Supporting Information – Mugabo-et-al_density-2013
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Appendix C – Age-structured projection matrix model
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Model assumptions
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We used age-structured matrix population models with post-breeding census and birth pulse
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dynamics (Caswell 2001). We first considered a one-sex model (female based) with three age-
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classes (juveniles, yearlings and adults) and the projection matrix A
 S1  F1

A=  S1
 0

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S 2  F2
0
S2
S 3  F3 

0 ,
S 3 
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where S1 is the annual juvenile survival, S2 the annual yearling survival, S3 the annual adult
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survival, F1 the juvenile fecundity, F2 the yearling fecundity and F3 the adult fecundity. Fecundity
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in each age class was the product of the stage-specific proportion of breeders (γi), total clutch size
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(fi), hatching success (hsi), and primary sex ratio (sri; the proportion of female offspring at birth).
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Estimates of demographic rates were obtained from our experimental data except for the primary
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sex ratio which was assumed to be balanced (Table C1).
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Juvenile survival, proportion of breeders among juveniles and the total clutch size of
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juveniles, yearlings and adults were density-dependent. In each case, we used the best density-
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dependent function supported by our experimental data, i.e., f ( S1 ) 
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juvenile survival, f ( 1 ) 
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(logistic regressions) and f ( Fi )  exp( ai  bi n) for the total clutch size (log-linear regression, see
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the section Statistical analyses in the main text), where ai is the intercept (at n=0), bi the slope of
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density dependence and n the density level (n=N/N0 where N is the initial density and N0 the
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initial density in populations of density level 1; see Table C1). Density dependence of
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demographic rates is modeled as a function of total population density and is independent of age-
1
for
1  exp  (a1  b1n)
1
for the proportion of breeders among juveniles
1  exp  (a1  b1n)
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Supporting Information – Mugabo-et-al_density-2013
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structure. Starting conditions of the simulations were similar to the release conditions of our
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experiment.
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A two-sex version of the density-dependent deterministic model was developed to predict
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the density and age and sex-structures at equilibrium. The two-sex models assumed a male and
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female coupled life cycle and a polygynous mating system. In two-sex models, potential breeders
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of all age classes mated randomly. The number of clutches per generation was calculated
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according to the mating function M(Nm, Nf)=min(hNm, Nf) where Nm is the number of males, Nf
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the number of females and h the harem size (here, h=4 ). For the unrestricted harem size, the
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mating function M(Nm,Nf) is equal to Nf when Nm is > 0 and to 0 when Nm = 0. A harem size of
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four corresponds to a conservative estimate of the average number of females inseminated by a
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single breeding male (Fitze et al. 2005). In this case, the number of breeding male is limiting
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when it is lower than one quarter of the number of breeding females. An unrestricted harem size
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implies that a single breeding male can inseminate all breeding females from the same
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population. In this case, the number of males is limiting only when there is no breeding male.
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Elasticity analyses
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We carried out density-independent and density-dependent elasticity analyses of λ and Neq for
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each demographic parameter and slope of density dependence (i.e., elasticities of lower-level
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parameters and not of matrix elements, Caswell 2001). Elasticities of λ and elasticities of Neq
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predict the proportional change in λ and in Neq respectively, given a small proportional change in
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a parameter of the model while all other parameters remain constant (Caswell 2001).
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Computations were carried out using ULM (Legendre & Clobert 1995). Elasticities of Neq are
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proportional to elasticities of λ if (i) the population has a stable equilibrium, (ii) density
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dependence is a function of the total number of individuals within the population, and (iii)
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density dependence operates in the same way in all age classes (Grant & Benton 2000). Here,
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only the first two conditions were met.
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Supporting Information – Mugabo-et-al_density-2013
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Demographic stochasticity
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The consequences of population density for the extinction risk were investigated using stochastic
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one-sex and two-sex models. We performed individual-based simulations by incorporating
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demographic stochasticity on all demographic events. Survival probabilities, breeding
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probabilities, mating probabilities, sex at birth and the number of mating events were drawn from
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binomial distributions while the number of offspring produced was drawn from Poisson
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distributions.
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References
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Caswell, H. (2001) Matrix population models, 2 edn. Sinauer Associates, Sunderland.
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Fitze, P.S., Le Galliard, J.-F., Federici, P., Richard, M. & Clobert, J. (2005) Conflict over
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multiple-partner mating between males and females of the polygynandrous common
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lizards. Evolution, 59, 2451-2459.
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60
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Grant, A. & Benton, T.G. (2000) Elasticity analysis for density-dependent populations in
stochastic environments. Ecology, 81, 680-693.
Legendre, S. & Clobert, J. (1995) ULM, a software for conservation and evolutionary biologists.
Journal of Applied Statistics, 22, 817-834.
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Supporting Information – Mugabo-et-al_density-2013
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Table C1. Parameter estimates for density-dependent and density-independent one-sex and two-sex models. Estimates of the proportion of
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breeders and hatching success are given on the logit scale f ( xi ) 
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given on the log scale f ( xi )  exp( xi ) (log-linear regression). In the density-independent model, estimates were calculated by pooling data from
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all enclosures. In the density-dependent model, xi ( n)  ai  bi n , where ai is the intercept (at n=0), bi the slope of the density dependence and n
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the density level. Estimates in bold are for females, estimates in italic are for males and other estimates are for both sexes.
Demographic rates ( xi )
1
(logistic regression) and estimates of the total clutch size are
1  exp(  xi )
Density-independent model
One-sex model
Two-sex model
Annual juvenile survival (S1)
0.400
0.400, 0.303
Annual yearling survival (S2)
Annual adult survival (S3)
0.527
0.281
0.527, 0.191
0.281, 0.164
Density-dependent model
One-sex model
Two-sex model
-0.915 + 0.204 × n
-0.915 + 0.204 × n
-1.353 + 0.204 × n
0.527
0.527, 0.191
0.281
0.281, 0.164
Juvenile fecundity (F1)
Percentage of breeders (γ1)
Total clutch size (f1)
Hatching success (hs1)
-0.393
1.06
0.836
2.120 - 0.986 × n
1.370 - 0.133 × n
0.836
Yearling and adult fecundity (F2-3)
Percentage of breeders (γ2-3)
Total clutch size (f2-3)
Hatching success (hs2-3)
2.290
f2 = 1.915, f3 = 1.923
hs2 = 0.846, hs3 = 0.914
2.290
f2 = 2.281 - 0.133 × n, f3 = 2.310 - 0.133 × n
hs2 = 0.846, hs3 = 0.914
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