation and additivity.

In this review, I focus on survival probability because it is the most direct vital rate that exploitation can affect and on compensation of annual variation in mortality patterns. I do not consider the studies of compensation via increased fecundity of survivors, nor those that focused solely on spatial variation in mortality patterns (but note that many methods pertaining to temporal variation can be applied to spatial variation too). First, I use variance formulae to introduce a metric for the rate of compensation–additivity, which is then used to describe the gradient of potential demographic response to changes in cause-specific mortality. Second, I synthesize the results of 10 case studies (four new analyses, six published studies), focusing on how the rate of compensation varied with age, with life history, with average cause-specific mortality and with population density. I build a predictive model based on these 10 cases, as a first attempt to predict the rate of compensation–additivity in less intensively studied species and populations.


All survival and mortality rates are annual, and I focus on temporal variation in mortality rates (vs. spatial variation). ‘Anthropogenic mortality’, denoted h, is the mortality cause under study (hunting, collisions, etc.) and is opposed to ‘natural mortality’, denoted n, which designates mortality due to all other causes: starvation, predation and diseases, but also mortality from anthropogenic causes other than the one under study. If anthropogenic mortality is seasonal, all years start at the onset of the season during which anthropogenic mortality occurs (e.g. survival is estimated from October 1 to September 30 in the case of waterfowl hunting). The methods also apply to the study of compensation for other causes of mortality, for example, h can represent mortality by predation.

The idea of a compensation rate was recently brought forward by Sedinger et al. (2010) and Servanty et al. (2010), who estimated a temporal correlation coefficient between anthropogenic and natural mortality probabilities, denoted math formula. Intuitively, if math formula, any increase in anthropogenic mortality is associated with a compensatory decrease in natural mortality. I developed (simple) formulae to quantify the level of compensation, which is given below.

Using the notations above, overall annual survival S is written as:

display math(eqn 1)

From the formulae for the variance of a sum of random variables and for the correlation coefficient of two variables, it follows that:

display math(eqn 2)

math formula the covariance between overall survival and anthropogenic mortality quantifies the extent to which overall survival depends on anthropogenic mortality. Following eqn (eqn 2), and based on the balance between math formula, the correlation between natural and anthropogenic mortality, Var(h) the variance in anthropogenic mortality and Var(n) the variance in natural mortality, there are five distinct situations regarding the sign and value of math formula. These five situations correspond to five sections that exhaustively cover the gradient of demographic responses to changes in cause-specific mortality.

  1. Over compensation: math formula :annual survival increases with anthropogenic mortality rate (math formula). Decreases in natural mortality are greater (in absolute value) than increases in anthropogenic mortality. This situation is mostly found in species with marked hierarchy or size variation; in these populations, the removal of highest-ranking or largest individuals can trigger an increase in the survival of low-ranking or small individuals that numerically overcompensate the number of removed individuals (Zipkin et al. 2008). Overcompensation may be more common in compensation via fecundity (O'Regan et al. 2012).
  2. Complete compensation:math formula: annual survival is independent from anthropogenic mortality (math formula), and changes in anthropogenic mortality rate have no impact at the population level.
  3. Partial compensation:math formula): the reduction in annual survival rate is smaller than the change in anthropogenic mortality rate (math formula).
  4. Complete additivity: math formula: natural and anthropogenic mortality are independent. Both mortality sources add to each other without interacting (math formula)
  5. Overadditivity (or superadditivity or depensatory mortality): math formula: natural mortality increases with anthropogenic mortality (math formula).

The above considerations highlight the role of the ratio between the temporal variances in both types of mortality. If math formula, then complete compensation is impossible. If Var(n) largely exceeds Var(h), then the population is in a good configuration for overcompensation to occur. Note that except in situation 4, n is altered from what it would be in the absence of anthropogenic mortality (mortality in the absence of anthropogenic influence: n0).

The rate of compensation–additivity C then indicates which situation the system is in.

display math(eqn 3)

C-values above 1 indicate overcompensation, while values below 0 indicate overadditivity. In the case of partial compensation, C varies between 0 (complete additivity) and 1 (complete compensation) and can be interpreted as the proportion of the fluctuations in anthropogenic mortality around its average that are compensated.

The estimation of C is based on the variation in h during the study period; thus, the C-estimate is associated with the expected value of anthropogenic mortality during the study, E(h). It can be considered valid over the range of ‘commonly’ observed h-values [e.g. one standard deviation interval around E(h)].

