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SUPPORTING INFORMATION
Appendix S1. Detailed methods of integral projection model.
Appendix S2. Detailed methods of model selection approach to estimate size- and climatedependent vital rates.
Table S1. Model selection results for influence of climate variables on vital rate functions.
Table S2. IPM parameters for non-size-dependent vital rates.
Figure S1. Histograms of climate variables (1991 – 2013) that explained a significant proportion
of variation in vital rate functions.
Figure S2. Coefficients of variation for each climate variable in each climate scenario sampled
over 50,000 years (resampled from climate values for 1991 – 2013).
Figure S3. Relationship between seedling recruitment in year t + 3 and seed production in year t.
Figure S4. Pairwise invasibility plot showing evolutionarily stable strategy for intercept of
flowering function (c0) and observed coefficient.
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Appendix S1. DETAILED METHODS—Integral projection model
Continuous size dynamics in the IPM are described by:
𝑁(𝑦, 𝑡 + 1) = 𝜙𝑇 𝑇(𝑡)𝜂𝑠 (𝑦) + 𝜙𝐷 𝐷(𝑡)𝜂𝐷 (𝑦) + ∫Ω 𝜙(𝑥)(1 − 𝜇(𝑥)) 𝛾(𝑦, 𝑥)𝑁(𝑥, 𝑡)𝑑𝑥, (1)
where growth is conditioned on the probabilities of surviving and not going dormant. Here, the
growth function γ(y, x) gives the probability distribution of future size given current size
according to γ(y, x) ~ 𝒩 (M(x), σ), where M(x) is the size-dependent mean and σ is the variance
of the future size distribution. Miller et al. (2012) reported that, due to costs of reproduction,
vegetative (V) and reproductive (R) plants followed different growth trajectories given current
size (γV(y,x) ≠ γR(y,x)). We account for this difference in the IPM by weighting the future size
distributions of vegetative (γV(y,x)) and reproductive plants (γR(y,x)) by their size-specific
probabilities of flowering:
 ( y, x )   V ( y, x )1   ( x )   R ( y, x )  ( x )
(2).
Additionally, we detected a reduction in growth associated with each fruit produced by flowering
plants, which is included in the reproductive growth function (γR(x)) (Miller et al. 2012).
New individuals enter the population as seedlings following a normal size distribution, S
~ 𝒩( x S ,σS) after survival from a tuber (probability ϕT) in the previous year. Dormant plants that
survive (probability ϕD) also recruit into the continuous size distribution the following year,
following a different normal distribution of sizes, D ~ 𝒩( xD ,σD). We do not include the
possibility of dormancy for more than one year (this was observed only twice). The dynamics of
dormant plants are given by:
𝐷(𝑡 + 1) = ∫Ω 𝜙 (𝑥)𝜇(𝑥)𝑁(𝑥, 𝑡)𝑑𝑥
(3).
Tubers are protocorms that survive to the next year (probability ϕP), with dynamics
described by: T(t + 1) = ϕPP(t). Protocorm production is estimated as the number of flowers
produced per plant in the previous year multiplied by the proportion of flowers that set fruit (),
the number of seeds per fruit (), and the seed-to-protocorm transition probability (ζ), integrated
over the range of plant sizes ():
𝑃(𝑡 + 1) = 𝜐𝛼𝜁 ∫Ω 𝛽 (𝑥)𝜔(𝑥)𝑁(𝑥, 𝑡)𝑑𝑥
(4).
Literature Cited
Miller, T. E. X., Williams, J. L., Jongejans, E., Brys, R. & Jacquemyn, H. (2012) Evolutionary
demography of iteroparous plants: incorporating non-lethal costs of reproduction into
integral projection models. Proceedings of the Royal Society of London B, 279, 28312840.
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Appendix S2. DETAILED METHODS— Estimating size- and climate-dependent vital rates
We took the following steps to choose the best size- and climate-dependent model to
describe each vital rate:
(1) We first examined whether vital rates varied significantly among years. If so, this suggests a
role for climate to explain some amount of inter-annual variation. For each size-dependent vital
rate function, we used likelihood ratio tests to compare three generalized linear models with the
following explanatory variables: only fixed effects of size in year t (loge(leaf area cm2)), fixed
effects of size and year (e.g., average survival varies from year to year), and fixed effects of size,
year, and their interaction (e.g., the influence of size on survival varies from year to year). If the
year term was not significant at α < 0.05, we did not further explore climate effects on the vital
rate. If the year × size interaction was not significant at α < 0.05, we explored climate effects
only on the intercepts of the vital rate functions. We used the appropriate error distribution for
each vital rate: probabilities of survival and flowering were fit with a binomial distribution, and
growth, seedling size and number of flowers with a Gaussian distribution, as in Miller et al.
