Marie Nevoux et al.
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Electronic supplementary material ESM 2. Modelling the influence of density-
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dependence on population size trajectories.
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Pre-breeding individuals (or floaters) constitute about 40% of the total population. Hence,
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we built a pre-breeding census matrix population model with tree stages (non-breeders,
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one year old breeders, two years old and older breeders), such as:
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½.F2.Sj.Rb1
nb1
Sa
nb2
Sa
½.F1.Sj.Rb1
½.F2.Sj.(1-Rb1)
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½.F1.Sj.(1-Rb1)
Sa
npb
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At time t, one year old breeders (b1) and older ones (b2) produce some offspring, with
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respective breeding success F1 and F2, that survive to t+1 with the probability Sj and
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either recruit into the breeding population at one year old (b1) with the probability Rb1 or
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enter the pre-breeding stage (pb) with the probability (1-Rb1). Breeders and pre-breeders
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survive from t to t+1 with the probability Sa and pre-breeders recruit into the breeding
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population at age two with the probability 1 (86.4% of the birds are observed to breed by
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age 2, thus we ignore the small proportion of older non-breeders). In Butler et al. (2009),
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recruitment to the breeding population was constrained by a carrying capacity threshold.
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Because we were interested in the intrinsic regulatory “power” of the demographic traits,
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Marie Nevoux et al.
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we did not restrict population size by an arbitrary limit in this model. Thus, Rb1 is defined
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by the observed proportion of individuals starting to breed at age 1. The population size
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at time t+1 is defined by N(t+1) = A.N(t), with N(t) representing the population vector at
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time t and A the population matrix; A corresponding to one of four models.
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0
A(t) =  0

Sa
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First, we ran the model M[F(N), Sj(N)] with both density-dependent breeding success and
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juvenile survival (i.e. all breeding success and survival terms dependent on population
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size) until stabilisation of the population size (i.e. N(t+1) = N(t). Using the number of
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females in each stage at equilibrium (npb(e), nb1(e) and nb2(e)) as a starting point, we
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2
.F1.Sj.1  Rb1 
1
2
.F1.Sj.Rb1
Sa
1
2
.F 2.Sj.1  Rb1 

1 .F 2.Sj.R
2
b1

 t
Sa
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investigated the effect of density-dependence on population regulation by progressively
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replacing density-dependent terms by constant ones. We compared the population size
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trajectories of the initial model (M[F(N), Sj(N)]) to a model with density-dependent
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breeding success and constant juvenile survival (M[F(N), Sj(.)]), a model with density-
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dependent juvenile survival and constant breeding success (M[F(.), Sj(N)]) and a density-
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independent model (M[F(.), Sj(.)]). Density-dependent breeding success (F(N)) and juvenile
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survival (Sj(N)) are defined as functions of the number of adults in the population to
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describe the influence of density-dependent process on the population dynamics, as
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described in the results. Density-dependent survival is modelled according to our best
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model which is coded with with a plateau and a linear decline above a population size of
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25 pairs (p(a) S(a1 * (plateau(<25), N(>25)) + a2), model 10 in ESM 3); the linear model of
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density-dependent survival (p(a) S(a1 * N), model 9 in ESM 3) generated very similar
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population size trajectories (results not presented).
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Marie Nevoux et al.
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Parameterisation of the matrix population models. All estimates and relationships are
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derived from this study (data collected between 1987/88 and 2007/08), or from Butler et
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al. (2009) when specified (*).
Parameter Description
npb(0)
nb1(0)
nb2(0)
npb(e)
nb1(e)
nb2(e)
nb1(t)
nb2(t)
F1(.)
F2(.)
Sj(.)
F1(N)
F2(N)
Value
Initial number of pre-breeding females† 0*
Initial number of breeding females,
aged one†
Initial number of breeding females,
older than one†
Number of pre-breeding females at
equilibrium‡
Initial number of breeding females,
aged one‡
initial number of breeding females,
older than one‡
Time-dependent number of breeding
females, aged one
Time-dependent number of breeding
females, older than one
Constant breeding success in stage 1
(number of fledglings)
Constant breeding success in stage 2
(number of fledglings)
Constant juvenile survival
Density-dependent breeding success in
stage 1
Density-dependent breeding success in
stage 2
4*
2*
4.0, as resulting from model M[F(N), Sj(N)]
7.3, as resulting from model M[F(N), Sj(N)]
36.1, as resulting from model M[F(N), Sj(N)]
as estimated by the model
as estimated by the model
0.855
1.484
0.500
exp(-0.231 + 0.605 - 0.015 * (nb1(t) + nb2(t)))
exp(-0.231 + 2 * 0.605 - 0.015 * (nb1(t) + nb2(t)))
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Marie Nevoux et al.
((nb1(t) + nb2(t)) < 25) * 0.663 + ((nb1(t) + nb2(t)) > 25) *
Sj(N)
Density-dependent juvenile survival
Sa
Adult survival
0.762
Rb1
Recruitment at age 1
0.645
logit-1(2.130 - 0.058 * (nb1(t) + nb2(t)))
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†
used in model M[F(N), Sj(N)].
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‡
used as initial number in models M[F(N), Sj(.)], M[F(.), Sj(N)] and M[F(.), Sj(.)].
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Reference
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Butler, S.J., Benton, T.G., Nicoll, M.A., Jones, C.G. & Norris, K. (2009). Indirect
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population dynamic benefits of altered life-history trade-offs in response to egg
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harvesting. Am. Nat., 174, 111-121.
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