1
APPENDIX A. Detailed description of data and methods
2
3
4
A.1 Acanthomintha ilicifolia
Acanthomintha ilicifolia, or San Diego thornmint, is a rare annual, aromatic herb in
5
Lamaiceae (mint family). It is endemic to San Diego County and northern Baja California (US
6
Fish and Wildlife Service 2009). A. ilicifolia can be two to six inches in height with rose-
7
highlighted, white, tubular flowers. It is highly specialized to openings on gabbro soils derived
8
from igneous rock and gray calcareous clay soils (US Fish and Wildlife Five-Year Review 2009)
9
and occurs in coastal sage scrub, chaparral, and native grasslands.
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11
A.2 Habitat suitability, land use change, and metapopulation maps
12
We estimated the habitat suitability of each cell over a uniform grid for A. ilicifolia as a
13
function of environmental conditions. Because data on A. ilicifolia locations are sparse and not
14
obtained from probability-designed surveys, we used two SDM methods, robust to small or
15
biased samples (Elith and Graham 2009). They are MaxEnt (Phillips et al. 2006) and Random
16
Forest (Prasad and Iverson 2006, Cutler et al. 2007). MaxEnt was run using 10,000 random
17
background points, 500 maximum iterations, and a convergence threshold of 0.00001. We used
18
the automatic feature type option for training and estimating response curves, with jackknifing to
19
determine variable importance. Random Forest was run using the package randomForest in the
20
R statistical programming environment. We used 2116 background (pseudo-absence) points
21
based on locations of vegetation plots and occurrences of other plant species in our database.
22
Five hundred trees were estimated using three predictors per tree. To estimate the AUC, we used
23
the averaged “out-of-bag” predictions from the 500 models.
1
24
Current location data for A. ilicifolia included 104 point locations of observations
25
obtained from the California Natural Diversity Database (CNDB) and collection records from the
26
Consortium of California Herbaria (CCH). Environmental predictors included climate (mean
27
January minimum temperature, mean July maximum temperature, and mean annual
28
precipitation), soil, and terrain variables important to predicting southern California plant species
29
distributions (Syphard & Franklin 2009; see Table A.1 below). For current climate conditions,
30
the values in each one hectare grid cell were derived from 1971-2000 Parameter-Elevation
31
Regressions on Independent Slopes Model data (PRISM, Daly et al 2006), spatially downscaled
32
to a 90 m Digital Elevation Model (Flint & Flint 2012) and re-sampled to 100 m resolution.
33
Both temperature variables were important in both SDMs, and precipitation was
34
important in Random Forest. In MaxEnt, a steep decline in probability of species presence is
35
predicted below 500 mm annual precipitation, above average minimum January temperature of 8
36
C, and above average maximum July temperature of 33 C, which would lead to predicted habitat
37
shifts under the warmer drier future climate scenario. Soil order and the other soil variables were
38
moderately important in both models. A. ilicifolia is known to be associated with open or bare
39
clayey patches (lenses) within otherwise dense, closed canopy shrub communities. However,
40
given the coarse resolution of the environmental predictors, these microhabitat features are not
41
being directly identified in the SDMs. Rather, the correlative models identify the soil types and
42
properties that support the shrubland community within which A. ilicifolia occurrences are
43
located. The presence points occurred primarily in alfisols (46 %), but also vertisols (22 %) and
44
entisols (11 %). This difference in scales between the SDM resolution and the small patches that
45
define subpopulations of A. ilicifolia is addressed by scaling the carrying capacity of suitable
46
habitat patches (next section A.3)
2
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48
49
Table A.1. Soil, terrain, and climate variables used to predict habitat suitability, and variable importance in MaxEnt and Random Forest models.
PRISM and STATSGO data are 1-km resolution. DEM and derivatives are 30-m resolution. All predictors were resampled to 100 m.
