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Plant Populations Track rather than Buffer Climate Fluctuations: Supporting Information
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Plant Populations Track rather than Buffer Climate Fluctuations
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Appendix S1: Demography data set and simulation methods
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We used published demographic data from 40 species, ranging from short-lived to long-
5
lived perennial plants (Table S1). We chose to use only one population per species and to
6
focus on the temporal variation in those populations. For each species we preferentially
7
selected an overall moderately growing population, because not all vital rates may be
8
well represented in declining populations (Silvertown et al. 1996).
9
List of the species used in the analyses. The name of the population used is given
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between brackets for those species for which the cited literature contains data on multiple
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populations. Years = number of annual transitions and dates of years for which data are
12
available. Dim. = matrix dimension (e.g. 6 means a 6x6 matrix). R-S-G = number of
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Reproduction, Survival and Growth rates that vary among years. L.S. = conditional total
14
life span and clonal conditional total life span between brackets (see Life Span section
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below). λ range = the minimum and maximum population growth rate, λ, in the set of
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matrices. V/2 is the total contribution of the variation in all vital rates to the stochastic
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population growth rate, log λs. V   ek el ck cl Corr(k , l ) . H is the sum of the covariance
k ,l
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contributions to V; H   ek el ck cl Corr(k, l ) . The matrices of Anthyllis vulneraria were
k l
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kindly provided by Dries Adriaens (unpublished data). Maria B. García and Olav
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Skarpaas kindly provided additional information about the transition matrices for
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Petrocoptis pseudoviscosa and Mertensia maritima, respectively.
Plant Populations Track rather than Buffer Climate Fluctuations: Supporting Information
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Dim.
R-S-G
Life span
λ range
V
H
6: 92-98
6
1-6-8
18.5
0.78-1.09
0.014
0.01022
Agrimonia eupatorium (B) (Kiviniemi 2002)
5: 93-98
4
1-4-3
76.5
0.96-1.14
0.00587
Allium tricoccum (Nault & Gagnon 1993)
4: 84-88
12.4 (∞)
1.03-1.13
Alnus incana (Divide Swamp) (Huenneke & Marks 1987)
3: 79-82
5
4-5-4
Anthyllis vulneraria (O) (Adriaens et al. Unpubl. Data)
3: 03-06
4
2-4-7
3.5
0.80-1.53
Arabis fecunda (control) (Lesica & Shelly 1996)
4: 87-91
4
4-4-10
5.1
0.30-1.63
Ardisia elliptica (Al-Khafaji et al. 2007)
3: 99-02
8
3-6-10
244.0
1.03-1.10
21: 78-00
5
1-5-11
15.0
0.68-1.43
Astragalus scaphoides (Haynes) (Lesica 1995)
5: 87-92
4
3-4-12
14.4
0.83-1.60
Astragalus tyghensis (10) (Kaye & Pyke 2003)
9: 91-00
5
4-2-13
15.6
0.78-1.30
Borderea chouardii (García 2003)
7: 95-02
6
3-5-12
129.8
0.98-1.02
Calathea ovandensis (2) (Horvitz & Schemske 1995)
4: 82-86
8
6-8-26
23.6
0.90-1.25
Centaurea jacea (N) (Jongejans & de Kroon 2005)
4: 99-03
6
6-3-15
5.9 (22.3)
0.87-1.06
Cimicifuga elata (Eugrass) (Kaye & Pyke 2003)
4: 92-96
5
1-5-11
23.5
0.89-1.67
Cirsium dissectum (B) (Jongejans et al. 2008)
4: 99-03
6
6-3-11
5.7 (∞)
1.06-1.26
Cryptantha flava (control) (Lucas et al. 2008)
3: 97-00
7
6-7-27
11.7
0.65-1.06
Fumana procumbens (Bengtsson 1993)
6: 85-91
6
2-6-15
15.1
