ONLINE APPENDIX A. Precipitation and demographic information during the studies
Annual precipitation (mm) recorded near each field site, classifying annual period as wet, normal or dry, according to the 75th and 25th
percentiles of the last 30 years of precipitation. Deterministic population growth rate (λ) ± 95% C. I. (999 iterations from jackknife) ≥
1 implies demographic viability, and < 1 indicates demographic decline.
Period
Cryptantha flava – Utah
Carrichtera annua – Israel
Precipitation
Type
λ
Precipitation
Type
λ
1997-1998
491.744
Wet
0.94 ± 0.009
-
-
-
1998-1999
466.852
Wet
0.97 ± 0.011
-
-
-
1999-2000
382.778
Normal
0.96 ± 0.009
-
-
-
2000-2001
348.742
Normal
0.67 ± 0.002
-
-
-
2001-2002
174.24
Dry
-
-
-
2002-2003
236.98
Dry
351.80
Wet
2.32 ± 0.008
2003-2004
252.984
Dry
0.93 ± 0.012
234.80
Normal
0.47 ± 0.001
2004-2005
533.65
Wet
0.97 ± 0.008
377.30
Wet
1.64 ± 0.005
2005-2006
382.78
Normal
0.73 ± 0.006
195.60
Dry
0.43 ± 0.002
2006-2007
313.94
Normal
0.71 ± 0.006
259.90
Normal
0.79 ± 0.002
319.28
Normal
181.20
Dry
0.450 ±
0.001
275.08
Dry
113.80
Dry
12.42 ±
0.070
2009-2010
321.56
Normal
0.84 ± 0.006
-
-
-
2010-2011
328.01
Normal
0.98 ± 0.008
-
-
-
75th percentile
391.22
2007-2008
2008-2009
0.59 ± 0.002
1.00 ± 0.006
1.18 ± 0.005
321.31
§
Last 30-year
average
350.75
25th percentile
295.21
264.43
212.46
2002 census was not carried out and the population growth rate (λ) represents the 2001-2003 population dynamics.
ONLINE APPENDIX B. Details of the IPMs and PMMs used in this study
Integral Projection Model (IPM) for Cryptantha flava
An IPM describes the dynamics of a population by linking size distribution of individuals
at time t , n(x, t) to their size distribution the next period t + 1, n(y, t + 1). This is accomplished
by explicitly incorporating underlying demographic processes, the vital rates. This linkage is
based on the contribution of established individuals, described by a kernel p(x,y), and their
offspring, described by a kernel f(x,y).
𝜔
𝒏(𝑦, 𝑡 + 1) = ∫𝛼 [𝑝(𝑥, 𝑦) + 𝑓(𝑥, 𝑦)] ∙ 𝒏(𝑥, 𝑡) ∙ 𝑑𝑥
Both demographic processes are integrated for the range of sizes observed in the field: α
= 1 rosette to ω = 60 rosettes. Although individuals larger than 60 rosettes exist in the field, their
frequency is low (0.6%) and we did not include them in the output shown here, though their
effects on the population were accounted for because we included all individuals, regardless of
size, in our vital rate function parameterization (below).
To parameterize the vital rate functions that integrate the IPM, we used data for each
annual period. Each IPM included six vital rates (see Figure 3 and Online Appendix C). The first
two define the kernel p: survival during the annual period t to t+1 (σ), and change in individual
size from t to t+1 (γ) together with its associated variance. The remaining four vital rates define
the kernel f: probability of flowering in year t (φ), number of flowering rosettes produced per
individual that year (χ), probability of seedling establishment (ε), and size distribution of
seedlings at the end of the annual period, in t+1 (ϑ) (Figure 3). Seedling establishment rates ε
also include seed production per flowering rosette and seed germination, and here we treat them
as a black box since we did not measure them separately in the field. Survival (σ) and flowering
probabilities (φ) were modeled using logistic regressions. The size distribution of seedlings (ϑ)
was modeled with a negative Poisson regression. The vital rates γ and χ were analyzed using
linear regressions. For each vital rate parameterization, we used linear, quadratic and cubic
functions and retained the one that resulted in the lowest Akaike Index Criterion score.