______________________________________________________________________________________________________________
50
51
Environmental Predictor
Source
MaxEnt
Random Forest
______________________________________________________________________________________________________________
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53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
Average annual precipitation (1971-2000) mm
PRISM
6
22
Average minimum January temperature (1971-2000) ˚C
PRISM
38
24
Average maximum July temperature (1971-2000) ˚C
PRISM
22
23
Soil order (13 categories:acidigneous, alfisol, alluvial,
badland, entisol, inceptisol, metamorphic, mollisol, rock,
roughstony, terrace, ultisol, vertisol, other)
STATSGO
7
14
Soil depth (m)
STATSGO
2
10
Soil available water capacity (cm/cm)
STATSGO
9
8
Soil pH
STATSGO
5
7
Slope angle (degrees)
USGS 30 m DEM
6
19
Potential winter solstice solar insolation (Watt/hr m2)
DEM using Solar Analyst
1
21
Potential summer solstice solar insolation (Watt/hr m2)
DEM using Solar Analyst
2
23
Topographic moisture index (unitless)
DEM
3
23
______________________________________________________________________________________________________________
Notes: STATSGO: State Soil Geographic data base for California, U.S. Department of Agriculture Natural Resources Conservation Service. URL
http://gis.ca.gov/catalog/BrowseRecord.epl?id=21237.
DEM: Digital Elevation Model;
USGS: U.S. Geological Survey;
Solar Analyst: an ArcView extension for modeling landscape scale solar radiation.
MaxEnt Estimated Variable Contribution: As noted in the MaxEnt software, to determine the estimate, “in each iteration of the training algorithm,
the increase in regularized gain is added to the contribution of the corresponding variable, or subtracted from it if the change to the absolute value
of lambda is negative.” Variable contributions should be interpreted with caution when the predictors are multicollinear.
Variable importance in the Random Forest model is estimated as the mean decrease in the Gini Index.
3
85
The output of each SDM, based on current location and climate data, is a cell map
86
displaying a 0-to-1 habitat suitability measure for each cell. The training accuracy of the
87
resulting suitability maps, measured in Area Under the Curve terms, was AUC = 0.907 for
88
MaxEnt and 0.870 for Random Forest.
89
For future climate, we used one emission scenario (A2, which assumes business as usual
90
CO2 emissions in a socio-economically heterogeneous world) and two general circulation model
91
projections (the Department of Energy and National Center for Atmospheric Research’s Parallel
92
Climate Model, or PCM, and the National Oceanic and Atmospheric Association's Geophysical
93
Fluid Dynamic Laboratory’s CM.2 model, or GFDL), downscaled following Flint and Flint
94
(2012) and bias corrected using the historically measured PRISM data. PCM was generally less
95
sensitive to climate forcings and predicted a slightly wetter and warmer climate. The more
96
sensitive GFDL predicted a substantially hotter and drier climate for California. These two
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scenarios were chosen because they fit historic climate data reasonably well for California, but
98
present contrasting forecasts for future climate (Cayan et al. 2008). Climate variables derived
99
from monthly projections for 2070-2099 were averaged to represent predicted climate ca. 2099
100
for each climate scenario. We constructed 2099 habitat suitability maps (for each assumed
101
climate projection) by substituting 2099 climate data into the SDM suitability functions
102
estimated from current climate data. To create a time series of dynamic habitat maps across 100
103
years, we linearly interpolated between current and future habitat suitability maps on a cell-by-
104
cell basis for each time step.
105
We developed spatially explicit, binary projections of urban development for the years
106
2000-2050 using the SLEUTH model (Syphard et al 2011). SLEUTH is a widely used cellular
107
automaton model that predicts the spatial extent of urban expansion (Clarke et al. 2007). The
4
108
predictive strength of the model results from a rigorous calibration process that fits parameter set
109
coefficients as a function of observed dynamics particular to development patterns in the study
110
region, using historic data representing urban extent and road networks (Clarke et al. 1996). For
111
each time step up to 2050, the spatial output of the urban growth model was overlapped with the
112
projected suitability maps. Areas where urban growth overlapped suitable habitat were changed
113
to unsuitable habitat. Thus, projected suitability maps based on climate change were sometimes
114
modified by land use change (see Fig. A.1).