0.86-1.09
Geum rivale (D) (Kiviniemi 2002)
3: 93-96
5
1-5-8
36.3
0.94-1.04
Haplopappus radiatus (1out) (Kaye & Pyke 2003)
9: 91-00
4
1-4-7
11.1
0.79-1.19
Harrisia fragrans (1) (Rae & Ebert 2002)
5: 88-93
6
4-6-14
30.5
0.83-1.00
0.008
84
0.001
93
0.000
55
0.120
61
0.445
55
0.000
20
0.050
10
0.106
61
0.024
70
0.000
58
0.020
16
0.003
21
0.082
93
0.004
69
0.065
71
0.008
43
0.006
73
0.052
00
0.008
73
40 Species (population)
Years
Actaea spicata (B) (Fröborg & Eriksson 2003)
Asplenium scolopendrium (P/F) (Bremer & Jongejans 2010)
15 17-11-94
29.9 (1150) 0.97-1.00
23
0.00050
-0.00204
0.03965
0.33364
-0.00125
0.03441
0.06321
0.01571
0.00001
0.01178
-0.01475
0.04416
-0.00071
0.05580
-0.00303
0.00269
0.01756
0.00543
Plant Populations Track rather than Buffer Climate Fluctuations: Supporting Information
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Hudsonia montana (Morris & Doak 2002)
4: 85-89
6
8-6-10
39.7
0.85-1.02
0.005
0.00409
Hypericum cumulicola (29) (Quintana-Ascencio et al. 2003)
5: 94-99
6
3-6-18
12.9
0.94-1.58
0.02981
Hypochaeris radicata (N) (Jongejans & de Kroon 2005)
4: 99-03
6
4-4-9
7.8 (8.3)
0.59-1.05
Kosteletzkya pentacarpos (Pino et al. 2007)
8: 96-04
3
2-3-3
92.1
1.00-1.27
Lathyrus vernus (L) (Ehrlén 1995)
3: 88-91
7
2-6-14
79.7
0.99-1.10
Lomatium bradshawii (FB0) (Kaye & Pyke 2003)
7: 88-97
5
2-5-14
13.1
0.74-1.47
Lomatium cookie (Middle) (Kaye & Pyke 2003)
5: 94-99
6
2-6-22
18.0
0.89-1.34
Mertensia maritima (Malmøya) (Skarpaas 2003)
4: 97-01
5
3-5-22
12.5
0.76-1.40
Mimulus cardinalis (Carlon) (Angert 2006)
3: 00-03
4
3-4-8
9.9
1.03-1.14
Mimulus lewisii (ML) (Angert 2006)
3: 00-03
4
3-4-8
24.6
0.93-1.05
Opuntia rastrera (nopalera) (Mandujano et al. 2001)
7: 91-98
10
9-6-55
74.5
0.94-0.98
Petrocoptis pseudoviscosa (Abi) (García 2008)
5: 95-00
6
4-5-13
66.0
0.98-1.04
Plantago media (B) (Eriksson & Eriksson 2000)
4: 93-97
5
1-5-6
39.2
0.94-1.00
Potentilla anserina (Eriksson 1988)
3: 81-84
6
6-6-15
10.3 (21.7)
0.68-1.08
Primula farinosa (B) (Lindborg & Ehrlén 2002)
3: 96-99
4
1-4-7
18.6
0.99-1.14
Primula veris (22) (Endels 2004)
3: 99-02
6
3-5-17
18.0
0.90-1.23
Primula vulgaris (Cycl1) (Endels et al. 2007)
3: 99-02
6
3-6-18
8.8
1.13-1.49
Ramonda myconi (Ingla2) (Picó & Riba 2002)
5: 92-97
5
3-2-12
53.7
0.97-1.04
Saxifraga cotyledon (large) (Dinnétz & Nilsson 2002)
4: 92-96
7
8-7-23
13.2 (304)
1.17-1.36
Succisa pratensis (B) (Jongejans & de Kroon 2005)
4: 99-03
6
4-3-8
36.1 (118)
1.23-1.33
0.103
17
0.066
21
0.004
32
0.002
68
0.081
79
0.034
70
0.081
21
0.001
51
0.003
59
0.000
73
0.000
72
0.003
43
0.046
22
0.007
23
0.016
09
0.012
57
0.001
30
0.004
77
0.000
83
52
0.05022
-0.00956
0.00109
0.05954
0.01972
0.06093
-0.05097
0.00228
0.00041
-0.00013
0.00058
0.02951
-0.00251
0.00492
-0.00710
0.00048
-0.00756
-0.00238
Plant Populations Track rather than Buffer Climate Fluctuations: Supporting Information
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The projection matrix models in these studies have the following form: n(t + 1) =
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An(t), where n(t) and n(t + 1) are vectors whose elements, ni, are the number of
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individuals that belong to the i-th category at time t and t + 1. The elements of matrix A,
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aij, give the contributions of the number of individuals in the j-th category (column) to the
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number of individuals in the i-th category (row) one year later. The dominant eigenvalue
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of A is the growth rate (λ) with which the population will increase (or decrease if λ<1)
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once the distribution of the individuals over the categories has stabilised and provided
29
that environmental conditions do not change through time (Caswell 2001).
