Functional Ecology
Appendix S2: Implementation of demographic rates in the models
1. Fecundity rates
Fecundity in Centaurea corymbosa has been the subject of detailed studies (Colas,
Olivieri & Riba 2001, Hardy et al. 2004a,b, Kirchner et al. 2005). From these studies it
appeared that the amount of seeds produced by any individual depended mainly on the
number of capitula produced by this individual while flowering. Thus, to implement realistic
variation in fecundity among years (environmental stochasticity) and among individuals
(demographic stochasticity), we looked for the best distributions describing the variation in
the number of capitula among years and individuals, as detailed below.
We fixed the number of ovules per capitulum at 43, the mean number observed in the
field (Table A1). We considered that each ovule had a probability φ to be fertilized and that
each fertilized ovule had a probability β to abort (Table A1). Then, we calibrated the
probability of a sound seed to become a seedling (the expected emergence rate) at ε=0.046,
which gave an asymptotic growth rate, with the life cycle presented in Fig. 1 of the main text
and a fixed generation time, just above unity: λ=1.054. Such an asymptotic growth rate
allowed stochastic populations to become extinct, which is a necessary condition for studying
extinction risk. It may be noted that, although little data from demographic quadrats are
available to allow accurate estimates of the emergence rate in C. corymbosa, the implemented
value seems very realistic. In summary, we implemented 43 ovules in each capitulum and
each ovule had a probability to become a seedling of 0.0253 (i.e. the following product: φ*(1β)*ε).
a. Variation in the mean number of capitula among years.
Because data on the number of capitula were available over over only three years, we could
not directly observe the distribution of the mean number of capitula among years. Rather, we
inferred this distribution by combining data on the rosette size of individuals just before
bolting (in March) over twelve years and data on the relationship between rosette size and the
number of capitula.
The mean rosette diameter of individuals before bolting did not deviate significantly
from normality over twelve years (Liliefors test: P = 0.3567), so that we considered this
distribution as normal although the sample was small.
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Heterogeneity of reproductive age increases the viability of semelparous populations
P. Acker, A. Robert, R. Bourget and B. Colas
Functional Ecology
We looked for the best function describing the relationship between the number of
capitula produced by a flowering plant and its rosette diameter just before bolting. A linear
regression gave a slope of 4.50 and an intercept of -29.26 (R² = 0.3951, P < 2.2 10-16), which
was biologically inconsistent because negative numbers of capitula are not possible. An
exponential regression using the method of least squares (nls function of R software),
according to the formula y = b * eax, fitted much better better with the data (b = 4.39, a =
0.14, R² = 0.4140, ΔAIC=10.56 in the benefit of the exponential model).
After transforming the normal distribution of the mean rosette diameter by the
exponential relationship described above, we obtained a log-normal distribution, according to
the properties of the log-normal law. Thus, to simulate the variation among years of the
expected mean number of capitula per flowering plant in a realistic way, we used a lognormal distribution with a mean of 26 capitula (γ in Table A1 of Appendix 1). The standard
deviation of this distribution varied according to the strength of environmental stochasticity
implemented in the models.
b. Variation in the number of capitula among individuals within years.
To describe the variation in the number of capitula among individuals, the log-normal
distribution was chosen according to its resemblance to the observed distributions which are
skewed to the right (according to Liliefors normality test applied on log-transformed data: Pvalue = 0.1699 in 1994, 0.0035 in 1995 and 0.3749 in 1996). Its standard deviation was
calculated as the average of the estimated standard deviations in 1994, 1995 and 1996 (23).
2. Variation in age at flowering and survival rates.
a. Variation in age at flowering
The probabilities pk of entering a life path (see Fig. 1 in the main text), which determined the
age at flowering k, followed distributions with different levels of variation, built with the
probability densities of normal distributions with a mean μ = 5 years old and with different
standard deviations σ. This allowed studying the effect of variation in age at reproduction. For
each age at flowering k (3 ≤ k ≤ 7), we took the probability densities nk of a normal
distribution at the point k:
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Heterogeneity of reproductive age increases the viability of semelparous populations
P. Acker, A. Robert, R. Bourget and B. Colas
Functional Ecology
𝑛𝑘 =
1
1 𝑘−5 2
)
𝜎
𝜎√2𝜋
𝑒 −2 (
divided by the sum of probability densities of this normal distribution of each of the possible
ages at flowering k:
𝑝𝑘 =
𝑛𝑘
7
∑𝑘=3 𝑛𝑘
.