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116
A.3 Metapopulation Maps
117
The threshold that resulted in maximum training sensitivity plus specificity was specified
118
so that only predicted habitat values above these values were predicted present. For MaxEnt and
119
Random Forest, the thresholds were 0.04 and 0.37, respectively. Metapopulation maps were
120
imported into the spatially explicit population modeling platform RAMAS GIS® 5.0 (Akçakaya
121
& Root 2005). Each year a new map was imported into RAMAS to reflect the changes in habitat
122
due to changes in climate. The carrying capacity of a patch was calculated as the sum of habitat
123
suitabilities, over those cells with suitabilities above the threshold, multiplied by 15,000
124
individuals for each hectare. Although more than 15,000 individuals could occur in a hectare
125
(the pixel size of our SDM predictions), A. ilicifolia is restricted to a particular soil type, namely
126
clay lenses, which are typically much smaller than a hectare. Thus, we used the value of 15,000
127
based on the largest population reported (Bauder 1994), occurring in less than a hectare in Poway
128
in 1994. This assumption was necessary given that SDM predictions do not have sufficiently
129
fine spatial and categorical resolution to predict the occurrence of every clay lens. Simulations
130
were computed for 12 scenarios described in the main text.
131
5
(a)
(b)
(c)
Suitability
Index
(d)
(f)
(i)
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133
134
135
136
137
138
139
140
141
142
(e)
(g)
(h)
(j)
Figure A.1. Habitat suitability maps for A. ilicifolia. The first five maps are MaxEnt derived:
a. Current map, without thresholding.
b. Current map, with thresholding. Black dots are observed presences.
c. Future map (2100) based on land use change only, with thresholding.
d. Future map (2100) based on PCM climate change, without thresholding.
e. Future map (2100) based on GFDL climate change, without thresholding.
The second five maps (f to j) are analogous but Random Forest derived.
6
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144
A.4 Two-stage demographic model
Each patch in the metapopulation is governed by its own two-stage demographic model.
145
The model structure is the same for all patches, governed by the same underlying parameters.
146
However, each patch model has its own history, depending on fire, carrying capacity constraints,
147
natural variability, management options to control invasive species or relocate individuals, and
148
random perturbations representing environmental and demographic stochasticity.
149
The two life stages for each demographic model are seeds and adult plants. The time step
150
is one year, assumed to start in spring just before plant death and seed set. Figure A.2 is a
151
schematic of the staging for a single patch. In a given year in the patch, each plant either dies
152
with no replacement or is replaced by another plant or seed or both, leading to one of the
153
transitions shown on Fig. A.2 (the magnitudes f, g, s, and d on Fig. A.2 are defined below).
154
Although every plant dies by the end of the year, a large fraction is effectively replaced by new
155
plants at the start of the next year. Plants can “survive” in this sense. The new plants can enter
156
the system from either a current year seed or from an earlier year seed in the species’ seedbank.
157
It is tempting to simplify to a one-stage model which follows only adults, leaving seeds behind
158
the scenes, and we present such a simplification in Section A.6. However, tracking both seeds
159
and adult plants adds two important dimensions of realism. First, since seed germination rates
160
increase with seed age, the seedbank is important to the population dynamics. Second, seeds and
161
adults respond differently to fire. Adults are mostly killed in a fire, in which case the next year's
162
population is heavily dependent on the seedbank.
163
7
Year 1
Year 2
Spring
Summer
Fall
Winter
Spring
...