𝑝(𝑥, 𝑦) = 𝜎(𝑥) ∙ 𝛾(𝑥, 𝑦)
𝑓(𝑥, 𝑦) = 𝜙(𝑥) ∙ 𝜒(𝑥) ∙ 𝜖 ∙ 𝜗(𝑦)
In order to calculate the population growth rates and elasticities involved in our
predicting scenarios (See Projecting demographic effects of changes in precipitation section) on
this demographic model, one needs to first discretize the IPMs. This means that the kernel k of
each IPM is converted (discretized) into a projection matrix K of large dimensions by imposing a
mesh onto k [1-3]. Because mesh size can affect model output [4], we discretized each of our
IPM kernels with 29 x 29, 59 x 59 and 119 x 119 mesh points over the state variable size,
resulting in discrete categories in increments of 2, 1 and 0.66 rosettes, respectively. Since the
calculated values of λ, stable stage distribution and relative reproductive output stayed fairly
constant regardless of mesh size (not shown), we opted for the 59 × 59 mesh. This mesh size
choice also allowed for a straightforward interpretation because each mesh point corresponds to
a one-rosette change in individual size.
Periodic Matrix Model (PPM) for Carrichtera annua
We described the life cycle of Carrichtera with a PMM of three seasons within the
annual period: dormant (k = 1 in figure 3.b), post-dormant (k = 2) and pre-dormant (k = 3)
seasons. The population dynamics of season i are represented by a periodic matrix Bk which,
similarly to an IPM, projects the population vector n from the beginning to the end of season k.
Bk projects the population back to the beginning of the dormant season.
Stages
𝑩1 = 𝑆𝑒𝑒𝑑𝑙𝑖𝑛𝑔
𝐷𝑜𝑟𝑚𝑎𝑛𝑡 𝑠𝑒𝑒𝑑
𝐺𝑒𝑟𝑚𝑖𝑛𝑎𝑏𝑙𝑒 𝑠𝑒𝑒𝑑
𝛿𝐺𝑒𝑟𝑚
0
Stages
𝑩2 = 𝑅𝑒𝑝𝑟𝑜𝑑𝑢𝑐𝑡𝑖𝑣𝑒
𝐷𝑜𝑟𝑚𝑎𝑛𝑡 𝑠𝑒𝑒𝑑
𝑆𝑒𝑒𝑑𝑙𝑖𝑛𝑔
𝜎
0
Stages
𝑩3 = 𝐺𝑒𝑟𝑚𝑖𝑛𝑎𝑏𝑙𝑒 𝑠𝑒𝑒𝑑
𝐷𝑜𝑟𝑚𝑎𝑛𝑡 𝑠𝑒𝑒𝑑
𝐷𝑜𝑟𝑚𝑎𝑛𝑡 𝑠𝑒𝑒𝑑
0
𝛿𝐵𝑎𝑛𝑘1
𝐷𝑜𝑟𝑚𝑎𝑛𝑡 𝑠𝑒𝑒𝑑
0
𝛿𝐵𝑎𝑛𝑘2
𝑅𝑒𝑝𝑟𝑜𝑑𝑢𝑐𝑡𝑖𝑣𝑒
𝜓𝐺𝑒𝑟𝑚
𝜓𝐵𝑎𝑛𝑘
𝐷𝑜𝑟𝑚𝑎𝑛𝑡 𝑠𝑒𝑒𝑑
𝜀𝐺𝑒𝑟𝑚
𝛿𝐵𝑎𝑛𝑘3
Note that the season length need not be the same for each Bk, and that the stage type need
not be the same from one season to the next [1].
In order to parameterize the Bk seasonal matrices, we used data for each annual period.