We then obtained a discrete distribution of all the possible ages at flowering k, but this
discrete distribution had a different standard deviation from the continuous normal law used
to construct it. We calculated the standard deviation s of the discrete distribution as:
7
𝑠 = √∑
𝑘=3
𝑝𝑘 (𝜇 − 𝑘)2 .
We built 6 distributions according to this method with standard deviations σ of 0.320,
0.375, 0.500, 0.750, 1.000 and 1.500 for the normal distributions, which led to discrete
distributions of age at flowering with standard deviations s of 0.121, 0.232, 0.464, 0.748,
0.961 and 1.195, respectively.
A last distribution was built following probability densities of a discrete uniform law. Since
there were 5 possible ages at flowering, the standard deviations of this distribution was:
𝑠=√
(5 − 1)(5 + 1)
= √2 = 1.414 .
12
The eight distributions of pk used in the simulations are presented graphically in Fig.
A1.
Figure 1. Distributions of the probabilities pk of entering a life path (defined by the age at
reproduction k) for each standard deviation s considered. For s=0 there is no demographic
heterogeneity and demographic heterogeneity increases with s.
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Heterogeneity of reproductive age increases the viability of semelparous populations
P. Acker, A. Robert, R. Bourget and B. Colas
Functional Ecology
b. Survival rates
In the models simulating populations with a constrained generation time, we fixed the
flowering expectancy of two-year-old rosettes (i.e. their probability to flower before flowering
in the future) at 𝑝𝑓|2 = 𝑠2 ∗ 𝑠3 ∗ 𝑠4 = 0.287. In these models, implemented survival rates
were calculated as explained in the Material and Methods of the main text, which allowed
keeping an identical mean generation time (5 years) in every population whatever the
implemented level of demographic heterogeneity.
In the models simulating populations with an unconstrained generation time,
populations with no demographic heterogeneity had the same flowering expectancy 𝑝𝑓|2 =
0.287. However, flowering expectancy increased with the level of demographic heterogeneity
in age at flowering in populations with an unconstrained generation time as explained in the
Material and Methods in the main text.
The values of implemented survival rates in both models are given Table A2.
Table 1. Survival rates implemented used in the models with unconstrained and constrained
generation times, according to life paths with a pre-determined age at flowering k.
Survival rates
(3 ≤ k ≤ 7)
Unconstrained
generation time
Constrained generation time
∀𝑘, 3 ≤ k ≤ 7
k=3
k=4
k=5
k=6
k=7
s0
0.480
0.480
0.480
0.480
0.480
0.480
s1
0.343
0.343
0.343
0.343
0.343
0.343
s2,k
0.624
0.287
0.536
0.660
0.732
0.779
s3,k
0.668
-
0.536
0.660
0.732
0.779
s4,k
0.689
-
-
0.660
0.732
0.779
s5,k
0.689
-
-
-
0.732
0.779
s6,k
0.689
-
-
-
-
0.779
References.
Colas, B., Olivieri, I. & Riba, M. (2001) Spatio-temporal variation of reproductive
success and conservation of the narrow-endemic Centaurea corymbosa (Asteraceae).
Biological Conservation, 99, 375–386.
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Heterogeneity of reproductive age increases the viability of semelparous populations
P. Acker, A. Robert, R. Bourget and B. Colas
Functional Ecology
Hardy, O.J., González-Martínez, S.C., Fréville, H., Bocquien, G., Mignot, A., Colas, B.
& Olivieri, I. (2004a) Fine-scale genetic structure and gene dispersal in Centaurea
corymbosa (Asteraceae) I. Pattern of pollen dispersal. Journal of Evolutionary Biology,
17, 795-806.
Hardy, O.J., González-Martínez, S.C., Colas, B., Fréville, Mignot, A. & Olivieri, I.
(2004b) Fine-scale genetic structure and gene dispersal in Centaurea corymbosa
(Asteraceae) II. Correlated paternity within and among sibships. Genetics, 168, 16011614.
Kirchner, F., Luijten, S.H., Imbert, E., Riba, M., Mayol, M., González-Martínez, S.C. &
Colas, B. (2005) Effects of local density on insect visitation and fertilization success in
the narrow-endemic Centaurea corymbosa (Asteraceae). Oikos, 111, 130–142.
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Heterogeneity of reproductive age increases the viability of semelparous populations
P. Acker, A. Robert, R. Bourget and B. Colas