Adult replacement, μ22 = E[f] E[s9] E[gd]
Addition to seedbank, μ12 = E[f ] E[s9] E[(1 - g)]
Adult
seed
production, f
Seeds
seed
survival, s9
Seeds to seedbank
seedling
survival, d
germination, g
Deaths
Seedbank germination, μ21 = E[g x d]
Adult
Seedling
germination, g
seed
survival, s12
Seed*
Seed
Seedbank survival, μ11 = E[s12]
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165
166
167
168
169
170
171
172
173
Figure A.2. Schematic of one-year transitions. The dotted lines and boxes are the overall
transition rates represented by the two stages of the demographic model. The solid lines and
boxes represent component transitions making up the overall transition rates in the demographic
model. Because data on overall A. ilicifolia transitions were often missing, we relied on the
component steps. The unboxed text labels the stages. The magnitudes f, g, d, and the
subscripted s’s are defined in the next section.
For a representative patch, let the numbers of seeds and plants at the start of year t be
174
denoted nseed(t) and nplant(t). The change in these numbers from year t to year t+1 is governed
175
by four “vital rates” {c11(t), c21(t), c12(t), c22(t)} via the following rules. Independently, each
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seed either stays a seed with probability c11(t), germinates into a plant with probability c21(t), or
8
177
dies with probability 1–c11(t)–c21(t). Independently, each plant dies and is replaced by one of its
178
own germinated seeds with probability c22(t), or dies and is not replaced with probability
179
1–c22(t). Further, each plant produces seeds equal in number to a Poisson draw with mean
180
c12(t). The various independent random draws for seeds and plants represent demographic
181
stochasticity.
182
183
184
185
186
187
The dynamic result of these rules can be roughly approximated by the vector-matrix
equation
nseed(t 1) 

 
nplant (t 1)
c11(t) c12 (t)  nseed(t) 
,

 
c21(t) c22 (t) nplant (t)
(A.1)
188
Although this equation is not exact, it is helpful in suggesting the workings of the model. The

parameter c11(t) is a seed survival rate, c12(t) is a fecundity rate at which plants generate seeds,
189
c21(t) is a germination rate, and c22(t) plant replacement rate. If the cij(t) were fixed instead of
190
time-varying (see next paragraph), and if demographic stochasticity (last paragraph) were
191
removed, Eq. A.1 would fully determine a time path for nseed(t) and nplant(t).
192
The values of the vital rates {c11(t), c21(t), c12(t), c22(t)} for each year t are modeled as
193
independent draws from a lognormal distribution, truncated at one, when necessary, for the three
194
probabilities {c11(t), c21(t), c22(t)}. Except for a fire year (discussed below), the means and
195
standard deviations for these lognormal draws are assumed to be:
196
197
198
199

11 12  0.047 7.9 
M  
  
,
21 22   0.11 0.80
11 12 
S  
 
21 22 
0.019 5.0 

 .
0.072 0.70
(A.2)
The randomness of cij(t) represents environmental stochasticity. The settings in Eqs. A.2 are

9
200
explained next in Section A.5. For a fire year, special further adjustments to C(t) are made, as
201
explained in Section A.6.
202
203
204
205
206
A.5 Parameter settings for M and S
Data for estimating [ij] and [σij] are very limited. Typically ij and σij values are not
reported directly, but must be inferred from sparse data using strong assumptions.
Specification of 11 and σ11. Bauder (1997 and unpublished data) counted seeds on A.
207
ilicifolia plants and in the upper 2 cm of soil under the plants. We computed the average ratio of
208
seeds-in-soil to seeds-on-plants per meter squared over plots. To avoid artificially inflating the
209
standard deviation, we only included plots with more than 200 seeds. Our reasoning was that the
210
seeds in the soil were created at least one year earlier from plants alive at the time. Thus, a plot
211
with very few seeds would not have produced enough seeds to add to the soil seed bank. The
212
resulting estimates of yearly mean seed survival rate and its standard deviation are:
213
214
11 = 0.047, σ11 = 0.019.