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Matrix properties are traditionally analysed at the level of matrix elements, but
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analyses at the level of the underlying vital rates are becoming more common, as these
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vital rates (e.g. survival, growth and reproduction rates) are more basic and biologically
33
more interesting (Franco & Silvertown 2004; Morris & Doak 2004). In this paper we
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therefore use vital rates, τk, in all our analyses. A vital rate can occur in the functions of
35
several matrix elements, and a matrix element can be composed of multiple vital rates. As
36
an example we show the transition matrix of the Pyrenean violet (Ramonda myconi),
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which has five stages (seedlings, vegetative adults and small, medium and large
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reproductive adults) (Picó & Riba 2002):
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2 12
3
4
5
 1(1   21 )



2 (1   32  12 )
3  23
 4  24
5  25
 1 21



0
2  32
3 (1   53   43   23 )
 4  34
5  35


0
0
3  43
 4 (1   54   34   24 )
5  45




0
0





(1






)
3 53
4 54
5
45
35
25 

40
(S1)
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in which σj is the survival rate of individuals in stage j, φj the reproduction rate of
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individuals in stage j, and γij the growth (positive or negative) rate of surviving
Plant Populations Track rather than Buffer Climate Fluctuations: Supporting Information
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individuals from j to i. In all matrices we modelled stasis within a class as one minus the
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growth rates away from that class. The matrices of eight species (Allium tricoccum, Alnus
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incana, Centaurea jacea, Cirsium dissectum, Hypochaeris radicata, Potentilla anserina,
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Saxifraga cotyledon and Succisa pratensis) in the database also contained clonal
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propagation rates. In the analyses, clonal propagation and sexual reproduction rates are
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grouped as reproduction rates.
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Life span
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We characterized each species by its conditional total life span (Cochran & Ellner 1992).
51
This is the expected life span of a new individual provided that that individual reaches a
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certain life stage at least once. We used the maximum of only those conditional life span
53
estimates that are biologically possible according to the reproduction and survival rates in
54
the matrix models: new individuals start in stages to which sexual and clonal
55
reproduction contribute and can go through stages into which those individuals can grow
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(conditional on survival). We corrected the life span estimator by subtracting 1 as
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suggested by Ehrlén and Lehtilä (2002). The life span calculations were performed on the
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mean matrices, which were constructed for each species with the means of each vital rate
59
over the observed years. For the species with explicit clonal propagation rates we also
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calculated a clonal conditional total life span by treating the clonal propagation rates as
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additional survival rates. These clonal life spans (between brackets in Table S1) were
62
higher than the reported non-clonal life span. In two cases (Allium tricoccum and Cirsium
63
dissectum) the clonal life span was projected to be infinity (∞) because the
64
survival+clonality matrices of those populations had a dominant eigenvalue larger than 1.
Plant Populations Track rather than Buffer Climate Fluctuations: Supporting Information
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In Figure 2c we corrected the correlation contributions of the separate species by
66
multiplying the correlation contributions by the conditional total life span squared. This
67
was done because V is inversely proportional to the square of generation time (Orzack &
68
Tuljapurkar 1989), and we expect conditional total life span to highly correlate with
69
generation time.
70
Elasticity values and Coefficients of Variation
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The main vital rate characteristics used in this paper are the coefficient of variation (CV
72
or c) and the elasticity value (e) of a vital rate. The former is calculated by the ratio of the
73
standard deviation and the mean of a vital rate k:
74
ck 
sd( k )
k
(S2)
75
The elasticity value (de Kroon et al. 2000) of τk is the proportional change in the
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population growth rate (λ) in response to small, proportional change in τk:
77
ek 
k 
 k
(S3)
78
In all analyses of relationships between vital rate characteristics we only used vital rates
79
with non-zero ck and ek values. This was done because such vital rates indicate that either
80
their variance was not measured or that these vital rates were irrelevant for the population
81
that we selected from the original article. Inclusion of vital rates with artificially zero ck
82
or ek values would distort the studied ck-ek and Vk-ek relationships.