Predictions about interpopulation variation in compensation–additivity rate

First, populations should be able to compensate more easily for anthropogenic mortality when this rate is low than when it is high. This should translate into a negative effect of E(h) on C. Second, birds and mammals are often ranked along a continuum of fast to slow life histories (Harvey & Zammuto 1985; Stearns 1992). At one end of the continuum, fast (short-lived) species exhibit low survival rates but high fecundity. At the opposite end of the continuum, slow (long-lived) species exhibit high survival rates but low fecundities. Other axes of variation in life histories exist, but the fast/slow continuum is the first component of most multivariate analyses of vertebrate traits (Gaillard et al. 1989; Saether et al. 2005; Bielby et al. 2007). Along this continuum, the fitness consequences of a reduction in survival probability increase with decreasing speed of life (Pfister 1998). ‘Environmental canalization’ (Gaillard & Yoccoz 2003) or ‘life-history buffering’ is a corollary of this relationship: the temporal variance in adult survival generally decreases with speed of life history, as adult survival tends towards one (Zammuto & Millar 1985; Stearns & Kawecki 1994). Populations of slow species are thus predicted to have lower ability than faster species to compensate for anthropogenic mortality, precisely because they evolved strategies aimed at minimizing adult natural mortality. I used generation time (T, the weighted mean age of reproducing females in the population) as a metric for the position of the study species on the life-history continuum (Gaillard et al. 2005). T is also a proxy for Var(n0) in adults (Gaillard & Yoccoz 2003). The prediction is that C should decrease with T.

Third, compensation is expected to be stronger if individual performance is initially suppressed by negative density dependence (Salo et al. 2010; e.g. competition for food sources, use of suboptimal habitat). The prediction is that C should be lower in populations that are below the carrying capacity of their environment than in populations that are at or above it. I used a two-mode density index, D, which was filled in using basic considerations about population trend. If a population showed a steady increase, it was considered under its carrying capacity. If it was stable or oscillating around a long-term mean, it was considered at or around carrying capacity. A population in steady decline could be above carrying capacity if the latter was recently impacted by environmental changes or below capacity if the decrease was due to overexploitation; in all considered species, enough information was available to separate the two hypotheses. In brief, I used a two-mode variable to characterize population density as ‘at or above carrying capacity’ or ‘below carrying capacity’.

Lastly, within populations, younger age classes were expected to compensate better, because they are more sensitive to density dependence and more heterogeneous (Forslund & Pärt 1995; Bonenfant et al. 2009).

Materials and methods

Extracting C from published estimates of cause-specific mortality

The correlation between anthropogenic and natural mortality can be extracted from temporal time series of cause-specific mortality estimates, typically obtained by radiotracking individuals and recording the date and cause of death (Heisey & Patterson 2006). Other types of data are summarized in Table 1 and Appendix S1. Capture–recovery data (CR), which have been collected for a large number of species over long time periods and large spatial scales, are particularly relevant. They consist of information on the time elapsed between marking of individuals and their recovery as dead from anthropogenic cause, with many missing data. The statistical analysis of CR data is based on survival probability S = 1 − n − h and Seber's recovery probability math formula, where λ is the probability that a dead marked individual is reported as such, given that it died from anthropogenic cause (Williams, Nichols & Conroy 2002). The correlation coefficient between h and n can then be extracted from estimates pertaining to S and r, provided the properties of λ are known (details in Appendix S1, part 1). One interesting property of this approach is thus that harvest rate h is estimated while correcting for imperfect reporting rate λ, although in this study I relied on strong assumptions about λ (e.g. constancy across species and space).

Table 1. Different types of individual-based data and associated statistical methods used to estimate the correlation between anthropogenic and natural mortality and the rate of compensation–additivity
AbbreviationType of dataEstimated parametersParameters to be estimated separatelyUpward sampling bias on CDownward competition bias on CReferences
  1. S stands for annual survival, r for recovery probability, n for annual natural mortality, h for annual anthropogenic mortality, λ for mark reporting rate (for animals dead from the anthropogenic source) and n0 for annual natural mortality in the absence of anthropogenic mortality. E and Var stand for expected value and temporal variance, respectively, when only one of these quantities is involved. In the ‘bias’ columns, Y (Y) and N, respectively, indicate that a bias is systematically present, not systematically present (there is sampling covariance but it may vary in direction) or is absent. More details about the statistical methods are found in the quoted references (among others).