(A.4)
215
216
Let s12 denote a random 12-month seed survival rate, and the settings E[s12] = 11 = 0.047 and
217
SD[s12] = σ11 = 0.019 will be used for s12, where E[.] and SD[.] represent the mean of [.] and
218
standard deviation of [.].
219
Specification of 21 and σ21. These parameters represent the joint outcome of (i) the seed
220
germination rate, g, and (ii) the survival rate of germinated seeds from seedling stage to
221
reproductive adult stage, d. The two parts of the joint outcome are separated because of the way
222
the data were collected. Variables g and d are random variables. The joint germination-and-
10
223
survival rate is gd. (See the dashed line paths on Fig. A.2). The parameters 21 and σ21 are then
224
21 = E[gd] and σ21 = SD[gd].
225
Bauder (1997) reports highly variable germination in lab experiments controlling seed
226
age, temperature, and photoperiod. The highest germination rates typically occurred at cooler
227
temperatures, at longer photoperiods (except at very high temperatures), and for seeds that were
228
at least twelve months old. (Seeds less than 12 months old had lower germination rates,
229
depending on the temperature and light treatment.) These results are likely a reflection of A.
230
ilicifolia's adaptation to germinate in Mediterranean winter and spring, when rainfall is high,
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roughly half a year after seed set. However, these lab experiments are not directly applicable to
232
field germination because the lab plants were supplied with ample water, yielding 70%
233
germination in more than half the treatments. Thus, we used germination values from field
234
experiments by Pavlik and Espeland (1994) for the closely related species Acanthomintha
235
duttonii. They reported a mean germination rate of 0.0670 with standard deviation 0.0590 for
236
sown seeds germinated on north-facing slopes, and 0.1510 and 0.0850 for south-facing slopes.
237
They also report much higher and less variable germination in the two preceding years: between
238
25-28% germination in 1992 and 34-45% germination in 1993. Thus, we used the higher
239
estimate for 1994 with the lower reported standard deviation:
240
241
242
243
E[g] = 0.1510, SD[g] = 0.0590.
(A.5)
Pavlik and Espeland (1994) report seedling survival means ± standard deviations of
244
0.5260 ± 0.3590 and 0.4670 ± 0.2930 on north-facing and south-facing slopes, respectively.
245
These are similar to experimental survival results by Bauder (unpublished data) of 0.6735 ±
11
246
0.3287 for Acanthomintha ilicifolia. We used the Bauder data because they applied to our target
247
species (Pavlik and Espeland 1994 studied Acanthomintha duttonii):
248
249
250
251
E[d] = 0.6735, SD[d] = 0.3280.
(A.6)
Combining the information in (A.5) and (A.6) almost yields a specification for 21:
252
253
21 = E[gd] = E[g] E[d] + Corr[g, d]SD[g]SD[d]
254
= 0.1510 x 0.6735 + Corr(g, d) x 0.0590 x 0.3280 = 0.1017 + 0.0193 Corr[g, d].
255
256
The final step in specifying 21 is to provide a value for the correlation Corr[g, d]. It is likely
257
that germination and seedling survival are correlated since the seasonal resources that favor
258
germination (e.g. water) also favor seedling survival. Unfortunately, we have no data on the
259
strength of correlation. We thus simply assumed a small correlation of 0.3, leading to the
260
specification
261
21 = E[gd] = 0.11.
262
263
The same information yields a specification for σ21 = SD[gd], based on a standard
264
265
266
approximation (Goodman 1960, Eq. 19) for the variance of two dependent variables
Var[gd] = E[g]2 Var[d] + E[d]2 Var[g] + 2 E[g] E[d] Corr[g, d] SD[g] SD[d].
267
268
Substituting from (A.5) and (A.6), and again using Corr[g, d] = 0.3, yields Var[gd] = 0.0052, or
269
σ21 = SD[gd] = 0.072. In summary, the specifications of the seed germination-and-survival
270
parameters are
271
21 = 0.11, σ21 = 0.072.