(2012).
(2) When the year and/or year × size interaction significantly improved model fit, we re-fit the
same model with year as a random effect (modifying the intercept and / or slope with respect to
size). We then tested for correlations between year random effects and each of the climate
variables: mean daily precipitation, mean daily temperature, and proportion wet days across each
season (spring, summer, fall, winter) and across one transition year (May 16, year t – May 15
year t+1). To constrain the total number of candidate models, we retained only those climate
variables that were significantly correlated with year random effects; we assume that these
climate variables were the main causal drivers of inter-annual variation in demography. We
examined year / climate variable pairs for 10 or 11 years (2003 – 2013). Over the study period,
mean seasonal precipitation varied two- to three-fold and mean seasonal temperature varied 1.2 –
2.6-fold (Fig. S1), providing sufficient variation to quantify climate effects.
In addition to correlations between climate and demography in the same year, we also considered
a one-year lag (climate in year t-1 affects demography in year t), since the size of the tuber in the
previous year is likely to affect the vital rates in the next season via its effect on plant condition
and demography in the current year (Snow & Whigham 1989). That is, tuber condition in t-1
affects demography in t-1 (no lag), and the resources the plant accumulates in t-1 are stored in
the new tuber that will directly influence plant condition and demography in year t. We did not
consider two-year lags, because the link between tuber condition from two years previous and
the current year is more tenuous, and we wanted to avoid spurious correlations. When the
correlations between climate and random year effects were statistically significant or marginally
so (α < 0.10), we included those climate variables on a short list for model selection.
(3) To choose the best size- and climate-dependent function for each vital rate, we compared a
set of models that included all the climate variables that were correlated with inter-annual
variability (i.e., on the short list), but with no year effects. We chose the model with the lowest
Akaike Information Criterion (AIC). Candidate models included the influence of climate
variables on the intercepts of vital rate models when we found significant year effects, and on the
slopes (size x climate interaction) we found significant year × size interactions (in Step 1). For
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the best model for each vital rate, we also tested whether adding the climate variable from the
second-best model significantly improved the fit (evaluated by ΔAIC > 2). When it did, we
retained the two climate variables in the final model. Note that with a subset of the data (2004 –
2012), models with more than two climate variables did not improve the model fit (ΔAIC < 2).
In the full data set, we did not fit vital rate functions with more than two climate variables to
avoid over-fitting models and to constrain the total number of parameters in the IPM. We also
did not fit models with interactions between climate variables to constrain the total number of
models in the model selection process, although we recognize that some interactions between
climate variables might be expected, even if others are nonsensical.
(4) Finally, to fully account for inter-annual variability that was unrelated to the best climate
predictor(s), we refit the best model for each vital rate with year as a random effect, and with size
and climate variables as fixed effects.
To calculate a measure of the contribution of the climate variable(s) to explaining
variation in each vital rate, we used a measure of proportional reduction in deviance: D = 1 –
(dev1/dev2), where dev1 and dev2 are the deviances of the fixed-effects model including and
excluding climate variables, respectively (Zheng 2000; Dalgleish et al. 2011).
There were two exceptions to the approach described above. First, since previous work
supports an effect of flowering on growth (Jacquemyn, Brys & Jongejans 2010; Miller et al.
2012), and since models with 3-way interactions (climate × size × flowering) failed to converge,
we restricted climate effects for growth to two-way interactions. Thus, we allowed for an
interaction between climate and reproductive status, such that the influence of a climate variable
on the intercept of the growth function could differ between vegetative and flowering plants.
Second, for the proportion of fruits set per flower, we found significant inter-annual variation but
climate explained a very small fraction of it (D = 0.02 for spring precipitation, the best climate
variable). Further, fruit set is known to be strongly limited by pollination and less so by direct
effects of abiotic factors (Jacquemyn & Brys 2010). For these reasons, we did not include
climate-dependence in this vital rate though we did include random inter-annual variation.