CRCapture–recovery data with only one type of recoveriesS, rλ, n0YY

Brownie & Pollock (1985); Williams, Nichols & Conroy (2002)

Appendix S1, part 1

CR-HCapture–recovery data plus a measure of anthropogenic mortality rateS, r, b, E(n0)Var(n0)YY

Barker, Hines & Nichols (1991); Gauthier et al. (2001)

Appendix S1, part 2

IPMCapture–recovery data plus population surveysS, rλ, n0(Y)YBesbeas et al. (2002); Péron, Nicolai & Koons (2012)
CRPCapture–recovery and Capture–recapture datan, h n 0 YYServanty et al. (2010)
CRRCapture–recovery data with two (or more) types of recoveriesn, h n 0 YY

Schaub & Lebreton (2004)

Appendix S1, part 3

CP-HCapture–recapture data with a measure of anthropogenic mortality rateS, b, E(n0)Var(n0)NYVeran et al. (2007); Rolland, Weimerskirch & Barbraud 2010;
KFKnown-fate data (telemetry, radiotracking)n, h, b, E(n0)Var(n0)NY

Heisey & Patterson (2006); Creel & Rotella (2010)

Appendix S1, part 2

The estimate of covariance between natural and anthropogenic mortality need, however, be divided into three components: a sampling component due to the fact that both mortality rates are estimated from the same data and individuals (Brownie & Pollock 1985), a process component due to the fact that two sources of mortality compete for a finite number of individuals (Schaub & Lebreton 2004) and the remaining of process covariance, which is the quantity of interest. The first two components are hereafter called sampling and competition bias. Sampling bias comes from the fact that individuals have to be dead to be recovered, thus high r values are associated to low S values. Sampling bias is towards greater additivity. The degree to which covariance estimates are affected by sampling bias depends on the method (Table 1). For example, using data collected independently of anthropogenic mortality (e.g. population surveys) reduces sampling bias. Competition bias is towards greater compensation. Under the complete additivity hypothesis, and assuming natural mortality occurs before anthropogenic mortality, survival probability is written math formula, where n0 is independent from h. Then, cov(nh) = −E(n0)Var(h) (Schaub & Lebreton 2004) and math formula, instead of = 0.

In the study cases coming from the literature (see second next section), after transforming time series of S and r estimates into an estimate of temporal covariance between n and h, I corrected for sampling bias a posteriori using the methods for separation of (co)variance components detailed by Burnham et al. (1987). The method is based on the estimated sampling variance–covariance matrix math formula. If the number of marked individuals alive at each time occasion are assumed equal or similar, then the sample covariance is math formula, where K is the number of sampling occasions. If time-specific sample sizes are not equal, one introduced weights, and there is a system of K equations to solve (Burnham et al. 1987, p. 262–265). In most cases, math formula was, however, not provided in the original study presenting the case study; thus, sampling bias was not quantifiable.

Then, I corrected for competition bias on the correlation between n and h (Bc), using the formulae by Schaub & Lebreton (2004):

display math(eqn 4)

I used very rough estimates of E(n0) and Var(n0) (values deemed plausible based on life-history considerations, or values extrapolated from the data when it was possible). A sensitivity analysis was conducted (Appendix S3). More rigorous methods have been used elsewhere, for example, Servanty et al. (2010) combined multiple expert guesses to produce distributions of the needed parameters, and Devineau (2007) collected survival estimates from both exploited and non-exploited waterfowl populations, from which math formula can be extrapolated for any waterfowl species as a function of its body mass.

Random effects for the estimation of C in the Bayesian Monte Carlo Markov Chain framework

If the raw data are available, the temporal correlation between n and h can be estimated directly using a model that features a multivariate normal distribution correlating natural and anthropogenic mortality (Link & Barker 2005; Servanty et al. 2010; Appendix S2)

display math(eqn 5)

math formula stands for bivariate normal distribution, subscript t stands for year and math formula, math formula, ρσnandσh are parameters to estimate. Total covariance between natural and anthropogenic mortality is then the estimate for math formula. Sampling bias was approximated using the sampling covariance between math formula and math formula, and competition bias was estimated using eqn (eqn 4).

For data sets that were available for (re)analysis, I fitted this type of model within the Bayesian Monte Carlo Markov Chain framework. This approach presents the advantage that there is no need to use weights to account for varying sample sizes in the computation of sampling bias.

The TEN study populations

I selected 10 data sets from eight species of birds and two mammals, based on (i) availability of either the raw data or year-specific estimates from published analyses, (ii) sample size of more than 500 individuals and (iii) study duration exceeding species’ generation time or more than 5 years in short-lived species. In addition, no more than one species per taxonomic genus was selected. The 10 species had generation time varying between 1·5 and 14 years and were subjected to anthropogenic mortality rates varying between 2% and almost 50%.