Plant Populations Track rather than Buffer Climate Fluctuations: Supporting Information
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83
Simulations with mean–dependent random variation
84
To investigate whether vital rate correlations, e-c relationships and changes in the
85
variance contributions Vk are significantly different from what would be expected from a
86
random null model, we ran computer simulations in which vital rates were randomly
87
picked from statistical distributions around their observed means. The variances of these
88
distributions were determined by the overall relationship between the vital rate means and
89
their variances across all species in the original dataset. Therefore, we first studied the
90
mean-variance relationships in the data. We did this for the reproduction, survival and
91
growth rates separately. The variances and means of the reproduction rates (n=149) were
92
analysed with a linear regression model on a log10-log10 scale. Since the variance of
93
survival (n=199) and growth (n=613) rates approaches zero when these vital rates
94
approach their limits at 0 and 1, we fitted hyperbolic regression models to the variance
95
data that were zero at mean-values of 0 and 1. The resulting regression models are shown
96
here:
a) Reproduction rates
b) Survival rates
y= -0.45 + 1.969 x
y= 0.156 (x-x^2)
0
y= 0.186 (x-x^2)
0.30
0.30
0.25
0.25
Variance
Variance
5
log Variance
c) Growth rates
0.20
0.15
0.20
0.15
0.10
0.10
0.05
0.05
0.00
0.00
-5
-2
0
2
log Mean
97
4
0.0
0.2
0.4
0.6
Mean
0.8
1.0
0.0
0.2
0.4
0.6
Mean
0.8
1.0
Plant Populations Track rather than Buffer Climate Fluctuations: Supporting Information
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98
Fig. S1.1: Relationships between the means and variances of the vital rates of 40
99
plant species, separately for reproduction rates (log10-log10 scale, n=149, R2adj= 0.935),
100
survival rates (n=199, R2adj=0.397) and growth rates (n=613, R2adj=0.415). In the case of
101
growth rates the quadratic model fitted equally well as a linear model also forced through
102
the origin (R2adj=0.42), but better than quadratic or linear models including an intercept
103
(both R2adj=0.27). However, for simulation purposes we preferred the quadratic function
104
because it more realistically predicts low variances for the few growth rates with high
105
means. In the case of survival rates the quadratic model fitted than a linear model, forced
106
through the origin (R2adj=0.14) or not (R2adj=0.09). The blue dashed line indicates the cut-
107
off point (the reproduction value (mean=2.914) above which the regression model
108
predicts that the variances of the reproduction rates will be larger than their means)
109
between low and high reproduction rates for which we used gamma and negative
110
binomial distributions in the simulation (see details below).
111
Each simulation contained the same species and the same number of annual
112
transitions per species as the original data set, while the variances and covariances were
113
random. The vital rates that did not vary in the real dataset were kept constant in the
114
constructed datasets. The random vital rates were drawn from beta distributions (survival
115
and growth rates), gamma distributions (low reproduction rates) or negative binomial
116
distributions (high reproduction rates). The distinction between low and high
117
reproduction rates was made because it can be problematic to realistically generate
118
random variation for both high and low reproduction rates with the same function
119
because high reproduction rates can be more overdispersed. Furthermore, we wanted to
120
generate discrete random values (Ver Hoef & Boveng 2007) for high reproduction rates
Plant Populations Track rather than Buffer Climate Fluctuations: Supporting Information
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121
because high reproduction rates often represent e.g. the number of seeds produces,
122
whereas low reproduction rates more often represent continuous ratios like the average
123
number of seedlings next year per flowering plant this year. The use of different
124
statistical distributions for low and high reproduction rates did not generate a disjunct
125
simulated mean-varaince relationship as can be seen in Fig. S1.2 below. These choices
126
for simulating reproduction rates had no consequences for our results: alternative model
127
formulations with log-normal distributions and without distinction between low and high
128
rates resulted in the very similar patterns and conclusions (see Appendix S2). We used
129
the value (2.914) above which the regression model (see Fig. S1.1) predicted higher
130
variances than the vital rate means as the cut-off point between low and high mean
131
reproduction values. For each individual vital rate, the statistical distribution, from which
132
random values were drawn, was shaped by the mean of that vital rate in the original data
133
set and by the variance that is predicted by the regression models for that vital rate mean
134
(see figure on previous page). For each newly constructed matrix we also checked the
135
sum of the different growth rates of individuals in size class j:

ij
. If this sum was
i
136
larger than 1 (which is biologically impossible) for a particular matrix column in a new
137
dataset, we drew new random numbers for all involved growth and retrogression rates.