12
(A.7)
272
273
Specification of 12 and σ12. These parameters represent the mechanism by which adults
274
in spring produce seeds which ultimately arrive in the next year’s seedbank. The mechanism is
275
the joint outcome of (i) the fecundity of plants in producing seeds, (ii) the survival of these seeds
276
over the nine months from seedset to the beginning of spring, and (iii) the avoidance of
277
germination by the seeds in question. The fecundity rate per plant is denoted f, the nine-month
278
survival rate by s9, and the non-germination rate by 1–g (see the dashed line paths on Fig. A.2).
279
The three variables f, s9, and g are random variables, as is the joint outcome variable f s9(1–g).
280
The parameters 12 and σ12 are 12 = E[f s9(1–g)] and σ12 = SD[f s9(1–g)]. We treat the three
281
variables f, s9, and g as independent.
282
The mean and standard deviation of g was already specified in (A.5) above. The mean
283
and standard deviation of fecundity f were specified from seed production data gathered by
284
Bauder (personal communication) and Taylor and Burkhart (1994, referenced in Bauder 1997).
285
On average, A. ilicifolia produced 69 seeds per plant in undisturbed plots in the 1995 growing
286
season (Bauder unpublished data). Although the standard deviation of the number of seeds per
287
plant was not reported, the coefficient of variation over plots of the mean number of seeds/plant
288
could be calculated (0.34). We included plots with at least ten plants per plot; otherwise a very
289
few plants would have dominated the coefficient of variation. Taylor and Burkhart (1994)
290
reported average seed production of 115 and 261 seeds per plant for 1992 and 1993; no standard
291
deviations were reported. These large differences in seed production for 1992, 1993, and 1995
292
were likely due to rainfall (Bauder 1997), which was high in 1992, extremely high in 1993, and
293
low in 1995. For E[f], we excluded the 1993 outlier and averaged 1992 and 1995, resulting in
294
E[f] = 92. We multiplied E[f] by the coefficient of variation in the mean number of seeds/plant,
13
295
0.34, to get a standard deviation of SD[f] = 31.28. Thus, the specified mean and standard
296
deviation for f are
297
298
299
E[f] = 92,
SD[f] = 31.
(A.8)
If seed survival were exponential and deterministic, the relation between the nine-month
300
and 12-month survival rates, s9 and s12, would be s9 = (s12)9/12. To determine the standard
301
deviation in s9, we used an approximate linear relationship between s9 and s12 and through it
302
approximated the variance of s9 from the variance of s12. In particular, we used the two-term
303
Taylor series expansion f(x) = f(a) + f '(a) (x – a) of s9 = (s12)9/12 about the yearly mean seed
304
survival rate, specified as E[s12] = 11 = 0.047 in (A.4) above:
305
306
307
308
s9 = 0.1009 + (9/12)(0.047)–3/12 (s12 – 0.047) = 0.02524 + 1.6108 s12.
From this equation and from E[s12] = 11 = 0.047 and SD[s12] = σ11 = 0.019, as given in (A.4):
309
310
E[s9] = 0.02524 + 1.6108 E[s12] = 0.1009.
311
SD[s9] = (Var[0.02524 + 1.6108 s12])1/2 = (1.61082 Var[s12] )1/2 = 0.03060.
312
313
314
315
316
317
318
319
320
Summarizing:
E[s9] = 0.1009, SD[s9] = 0.0306.
(A.9)
Now the pieces are in place to specify 12 = E[f s9(1–g)] and σ12 = SD[f s9(1–g)]. For
12, use (A.8), (A.9), and (A.5):
12 = E[f s9(1–g)] = E[f] E[s9] E[1–g] = 92 x 0.1009 x (1–0.151) = 7.9.