Of these 10 study cases, four were new analyses, and six were published studies. The four new analyses were for Redhead (Aythya americana, REDH), Greater sage-grouse (Centrocercus urophasianus, SAGR), Sandhill crane (Grus canadensis, SACR) and Mourning dove (Zenaida macroura, MODO). The CR data were extracted from the GameBird database (Bird Banding Lab, USGS Patuxent Wildlife Research Center). The GameBird database contains banding and recovery records from throughout the United States and Canada and for a wide range of game birds. The data extraction procedures, the statistical models, as well as the WinBUGS codes are provided in Appendix S2.

The six other case studies came from the literature. In two of these cases, the correlation between natural and anthropogenic mortality and the temporal variances were already estimated by the original authors (Wildboar Sus scrofa, WBO (Servanty et al. 2010), and White storks Ciconia ciconia WHST). In the last four cases – Common teal Anas crecca COTE, Greater snow goose Chen c. caerulescens SNGO, Northern bobwhite Colinus virginiana NOBO and Grey wolf Canis lupus WOL – compensation–additivity rate was obtained by transforming the year-specific estimates provided by the original study (Appendix S1). In the French COTE study, uncertainties existed about band reporting rate λ, but the case was included because it represented harvest by a geographically different group of hunters (compared with North American species such as REDH and SNGO) and because a sensitivity analysis showed that the result were qualitatively not affected by the value of λ within a range of plausible values (Appendix S3).

Statistical method for the comparative analysis

The comparative analysis was restricted to C-estimates for adults. Following the predictions, a linear model was built with the interactive effects of E(h) and D and the additive effect of T, on compensation–additivity rate C. It was fit to the data using a phylogenetic generalized least square model, with a Brownian drift along the taxonomy-based phylogeny (Martins & Hansen 1997). It was fit using the function gls in r-packcage NLME (Pinheiro et al. 2008; R-Development-Core-Team 2010) and the function corBrownian in r-package APE (Paradis, Claude & Strimmer 2004). The correlation structure was meant to account for the fact that the data set included both closely related species (e.g. three Anatidae) and phylogenetically distant species (birds and mammals) and that some taxa may compensate better than other. This was not the case, however, as the same analysis without correction for phylogenetic inertia yielded very similar results (not shown).


Age variation in compensation rate

In Redheads, the compensation rate C was significantly lower in hatch-year birds (95% CI: 0·31; 0·48) than in after hatch-year birds (95% CI: 0·53; 0·62). In sage-grouse, the available information did not allow rejection of the additivity hypothesis in hatch-year birds (95% CI for C: −0·60; 0·16), while the reverse was true in after hatch-year birds (95% CI: 0·32; 0·66). In doves, the C-estimate was (marginally) lower in hatch-year (95% CI: 0·01; 0·12) than in after hatch-year birds (95% CI 0·11; 0·29). By contrast, in slower-lived storks, partial compensation was found in juveniles but not in adults (Schaub & Lebreton 2004). In even slower-lived cranes, both age classes showed overadditivity (with extensive overlap between age-specific 95% CIs). The prediction of a higher compensation rate in young age classes was therefore not met.

Comparative analysis of compensation–additivity rate in adults

The model results indicated that both generation time (slope ± standard error SE: −0·05 ± 0·01, anova P-value: 0·01; Fig. 1) and density (effect of ‘below carrying capacity’ compared with ‘at or above capacity’: −0·49 ± 0·10, P-value: 0·005; Fig. 1) had significant effects on compensation–additivity rate. The effect of average anthropogenic mortality was marginally significant in populations around or above carrying capacity (slopes ± SE: −0·19 ± 0·54, P-value: 0·06; Fig. 1), but not significant in populations below carrying capacity (slopes ± SE: 0·62 ± 0·60, P-value: 0·35; Fig. 1).

Figure 1.

Comparative model of the rate of compensation–additivity C (colour scale) as a function of average anthropogenic mortality E(h) (y-axis), generation time T (x-axis), and whether the population is around or above carrying capacity (left panel) or below carrying capacity (right panel). The white circles indicate the locations of the study populations in the plane defined by variation in E(h) and T. Species abbreviations are as in Table 2. The bold white line is an isocline that separates compensation (left) from overadditivity (right). Thin white lines correspond to other isoclines separated by 0·25 changes in compensation–additivity rate. The map of standard deviation around model predictions is omitted because several elements required to compute it [such as sampling covariance between E(h) and C] were not available.