Literature Cited
Dalgleish, H. J., Koons, D. N., Hooten, M. B., Moffet, C. A. & Adler, P. B. (2011) Climate
influences the demography of three dominant sagebrush steppe plants. Ecology, 92, 7585.
Jacquemyn, H. & Brys, R. (2010) Temporal and spatial variation in flower and fruit production
in a food-deceptive orchid: a five-year study. Plant Biology, 12, 145-153.
Jacquemyn, H., Brys, R. & Jongejans, E. (2010) Size-dependent flowering and costs of
reproduction affect population dynamics in a tuberous perennial woodland orchid.
Journal of Ecology, 98, 1204-1215.
Miller, T. E. X., Williams, J. L., Jongejans, E., Brys, R. & Jacquemyn, H. (2012) Evolutionary
demography of iteroparous plants: incorporating non-lethal costs of reproduction into
integral projection models. Proceedings of the Royal Society of London B, 279, 28312840.
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Snow, A. A. & Whigham, D. F. (1989) Costs of flower and fruit production in Tipularia discolor
(Orchidaceae). Ecology, 70, 1286-1293.
Zheng, B. (2000) Summarizing the goodness of fit of generalized linear models for longitudinal
data. Statistics in Medicine, 19, 1265-1275.
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Table S1. Model selection results for influence of climate variables on vital rate functions. Listed climate variables were significantly correlated
with year random effects estimates. Climate variables that were used in the IPM are indicated in bold.
Vital rate
Model
Growth (n = 3198)
Winter temperature (lag 1) + Summer precipitation (lag 1) +
Summer precipitation (lag 1) × Flowering
Winter temperature (lag 1) + Summer precipitation (lag 1)
Winter temperature (lag 1)
Winter temperature (lag 1) + Winter temperature (lag 1) × Flowering
Summer Precipitation (lag 1) + Summer Precipitation (lag 1) ×
flowering
Summer proportion wet days (lag 1) + Summer proportion wet days (lag
1) × flowering
Summer proportion wet days (lag 1)
Summer Precipitation (lag 1)
Spring Precipitation (lag 1) + Spring Precipitation (lag 1) × flowering
Spring Precipitation (lag 1)
Fall temperature (no lag)
Fall temperature (no lag) + Fall temperature (no lag) × flowering
No climate
Probability of flowering (n = 3693)
Spring precipitation (lag 1) + Winter precipitation (no lag) + Spring
precipitation × size
Spring precipitation (lag 1) + Winter precipitation (no lag) + Spring
precipitation × size + Winter precipitation (no lag) × size
Spring precipitation (lag 1) + Spring precipitation × size
Spring proportion wet days (lag 1) + Spring proportion wet days (lag 1)
× size
Year proportion wet days (lag 1) + Year proportion wet days (lag 1) ×
size
ΔAIC
AIC weight
0
0.870
3.8
58.8
59.5
92.2
0.130
0.000
0.000
0.000
92.6
0.000
95.6
96.9
98.9
101.5
160.7
162.7
202.1
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0
0.674
1.5
0.326
101.1
112.4
0.000
0.000
161.4
0.000
6
Spring proportion wet days (lag 1)
Spring precipitation (lag 1)
Year proportion wet days (lag 1)
Winter precipitation (no lag) + Winter precipitation (no lag) × size
Winter precipitation (no lag)
No climate
175.6
180.7
198.7
200.1
225.6
230.7
0.000
0.000
0.000
0.000
0.000
0.000
Number of flowers (n = 1446)
Year average temperature (no lag)
Fall temperature (no lag)
No climate
0
34.7
63.5
1
0.000
0.000
Recruit size (n = 396)
Spring precipitation (lag 1)
Spring precipitation (lag 1) + Winter precipitation (no lag)
Spring proportion wet days (lag 1)
Winter precipitation (no lag)
Spring proportion wet days (no lag)
No climate
0
1.5
6.3
12.6
14.8
24.8
0.660
0.312
0.028
0.001
0.000
0.000
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Table S2. IPM parameters for non-size-dependent vital rates.