Vital rates were estimated from mean field values. Seedling survival (σ) was directly estimated
from permanent quadrat censuses. The remaining vital rates were calculated using the following
approximations. First, we calculated per capita fecundity (ψ, the summation of column 1 in B3)
from individual biomass records and the regression between seed number and reproductive
individual biomass (F1,26 = 725.8, P < 0.001, R² = 0.97). As an estimate for the missing biomass
record of 2005, we used the average biomass of all other sampling years. Second, we used the
aforementioned seed bank experiment to determine (i) the percentage of non-dormant seeds by
dividing the number of seeds germinating the first annual period by the total number of seeds
germinating over all three annual periods, (ii) the proportion of dormant seeds by dividing the
number of seeds germinating in the second and third annual periods by the total number of seeds
germinating, (iii) the proportion of dormant seeds that became germinable after one annual
period by dividing the number of seeds germinating in second annual period by the total number
of dormant seeds, and (iv) the proportion of dormant seeds that remain dormant for more than
one annual period by dividing the number of seeds germinating in the third annual period by the
total number of dormant seeds. Since these proportions were always measured under the same
experimental conditions, we took the mean proportions over all sampling annual periods as
constant values for estimates of seed bank processes (online appendix A). Proportions (i) and (ii)
were used to split the per capita fecundity into contribution to germinable seeds (ψGerm) and
contribution to dormant seeds (ψBank). Proportion (iii) was used as an estimate for the transition
from dormant to germinable seeds (εGerm). Proportions (iv) were used as estimates for the stasis
of dormant seeds for each season (δBank1, δBank2, and δBank3). Germination (δGerm) was estimated as
the ratio between number of seedlings and germinable seeds. Strictly speaking, δGerm is a
combination of germination, seed mortality, and seed dispersal. Because we did not measure
these processes separately, they are treated in the vital rate δGerm as a black box, although we
must note that seed dispersal is negligible in this system [5] while granivores are known to
consume large fractions of the total seed crop produced in deserts [6]. In our model, we assumed
that germinable seeds that did not germinate after the dormant season were lost from the
population. Likewise, we assumed that seed bank seeds can only become germinable seeds right
before the dormant season (note that for B1 and B2, only the main diagonal elements ≠ 0).
Back-multiplying all seasonal matrices Bk in the same annual period, from May of year t
to May of year t + 1, results in the A matrix, which describes the annual population dynamics
such that
𝒏(𝑡 + 1) = 𝑩3 ∙ 𝑩2 ∙ 𝑩1 ∙ 𝑛(𝑡) = 𝑨 ∙ 𝒏(𝑡)
where n(t) describes the number of individuals at the beginning of the annual cycle. The matrix
elements of A are consequently a function of the season-specific vital rates:
Stages
𝑨 = 𝑩3 ∙ 𝑩2 ∙ 𝑩1 = 𝑅𝑒𝑝𝑟𝑜𝑑𝑢𝑐𝑡𝑖𝑣𝑒
𝐷𝑜𝑟𝑚𝑎𝑛𝑡 𝑠𝑒𝑒𝑑
𝑅𝑒𝑝𝑟𝑜𝑑𝑢𝑐𝑡𝑖𝑣𝑒
𝛿𝐺𝑒𝑟𝑚 ∙ 𝜎 ∙ 𝜓𝐺𝑒𝑟𝑚
𝛿𝐺𝑒𝑟𝑚 ∙ 𝜎 ∙ 𝜓𝐵𝑎𝑛𝑘1
𝐷𝑜𝑟𝑚𝑎𝑛𝑡 𝑠𝑒𝑒𝑑
𝜓𝐵𝑎𝑛𝑘1 ∙ 𝜓𝐵𝑎𝑛𝑘2 ∙ 𝜀𝐺𝑒𝑟𝑚
𝛿𝐵𝑎𝑛𝑘1 ∙ 𝛿𝐵𝑎𝑛𝑘2 ∙ 𝛿𝐵𝑎𝑛𝑘3
ONLINE APPENDIX C. Vital rate parameters
The following tables contain the vital rates used to parameterize the stochastic integral projection model of Cryptantha flava
and the stochastic periodic matrix model of Carrichtera annua.