Table 2. Data sets included in the interspecific analysis of compensation patterns
Abbreviations T D SpeciesLocationPopulation segmentPeriodType of dataContextSample sizeReference of original study
  1. Type of data is abbreviated as in Table 1. T is the generation time for little to unexploited populations. Density variable D indicates whether the population was below (B), or at or above (A) carrying capacity. See Appendix S3 for details. ‘Population segment’ indicates which types of individuals were included in the comparative analysis (Fig. 1). ‘Context’ indicates how the population is exploited; SH stands for sport hunting, WC stands for wildlife control and BC stands for involuntary take.

  2. a

    These data sets included both spatial and temporal variation.

NOBO1·5ANorthern bobwhite Colinus virginianusBabcock-Webb WMA, Florida, USAdults2002–2009aKFSH2066Rolland et al. (2010)
MODO2·7BMourning dove Zenaida macrouraEastern and Central flyways (excluding non-hunting states)Adult females1960–1980CRSH564 048Otis & White (2002); Appendix S2, part 2
COTE2·8ACommon teal Anas creccaCamargue, FranceAdults1954–1975CRPSH55 175Devineau et al. (2010); Appendix S3
GRSA3·1AGreater sage-grouse Centrocercus urophasianusNorth Park area, Colorado, USAdult females1973–1989CRSH6021Zablan, Braun & White (2003); Appendix S2, part 2
REDH3·5ARedhead Aythya americanaMid-continental populationAdult females1950–2009IPMSH83 340Péron, Nicolai & Koons (2012); Appendix S2, part 3
WBO3·7BWildboar Sus s. scrofaChâteauvillain-Arc-en-Barrois, FranceAdult females1982–2007CRPSH1255Servanty et al. (2010)
WOL4·2BGrey Wolf Canis lupusNorth-western USAdults1982–2004aKFSH-WC711Murray et al. (2010)
SNGO7·1BGreater snow goose Chen caerulescens atlanticusBylot Is., Nunavut, CanadaAdult females1990–1998CR-HSH-WC3890Gauthier et al. (2001)
WHST7·1AWhite stork Ciconia ciconiaSwitzerlandAdults1984–1999CRRBC2912Schaub & Lebreton (2004)
SACR14·6ASandhill crane Grus canadensisNorth and Western North AmericaAdults1970–2009CRRSH3291Appendix S2, part 1


The overall pattern of between-population variation in compensation–additivity rate of adults was congruent with the predictions, despite (i) the varying need for working hypotheses and assumptions across the different case studies (Appendix S3 for an analysis of sensitivity), (ii) the relatively small sample size (10 species) and (iii) the only partial coverage of the parameter space in Fig. 1. Generation time was a strong predictor of compensation rate. This result highlights that, in addition to having naturally lower population growth rates and thus being less able to sustain exploitation (Reynolds, Webb & Hawkins 2005), long-lived species are also less able to compensate for increases in anthropogenic mortality by decreases in natural mortality. The effect of density was also clear-cut and in the direction expected. The effect of density may, for example, explain that in mourning dove, the compensation rate was only moderate, despite the fast life history. During the study period, doves were colonizing new, human-altered areas that have ample food resources; this probably alleviated density dependence and reduced the potential for compensation (see also O'Regan et al. 2012 in another Columbidae).

A small validation procedure using C-estimates from three studies that did not match the selection criteria yielded mixed results (Sooty albatross Phoebetria fusca, White-tailed deer Odocoileus virginianus, Raccoon Procyon lotor; Appendix S4). The predicted sign of the compensation–additivity rate was always consistent, but the predicted value was consistently higher than the value extracted from the original study. This may come from the fact that the validation studies fell within areas of the parameter space in Fig. 1 that were not documented by any of the 10 main studies. In particular, populations used for validation had higher anthropogenic mortality rate than populations with similar life history in the main data set. As the predictive model had an only weak effect of anthropogenic mortality on compensation rate, the prediction for the validation studies may have been biased high. Future work should be aimed at better documenting compensation–additivity in populations with anthropogenic mortality rates above 20%.

Regarding age effects on compensation–additivity rate, my results did not confirm the prediction that young age classes, because they are more heterogeneous and more sensitive to density dependence, would compensate more. A potential explanation would be that young age classes in general experienced higher average anthropogenic mortality. The method might, however, have been flawed: most density dependence in young age classes is expected to come from competition with adults, and correlating age-specific h and n might not capture that process (correlating n in young individuals to h in adults might prove more sensible).