Vital rate
Parameter value
Number of seeds per fruit
 = 6000
Seed germination rate
ζ = 0.015
Protocorm survival
ϕP = 0.0149
Tuber survival
ϕT = 0.0594
Dormant plant survival
Size distribution of plants
ϕD = 1.0
D ~ 𝒩( xD = 4.667, σD = 0.959)
emerging from dormancy
170
171
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Fig. S1 Histograms of climate variables (1991 – 2013) that explained a significant proportion of variation in vital rate functions. Dashed vertical
lines show location of ±1 standard deviation of the mean. Years with climate values ±1 standard deviation of the mean were used in the climate
scenarios to calculate the evolutionarily stable flowering size, and λs when the population maintains the currently observed flowering size or
perfectly tracks the ES size.
Winter precipitation
Winter temperature
Yearly temperature
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4
Frequency
5
4
0.5 1.0 1.5 2.0 2.5 3.0
178
mm/day
1.0
2.0
3.0
mm/day
4.0
1.0
2.0
3.0
mm/day
4.0
0
0
0
0
0
1
2
2
2
2
3
Frequency
6
4
Frequency
4
Frequency
6
4
2
Frequency
6
8
6
8
8
7
Summer precipitation
8
Spring precipitation
10
172
173
174
175
176
177
1
2
3
4
5
6
degrees C
7
8
9.5
10.5
11.5
12.5
degrees C
179
180
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Fig. S2. Coefficients of variation for each climate variable in each climate scenario sampled over 50,000 years (resampled from
climate values for 1991 – 2013). Extreme years (with climate values ±1 standard deviation of the mean) were drawn at the frequency
indicated on the horizontal axis and the remaining years at 1 - frequency. Climate variables as indicated in legend, except for climate
variable adjusted in scenario is drawn with orange. Changing frequencies of spring precipitation for A) extreme dry springs and F)
extreme wet springs; summer precipitation for B) extreme dry summers and G) extreme wet summers; winter precipitation for C)
extreme dry winters and H) extreme wet winters; winter temperature for D) extreme warm winters and I) extreme cold winters; and
annual temperature for E) extreme warm years and J) extreme cold years. Vertical lines at 25% and 75% indicate bounds of
frequencies used in climate scenarios.
80
100
Freq. of extreme wet springs
60
80
100
Freq. of extreme dry winters
H
40
60
80
100
Freq. of extreme wet summers
0.4
0.2
0.1
0.0
20
40
60
80
100
0
20
40
60
80
100
Freq. of extreme wet winters
0
Freq. of extreme warm winters
I
0.3
0.3
0.2
0.1
20
0
20
40
60
80
100
Freq. of extreme warm years
J
0.4
40
0.0
0
0.3
0.4
0.2
0.1
20
0.3
G
0.4
60
0.0
Freq. of extreme dry summers
0.2
40
0
0.2
100
0.1
80
E
0.0
60
0.1
20
0.3
0.4
0.2
0.1
40
0.0
0
191
20
0.3
0.4
0.3
0.2
0.1
0.0
Coefficient of variation
Freq. of extreme dry springs
F
0.0
0
0.4
100
0.2
80
0.1
60
D
0.0
40
0.4
20
C
0.3
0.4
0.3
0.2
0.1
0
190
B
0.0
0.1
0.2
0.3
0.4
A
0.0
Coefficient of variation
181
182
183
184
185
186
187
188
189
0
20
40
60
80
100
Freq. of extreme cold winters
0
20
40
60
80
100
Freq. of extreme cold years
Spring Precipitation
Summer Precipitation
Winter Precipitation
Winter Temperature
Annual Temperature
10
60
50
40
30
20
10
0
Seedling recruitment in year t+3
Figure S3. Relationship between seedling recruitment in year t + 3 and seed production in year t.
Note that in years t + 1 and t + 2, Orchis purpurea exists only in an underground protocorm and
tuber stage, respectively. Filled and open points represent two demographic census sites. The
recruitment function (line) was fit to data pooled across sites. The fitted function is: seedlingst+3
= 48.19*seedst / (1101558 + seedst).
0
500000
1500000
2500000
Seed production in year t
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Figure S4. Pairwise invasibility plot for the intercept (c0) of the probability of flowering function
((x)) for all climate years (1991 – 2013, each with equal probability of selection). With
increasing c0, plants flower at larger sizes. White and shaded areas show combinations for
resident and invader strategies for which invaders can versus cannot, respectively, invade the
resident. Observed intercept indicated by dot, and vertical and horizontal error bars show one
standard error of the estimate of the coefficient (c0).
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