Cryptantha flava:
Period
Survival
Change in individual size
σ
γ
Variation of change in
individual size
Δγ
a
b1(σ)
b2(σ)2
b3(σ)3
a
b1(γ)
b2(γ)2
b3(γ)3
a
b1(Δγ)
b2(Δγ)2
b3(Δγ)3
1997-1998
0.00
0.39
-0.02
0.00
2.74
0.72
0.00
0.00
1.38
0.32
0.00
0.00
1998-1999
0.08
0.07
0.00
0.00
2.47
0.52
0.00
0.00
0.82
0.22
0.00
0.00
1999-2000
0.22
0.29
0.00
0.00
3.31
0.84
0.02
0.00
1.77
0.37
0.00
0.00
2000-2001
-0.93
0.37
-0.02
0.00
3.94
0.77
0.00
0.00
2.27
0.24
0.00
0.00
2001-2003
-1.18
0.18
-0.01
0.00
6.76
0.82
0.00
0.00
6.19
-0.37
0.03
0.00
2003-2004
-0.92
0.35
-0.01
0.00
2.80
1.56
-0.01
0.00
2.40
0.39
0.00
0.00
2004-2005
-0.24
0.29
-0.01
0.00
6.09
0.36
0.02
0.00
4.45
-0.08
0.01
0.00
2005-2006
-0.71
0.26
-0.01
0.00
2.19
0.67
0.00
0.00
2.71
0.02
0.01
0.00
2006-2007
-0.29
0.02
0.00
0.00
3.28
0.15
0.05
0.00
2.06
0.34
0.00
0.00
2007-2008
-0.90
0.33
-0.01
0.00
1.52
1.32
-0.03
0.00
1.73
0.21
0.00
0.00
2008-2009
-0.59
0.51
-0.03
0.00
4.74
0.86
0.00
0.00
3.32
0.15
0.00
0.00
2009-2010
0.31
-0.09
0.01
0.00
3.73
0.94
0.00
0.00
-0.19
0.80
-0.01
0.00
2010-2011
0.27
0.13
0.00
0.00
6.17
0.71
0.00
0.00
3.78
0.28
0.00
0.00
Cryptantha flava (cont’d):
Period
Flowering probability
Recruitment
ϕ
χ
Establishment
probability
ε
a
b1(ϕ)
b2(ϕ)2
b3(ϕ)3
a
b1(χ)
b2(χ)2
b3(χ)3
1997-1998
-5.37
0.98
-0.05
0.00
0.53
0.09
0.00
0.00
1998-1999
-3.29
0.71
-0.03
0.00
0.16
0.15
0.00
1999-2000
-4.29
0.65
-0.03
0.00
0.48
0.04
2000-2001
-4.14
0.94
-0.06
0.00
0.15
2001-2003
-7.03
2.43
-0.24
0.01
2003-2004
-2.40
0.02
0.00
2004-2005
-4.74
0.92
2005-2006
-5.05
2006-2007
Size distribution of recruits
ϑ
b2(ϑ)2
b3(ϑ)3
a
b1(ϑ)
0.04
31.07
-49.92 26.15
-4.25
0.00
0.14
9.19
-6.27
2.02
-0.23
0.00
0.00
0.28
5.69
-1.75
0.00
0.00
0.15
0.00
0.00
0.04
3.72
-0.86
0.00
0.00
-0.38
0.17
0.00
0.00
0.02
2.75
-0.66
0.00
0.00
0.00
0.80
-0.01
0.00
0.00
0.88
5.42
-2.14
0.00
0.00
-0.06
0.00
0.65
0.05
0.00
0.00
0.04
39.13
-61.97 30.75
-4.77
0.93
-0.03
0.00
0.66
0.07
0.00
0.00
0.07
7.22
-3.11
0.00
0.00
-4.82
0.70
-0.02
0.00
0.28
0.04
0.00
0.00
0.49
7.55
-3.12
0.00
0.00
2007-2008
-6.26
1.52
-0.11
0.00
-0.36
0.10
0.00
0.00
0.31
9.33
-8.85
3.10
-0.36
2008-2009
-3.19
0.25
0.00
0.00
-0.30
0.10
0.00
0.00
0.40
9.56
-9.50
3.59
-0.43
2009-2010
-9.01
2.28
-0.18
0.00
-0.71
0.22
-0.01
0.00
0.10
-19.63 33.83
-11.56
0.00
2010-2011
-5.16
1.06
-0.05
0.00
-0.06
0.09
0.00
0.00
0.18
6.04
0.00
0.00
-2.74
a and bi represent the intercept and slopes, respectively, of the polynomic functions that parameterize the integral projection models.