14
321
322
By a standard formula for the variance of a product of three independent variables (Goodman
323
1962, Eq. 15):
324
325
(σ12)2 = Var[ f s9 (1–g)] = {[CV(f)]2 + [CV(s9)]2 + [CV(g)]2
326
+ [CV(f)]2[CV(s9)]2 + [CV(f)]2[CV(g)]2 + [CV(s9)]2[CV(g)]2
327
+ [CV(f)]2[CV(s9)]2[CV(g)]2} E[f s9 (1–g)]2.
(A.10)
328
329
Here CV(x) denotes the ratio of the standard deviation to the mean (the coefficient of variation).
330
Substituting (A.8), (A.9), and (A.5) in (A.10) yields
331
332
(σ12)2 = Var[ f s9 (1–g)] = ( 0.1156 + 0.0919 + 0.1526
333
+ 0.1156 x 0.0919 + 0.1156 x 0.1526 + 0.0919 x 0.1526
334
+ 0.1156 x 0.0919 x 0.1526 ) (92 x 0.1009 x (1–0.151) )2.
335
336
337
338
339
= 25.1220.
Thus, σ12 = 5.0. Summarizing,
12 = 7.9, σ12 = 5.0.
(A.11)
340
341
Specification of 22 and σ22. Finally, consider 22 and σ22, the mean and standard
342
deviation of the replacement rate of adult plants. By Fig. A.2, this replacement rate is the
343
product of (i) seed production f, (ii) seed survival to winter s9, (iii) germination g, and (iv)
344
seedling survival d. Since f, s9, and gd are assumed independent, the expected value of the
345
replacement rate is 22 = E[fs9 g d] = E[f]E[s9]E[gd], where [E[f] = 92, E[s9] = 0.10, and E[gd]
15
346
= 21 = 0.11 are given in Eqs. (A.8), (A.9), and (A.7), respectively. This would yield 22 = 92 x
347
0.10 x 0.11 = 1.02. However, a further adjustment is made. As noted above, Bauder (1997)
348
reports that germination rates are reduced for seeds less than a year old. At the most realistic
349
temperature conditions considered (12 hours at 10 degrees and 22 degrees) and 24-hour light,
350
seeds that were zero and two months old had germination rates of 70%. Seeds that were more
351
than six months old had germination rates of >90%. At the same temperature in dark conditions,
352
seeds that were zero and seven months old had germination rates of <85% and 12 and 14 month
353
old seeds had germination rates of 80 and 90%, respectively. Thus, we approximate that younger
354
seeds have germination rates that are 80% of older seeds. With this adjustment and slight
355
rounding, our setting is
356
22 = 92 x 0.10 x 0.11 x 0.8 = 0.80.
(A.12)
357
For the variance of s9 f (g d), we again apply the Goodman formula of Eq. A.10 to, applying it to
358
the means and standard deviations for f, s9, and gd from Eqs. (A.8), (A.9), and (A.7):
359
360
Var[s9 f (g d)] = [(0.0919)2 + (0.1156)2 + (0.1527)2 + (0.0919)2(0.1156)2 + (0.0919)2(0.1527)2
361
+ (0.1156)2 (0.1527)2 +(0.0919)2(0.1156)2(0.1527)2] 0.802 = 0.4897.
362
363
364
365
The square root of this variance is σ22 = 0.70. Summarizing,
22 = 0.80, σ22 = 0.70.
(A.13)
Because the standard deviation of the adult replacement rate c22(t) is so high, we expect
c22(t) occasionally to exceed one. In some models of this general type, a vital rate like c22(t) is a
16
366
“survival” rate not allowed to exceed one. However, in our context, c22(t) is a replacement rate,
367
which may exceed one since an adult plant may have multiple seeds. A high standard deviation
368
σ22 = 0.70 is validated by field observations showing year-to-year variation in A. ilicifolia
369
populations (see Figure A.3).
370
371
372
373
374
375
376
377
378
379
380
Figure A.3. A. ilicifolia populations from 2000-2005. The Sabre Spring
population was divided by 10 so that it would fit on this figure. Data from
Mike Kelly presented in the 2005 City of San Diego's Rare Plant
Monitoring Report for Acanthomintha ilicifolia.
