Appendix S1
How I used the result from published studies
When capture-recapture and capture-recovery models were fit to data using the frequentist
(likelihood maximization) approach, authors reported year-specific estimates of the model
parameters. In the following I describe how I extracted the correlation between n and h from
these year-specific estimates.
The appendix is based on populations in which a birth pulse is followed by a period of
anthropogenic mortality. It is formulated using sport hunting as an example, but results are
valid for other forms of anthropogenic mortality (fishing, collision, etc.). This appendix does
not provide details about the building of likelihood functions.
1) Reparameterization of Seber’s capture-recovery model
Seber’s capture-recovery model {Brownie, 1985 #40; Williams, 2002 #73}, denoted CR in
Table 1 of the main text, applies to datasets in which recoveries occur only from individuals
dead from the hunting. The parameters of the Seber model are the annual survival probability
𝑆 = 1 − 𝑛 − ℎ and Seber’s recovery probability 𝑟 =
ℎ
ℎ+𝑛
𝜆, where 𝜆 is the probability that a
dead marked individual is reported as such, given that it died from anthropogenic cause. An
alternative formulation is Brownie’s recovery probability 𝑓 = ℎ𝜆. Brownie’s parameterization
renders the computation of 𝑐𝑜𝑟𝑟(𝑛, ℎ) easier, but Seber’s parameterization makes the initial
capture-recovery model design and fitting more straightforward, especially since it provides a
better framework for examining the effects of covariates on S independently of the recovery
probability REF (Gauthier and Lebreton 2008).
To study compensation, one wants to extract the correlation between n the annual natural
mortality and h the annual hunting mortality. To do that, one needs to know 𝜆 the band
reporting rate. The expected value and temporal variance of this parameter were obtained
from external studies (reward bands, radio-transmitters; see Appendix S3).
The reparameterization is then performed via the delta-method which provides the following
system of five equations, where E, cov and Var stand for temporal expected value, temporal
covariance, and temporal variance:
Eq. S1
𝑐𝑜𝑣(𝑆, 𝑟) ≃ −
𝐸(𝜆)
2 (𝑐𝑜𝑣(𝑛, ℎ)(𝐸(𝑛) − 𝐸(ℎ)) + 𝑉𝑎𝑟(ℎ)𝐸(𝑛) − 𝑉𝑎𝑟(𝑛)𝐸(ℎ))
(𝐸(ℎ) + 𝐸(𝑛))
𝑉𝑎𝑟(𝑆) = 𝑉𝑎𝑟(𝑛) + 𝑉𝑎𝑟(ℎ) + 2𝑐𝑜𝑣(𝑛, ℎ)
2
𝐸(ℎ)
𝐸(𝜆)2
2
2
𝑉𝑎𝑟(𝑟) ≃ (
) 𝑉𝑎𝑟(𝜆) +
4 (𝐸(𝑛) 𝑉𝑎𝑟(ℎ) − 2𝑐𝑜𝑣(𝑛, ℎ)𝐸(ℎ)𝐸(𝑛) + 𝐸(ℎ) 𝑉𝑎𝑟(𝑛))
𝐸(ℎ) + 𝐸(𝑛)
(𝐸(ℎ) + 𝐸(𝑛))
𝐸(𝑆) = 1 − 𝐸(𝑛) − 𝐸(ℎ)
𝐸(ℎ)
𝐸(𝑟) ≃
𝐸(𝜆)
{
𝐸(ℎ) + 𝐸(𝑛)
From this system of equation it follows that (MAPLE output not simplified or factorized):
Eq. S2
Var(λ)Var(r)E(r)E(λ) − E(λ)3 cov(S, r) + Var(λ)E(λ)cov(S, r)E(S) − 2E(λ)2 E(r)cov(S, r)E(S) − 2E(r)Var(λ)cov(S, r)E(S)
2
𝑐𝑜𝑣(𝑛, ℎ) = −Var(S)E(λ) + Var(λ)E(r)2 + Var(λ)E(S)2 E(r)2 − 2Var(λ)E(S)E(r)2 − Var(λ)Var(r)E(r)2 − Var(r)E(λ)2E(r) + Var(r)E(λ)3 E(r) + E(λ)3 cov(S, r)E(S)⁄E(λ)3 (E(λ)3 + V
−Var(λ)E(λ)cov(S, r) + 2E(λ)2 E(r)cov(S, r) + 2E(r)Var(λ)cov(S, r) + 2Var(S)E(S)E(λ)2 − Var(S)E(S)2 E(λ)2
2
2E(λ)2 E(r)cov(S, r)E(S) + 2E(r)Var(λ)cov(S, r)E(S) + Var(S)E(λ)2 − Var(λ)E(r)2 − Var(λ)E(S)2 E(r)2 + 2Var(λ)E(S)E(r)2
𝑉𝑎𝑟(ℎ) =
+Var(λ)Var(r)E(r)2 + Var(r)E(λ)2 E(r)2 − 2E(λ)2 E(r)cov(S, r)
⁄E(λ)2 (E(λ)2 + Var(λ))
2
2
2
−2E(r)Var(λ)cov(S, r) − 2Var(S)E(S)E(λ) + Var(S)E(S) E(λ) )
−2Var(λ)Var(r)E(r)E(λ) + 2E(λ)3 cov(S, r) + Var(r)E(λ)4 − 2Var(λ)E(λ)cov(S, r)E(S) + 2E(λ)2 E(r)cov(S, r)E(S) + 2E(r)Var(λ)cov(S, r)E(S)
𝑉𝑎𝑟(𝑛) =
⁄E(λ)2 (E(λ)2 + V
+2Var(λ)E(S)E(r)2 + Var(S)E(λ)2 − Var(λ)E(r)2 − Var(λ)E(S)2 E(r)2 − 2Var(r)E(λ)3 E(r) + Var(λ)Var(r)E(r)2 + Var(λ)Var(r)E(λ)2
2
2
3
2
2
2
2
+Var(r)E(λ) E(r) − 2E(r)Var(λ)cov(S, r) − 2E(λ) cov(S, r)E(S) + 2Var(λ)E(λ)cov(S, r) − 2E(λ) E(r)cov(S, r) − 2Var(S)E(S)E(λ) + Var(S)E(S) E(λ)
𝐸(ℎ) = −E(r)(−1 + E(S))/E(λ)
{
𝐸(𝑛) = 1 − 𝐸(𝑆) + E(r)(−1 + E(S))/E(λ)
2) Capture-recovery model with independent measure of anthropogenic mortality
rate
This model (denoted CR-H in Table 1 of the main text) applies to datasets similar to those in
part 1, but for which in addition a measure of hunting pressure is available each year. It is a
variant of the CR model. The measure of hunting pressure, denoted H, can be the ratio
between total number of individuals harvested and total population size {Gauthier, 2001
#231}, or an independent assessment of the rate at which individuals are killed per unit of