Carrichtera annua:
Germinable
Period
seed
survival
Germinable
Juvenile seed persurvival
capita
Seedbank stasis
Seed per-
Seed
capita
emergence
contribution from
production
to seedbank
seedbank
δGerm
σ
ψGerm
δBank1
δBank2
δBank3
ψBank
εGerm
2002-2003
0.10
0.82
24.21
1.00
1.00
0.38
14.45
0.62
2003-2004
0.03
0.70
4.68
1.00
1.00
0.38
2.80
0.62
2004-2005
0.10
0.67
21.91
1.00
1.00
0.38
13.08
0.62
2005-2006
0.02
0.38
6.63
1.00
1.00
0.38
3.96
0.62
2006-2007
0.05
0.89
10.89
1.00
1.00
0.38
6.50
0.62
2007-2008
0.04
0.40
4.86
1.00
1.00
0.38
2.90
0.62
2008-2009
0.13
0.89
86.51
1.00
1.00
0.38
51.64
0.62
Each vital rate is described in the context of the life cycles of Cryptantha and Carrichtera in figure 3 of the manuscript.
ONLINE APPENDIX D. Present, back-projected and projected precipitation
Annual precipitation (from May of year t to April of year t+1) recorded by the closest
permanent weather stations to both field sites, and back-projected by the super-high resolution
climate change model used in this manuscript [7].
Cryptantha flava - Utah
Carrichtera annua - Israel
Permanent
Permanent
station
(current)
Backprojected
station
(current)
Backprojected
Maximum
533.65
354.12
456.04
430.62
75th percentile
399.35
305.93
321.31
262.04
Average
352.61
267.67
264.43
238.09
25th percentile
288.16
234.27
212.46
204.38
Minimum
174.24
168.64
113.80
119.85
1979-1980
-
308.11
456.04
265.13
1980-1981
-
246.31
243.45
237.20
1981-1982
280.92
298.19
196.32
245.60
1982-1983
318.52
256.54
338.37
239.57
1983-1984
477.27
234.52
140.51
187.99
1984-1985
346.71
214.01
259.98
140.86
1985-1986
501.90
239.74
211.79
261.01
1986-1987
332.74
294.70
304.50
214.78
1987-1988
309.12
354.12
315.05
296.81
1988-1989
210.82
241.96
235.92
248.28
1989-1990
290.58
337.68
259.17
205.46
1990-1991
375.67
345.40
337.14
299.88
1991-1992
239.27
212.55
446.06
234.23
1992-1993
392.18
305.21
265.13
311.13
1993-1994
368.81
233.54
228.23
201.13
1994-1995
420.88
168.64
358.50
215.11
1995-1996
388.37
312.76
214.48
218.90
1996-1997
440.18
332.58
232.08
223.91
1997-1998
491.74
260.42
244.11
178.06
1998-1999
466.85
234.63
127.34
119.85
1999-2000
382.78
284.01
173.08
430.62
2000-2001
348.74
275.93
307.93
303.18
2001-2002
174.24
217.62
323.40
176.12
2002-2003
236.98
214.83
351.80
259.39
2003-2004
252.98
-
234.80
-
2004-2005
533.65
-
377.30
-
2005-2006
382.78
-
195.60
-
2006-2007
313.94
-
259.90
-
2007-2008
319.28
-
181.20
-
2008-2009
275.08
-
113.80
-
2009-2010
321.56
-
-
-
2010-2011
328.01
-
-
-
Annual precipitation projected by the super-high resolution climate change model used
[7].
Cryptantha flava - Utah
Carrichtera annua - Israel
Projected
Projected
Maximum
491.83
318.84
75th percentile
349.26
251.38
Average
305.10
195.47
25th percentile
248.76
160.19
Minimum
195.08
84.62
2075-2076
268.74
174.99
2076-2077
248.82
186.35
2077-2078
491.83
206.95
2078-2079
282.37
182.93
2079-2080
291.82
212.74
2080-2081
238.27
250.65
2081-2082
471.35
268.89
2082-2083
280.15
106.76
2083-2084
224.10
265.66
2084-2085
459.83
105.15
2085-2086
276.46
204.49
2086-2087
230.24
179.90
2087-2088
269.81
204.73
2088-2089
352.44
292.48
2089-2090
323.85
318.84
2090-2091
269.12
253.59
2091-2092
282.57
115.81
2092-2093
248.57
198.80
2093-2094
399.56
279.52
2094-2095
195.08
102.72
2095-2096
262.79
108.59
2096-2097
348.20
188.34
2097-2098
378.82
197.88
2098-2099
227.72
84.62
References
1.
Caswell, H. 2001 Matrix population models: construction, analysis, and interpretation,
2nd edn. Sunderland, MA, USA: Sinauer Associates.
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