138
This procedure was repeated until they no longer summed to more than one.
139
To illustrate that this random vital rate generation method resulted in very similar
140
vital rate distributions as in the actual data, we here show (in the same format as the data-
141
based Fig. S1.1 above) the vital rate means and variances generated in a single, randomly
142
selected simulation (Fig. S1.2):
Plant Populations Track rather than Buffer Climate Fluctuations: Supporting Information
a) Reproduction rates
b) Survival rates
y= -0.61 + 1.975 x
0
y= 0.19 (x-x^2)
0.30
0.30
0.25
0.25
0.20
0.20
Variance
Variance
log Variance
c) Growth rates
y= 0.165 (x-x^2)
5
10/16
0.15
0.15
0.10
0.10
0.05
0.05
0.00
0.00
-5
-2
0
2
4
log Mean
0.0
0.2
0.4
0.6
Mean
0.8
1.0
0.0
0.2
0.4
0.6
0.8
Mean
143
144
We performed 10,000 independent simulations. The random redistribution of the
145
variance in the vital rates was successful in the simulations as can be seen from the series
146
of statistics in Figure S1.3. The range of variance-mean regression slopes in the
147
simulations encompassed the slopes in the analyses of the data (see Repr.Slope,
148
Surv.Slope and Grow.Slope on the next page). The scatter around the regression models
149
is, however, underrepresented in the simulations: the sums of squares of the residuals are
150
larger in the data. This might indicate that variation in vital rate variances cannot be
151
completely explained by vital rate means, but that environmental or intrinsic factors have
152
additional explanatory power. Also, the intercept of the reproduction rates regression
153
model is lower in the simulations than in the data, leading to a higher value at which the
154
regression model predicts the variance to be equal to the mean (i.e. the vertical blue
155
dotted line in Fig. S1a). However, these relative differences in the simulations did not
156
lead to important differences in the average vital rate variances: as can be seen in Figure
157
3 in the main text, the simulation regression lines occur at similar levels of ck and Vk as
158
the data regression line. This suggests that simulations have reproduced the mean-
159
variance relationships well enough.
1.0
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Grow .Mean.q75%
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0.0015
Grow .Var.q25%
# simulations
# simulations
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0.010
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0.03
Surv.Var.q75%
1500
1000
500
0
0.00 0.03
Grow .R.adj
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Surv.Mean.q25%
Surv.Var.q50%
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0.0 0.2 0.4
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Grow .SSresiduals
# simulations
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Repr.Var.q50%
Surv.R.adj
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# simulations
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0.0000
Surv.Mean.q75%
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Surv.SSresiduals
0.95
# simulations
# simulations
Surv.Mean.q50%
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# simulations
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Surv.Slope
# simulations
# simulations
Repr.Var.q75%
0.20
0.96
Repr.R.adj
0.000 0.008
Repr.Mean.q75%
# simulations
20
8 12
2500
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1500
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0
4
# simulations
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Repr.SSresiduals
3000
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0
Repr.Mean.q50%
# simulations
# simulations
Repr.Mean.q25%
1.0
0.88
# simulations
0.0
100
# simulations
0.00 0.15
Repr.Slope
2500
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2000
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0
# simulations
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Repr.Intercept
# simulations
# simulations
Repr.mean=var
-0.2
# simulations
-0.8
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Figure S1.3. In each panel the vertical blue line corresponds with the value in the original
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data set, while the histogram displays the variation in that parameter among 10,000
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simulated data sets. ‘Repr’=reproduction rates, ‘Surv’=survival rates, ‘Grow’=growth
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rates. ‘mean=var’= the point on the regression line where the variance equals the mean
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(indicated by the blue dotted line in the figures above). ‘Intercept’ and ‘Slope’ indicate
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regression parameters. ‘SSresiduals’ = Sum of Squares of the residuals of regression
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models as in Fig S1. ‘R.adj’ = Adjusted R2 of regression models as in Fig. S1. ‘q25%’,
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‘q50%’ and ‘q75%’ are quantiles of the distributions of the means and variances.
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