hunting effort multiplied by a measure of that hunting effort {Baird, 2000 #241}. This method
also applies to known-fate data (radio tracked individuals), in which case H is the proportion
of individuals known to have died from hunting (e.g., {Creel, 2010 #233}. Then, the
relationship between hunting pressure and overall survival probability S can be implemented
as a built-in regression, which is generally assumed linear {Gauthier, 2001 #231; Barker,
1991 #1019; Creel, 2010 #233}:
Eq. S1
𝑆 = (1 − 𝑛0 )(1 − 𝑏 ∙ 𝐻 )
where n0 (one minus the intercept, mortality in the absence of hunting) and b (slope of the
regression) are parameters to be estimated. b is a direct measure of the level of compensation.
Complete additivity corresponds to b = 1 and complete compensation corresponds to b = 0.
The compensation-additivity rate can be shown to be 𝐶 = 1 − (1 − 𝑛0 ) ∙ 𝑏. The associated
correlation coefficient 𝑐𝑜𝑟𝑟(𝑆, 𝐻) can then be used to compute 𝑐𝑜𝑟𝑟(𝑛, ℎ) using:
Eq. S2
𝑐𝑜𝑟𝑟(𝑛, ℎ) ≃
−𝑐𝑜𝑟𝑟(𝑆, 𝐻)√𝑉𝑎𝑟(𝑆)𝑉𝑎𝑟(𝐻) − 𝑉𝑎𝑟(𝐻)
√𝑉𝑎𝑟(𝑆) + 𝑉𝑎𝑟(𝐻) + 2 ∙ 𝑐𝑜𝑟𝑟(𝑆, 𝐻)√𝑉𝑎𝑟(𝑆)𝑉𝑎𝑟(𝐻)√𝑉𝑎𝑟(𝐻)
As highlighted by {Lebreton, 2005 #74}, if H is an imprecise measure, this can lead to a
downwards bias in the absolute value of b (i.e., bias towards compensation). If H is assumed
to be very imprecise, one may want to use hunting effort only instead of H, e.g., the number
and activity level of hunters {Rolland, 2010 #272}, or even a proxy for hunting effort, e.g.,
the maximum number of individuals that hunters are legally allowed to harvest (bag limit;
Péron et al. submitted). Then, the result of the regression still informs the presence of
compensation or additivity, but generally cannot be used to obtain 𝑐𝑜𝑟𝑟(𝑛, ℎ).
3) Cause-specific capture-recovery model
This multistate model {Schaub, 2004 #364; Servanty, 2010 #104} applies to datasets where
the recoveries from hunting are supplemented by either live reencounters (observations of live
individuals) or recoveries from other mortality causes (e.g., road-kills, power-line
electrocutions, sick and injured individuals). The latter data often stem from reports by the
general public. For clarity they are called retrieval data hereafter. In the first case (reencounter
data), the model is noted CRP in Table 1 of the main text. In the second case (retrieval data)
the model is noted CRR in Table 1 of the main text.
Parameters n, h, and λ are as in previous cases the annual natural mortality, hunting mortality,
and reporting rate of hunter-killed individuals. The difference is that n and h are directly
estimable (not confounded with λ), and thus the correlation of n and h is directly
implemented. New sets of parameters are p the probability of detecting live individuals, and
𝜆𝑛 , which is the probability for a marked individual that died from causes other than hunting
to be reported as such.
Note that identifiability or estimation issues may arise when using this type of model
{Catchpole, 2001 #253}. The version of this model in which all parameters are time-constant
is not identifiable. {Schaub, 2004 #364} report that models where at least h and 𝜆𝑛 (or
conversely n and λ) are time-dependent are identifiable, but warn against the risk of flawed
results since the non-identifiable time-independent model is nested within the identifiable
time-dependent model.
4) Capture-recapture model with independent measure of hunting pressure
This model (denoted CP-H in Table 1 of the main text) applies to datasets in which only live
reencounters exist, but each year an independent measure of hunting pressure is available.
This type of dataset is more typical of by-catch exploitation {Rolland, 2010 #263; Veran,
2007 #247} than sport hunting, since it implies that harvested individuals are never reported
as such (λ = 0). The modeling framework is the standard capture-recapture model {Lebreton,
1992 #26}, in which the estimated survival corresponds to “apparent” survival, i.e., the
product of true survival and site-fidelity (individuals may be alive but have exited the
monitored area). The usual notation for apparent survival is φ. The other parameter set is
reencounter probability p. If one assume no interaction between anthropogenic mortality and
site-fidelity, the regression is the same as in part 2 above (replacing notation S by φ).