These results have implications beyond harvest management. In particular, they suggest that sensitivity of population growth to predation depends on the density and life-history strategy of the prey species. Species coexistence can thereby be driven by predation. For example, superior but slow-lived competitors may be kept in check by predation, while inferior fast-lived competitors increase in numbers; invasion may be slowed down by increased additivity of predation mortality at low densities.

The issue of crippling losses

Crippling losses correspond to all deaths attributable to the anthropogenic cause at stake, but not counted as such (e.g. for hunting mortality: wounds, lead poisoning; Schulz, Padding & Millspaugh 2006; Guillemain et al. 2007). Crippling losses are a source of concern for the estimation of compensation-additivity rate because the impacted animals are classified as dying from natural causes. If γ individuals are crippled per individual that is available for reporting, then the bias in math formula is math formula (bias towards overadditivity, tending towards zero when the actual C-value tends towards 1). The Redhead case study offered particular insight into the effect of crippling losses in compensation–additivity analyses. The compensation rate for Redheads, estimated without correcting for crippling loss, was significantly different from 1 (Table 3). If the true compensation rate (i.e. compensation for anthropogenic mortality including deaths due to crippling) was 90% (almost complete compensation), then, following the above expression for the bias on C, crippling loss rate needed to exceed γ = 3 crippled bird per shot and retrievable bird. Reported values range from γ = 0·11 to 0·30 in a set of North American waterfowl (derived from Schulz, Padding & Millspaugh 2006 and references therein). Thus, the conclusion that compensation is only partial in Redhead was robust to the occurrence of crippling losses.

Table 3. Parameter estimates for the 13 study cases
SpeciesAnthropogenic mortalityNatural mortalityComplete compensation possibleBiases on math formula math formula math formula
math formula math formula math formula math formula SamplingCompetitionUncorrectedCorrected
  1. ‘Complete compensation possible’ (Yes/No) indicates whether math formula is lower than math formula. Estimates of the correlation coefficient between natural and anthropogenic mortality corr(nh) are provided ‘uncorrected’ and ‘corrected’ for the sampling and competition biases (see Part 2). Other notations are as in the main text.

  2. a

    Sampling bias is considered absent due to the study design.

  3. b

    Sampling bias could not be estimated using the results from the original study, but given the data structure, it is considered reduced (see dedicated section in the main text).


Observational vs. experimental approaches

This review was motivated by recent advances in statistical methods and by the availability of long-term monitoring data. Compared with more experimental approaches where density and mortality rates are controlled (Sandercock et al. 2011), this observational approach has two major limitations. (i) Variation in density and anthropogenic mortality may not be large enough to observe the whole palette of demographic responses to anthropogenic mortality. Populations known to compensate effectively under ‘natural’ or current conditions may not perform as well if mortality is drastically increased. (ii) Monitoring data may reflect the influence of confounding factors more than the process of compensation. For example, waterfowl harvest regulations have long been adapted in response to breeding population size, leading to the partial confounding of density dependence and changes in hunting mortality. However, the long-term monitoring data that were used in this study represent an opportunity to study the demographic response to ‘real-life’ changes in mortality pattern, at spatial and temporal scales largely unmatched by experimental data.

Predictive life-history model

A potentially interesting application would be the prediction of compensation–additivity rate in exploited wildlife populations that are logistically difficult to study because of remoteness or low densities. Anthropogenic wildlife mortality has caused many species' extinctions or near-extinctions in the past (Diamond 1984; Redford 1992; MacKenzie, Mosegaard & Rosenberg 2009), and concerns are rising today, especially in the tropics and for species about which little is known (e.g. Jenkins et al. 2011). Thus, using the results from well-studied populations to extrapolate the dynamics of poorly studied ones can constitute a rapid, cost-effective first step in building a population model of use to decision-makers. If compensation is expected, this can allow for more liberal regulations, with which stakeholders are more likely to comply (Bunnefeld, Hoshino & Milner-Gulland 2011). If overadditivity is expected, this constitutes a warning against unsustainable harvest rates and loss of resource to overadditive natural mortality.


This work was funded by a Quinney postdoctoral fellowship at Utah State University. I am also grateful to all people involved in data collection and management of the GameBirds database, to V. Rolland and O. Devineau for sharing some of their files, to D. N. Koons, M. Schaub, B. K. Sandercock and two anonymous referees for their comments on earlier versions of this article.