A.6 Fire occurrence and fire year vital rates
For each patch and year, the probability of fire was assumed to depend on the time since
the last fire according to a discrete time Weibull hazard function:
381
382
λ[T(t)] = cT(t)c–1/bc .
383
384
Here λ[T(t)] denotes the probability of a fire in year t given that the last fire occurred T(t) years
385
earlier; b and c are scale and shape parameters (Polakow et al. 1999). We set c = 1.42,
17
(A.14)
386
suggesting a relatively low influence of time since last fire, as is common in chaparral (Polakow
387
et al. 1999). In simulations, we chose b to represent average fire return intervals from 20 to 120
388
years, in keeping with historic fire rates (Wells et al. 2004). There was also a no-fire scenario.
389
At the start of a simulation, each patch was given an initial value T(0) drawn from the Weibull
390
distribution. Fires were assumed to burn entire patches, but fire on one patch was assumed
391
independent of fire on other patches. The largest patch in our model (under the Random Forest-
392
GFDL climate change scenario) was 180,000 hectares, roughly the same size as the six largest
393
(>100,000 ha) southern California fires that have occurred since 2001.
394
In a year t in which a fire occurs, a vital rates matrix C(t) is generated as described in
395
Section A.4, but then immediately reduced to reflect the effects of the fire. The reductions are
396
element by element, using the multipliers:
397
398
399
400
 f11

f 21
f12  0.3 0.05 
  
.
f 22  0.1 0.01
(A.4)
That is, c11(t) is multiplied by f11 = 0.3, c12(t) by f12 = 0.05, and so on. The specification of the

401
fij was fairly subjective since there are very few studies of the impact of fire on A. ilicifolia.
402
Sensitivities of ultimate model predictions to the fij are given on Table 1(c) of the main text.
403
The reasoning behind the four fij is given in the next four paragraphs.
404
Element f12. We assume that fires occur in late fall, after the year's adult A. ilicifolia
405
plants have set seed and died. Thus, we expect that fire does not affect the initial seed set.
406
However, it is likely that fire reduces the number of seeds per plant that survive to the next year
407
because new seeds are unlikely to be protected by burial and leaf litter. In Eq. A.4, we assumed
408
a 95% reduction in the seed set, resulting in f12 = 0.05.
18
409
Element f11. The only evidence that the seedbank persists in a fire is from observations of
410
an A. ilicifolia population on the Crestridge Ecological Reserve (32.83 N, 116.86 W) that burned
411
in 2003 (Patricia Gordon-Reedy, unpublished data). Following the fire, all plants were killed, as
412
noted by the reserve landowner. However, detailed censusing of A. ilicifolia did not resume until
413
2009. In 2009, the population appeared to be absent, but more extensive surveys in 2010
414
discovered an extant population, roughly the same size as the pre-fire population. Because the
415
Crestridge site is very remote, it is highly unlikely that the 2010 population emerged from
416
colonists from another site. By this anecdotal evidence, A. ilicifolia seedbanks persist after a
417
fire. Lacking quantitative measurements of post-fire seedbank survival, we assumed a 70%
418
reduction in seed survival in the seedbank, implying f11 = 0.3. The higher proportional survival,
419
as compared to the new seed set, is due to the presumed protection of soil and leaf litter.
420
Element f21. There is no information on the effect of fire on germination. However, the
421
fact that no individuals were seen in 2004 and 2009 (albeit, with minimal sampling effort)
422
suggests that germination is reduced in a fire year, and possibly for multiple years following the
423
fire. Thus, we reduced germination by 90%, implying f21 = 0.1.
424
Element f22. Finally, it is assumed that very few plants return the year following a fire.
425
Thus, the replacement of plants in a fire year was set at only 1% of its value in a non-fire year,
426
implying f22 = 0.01. We did not set this value to zero because fires can be heterogeneous across
427
the landscape, especially in the relatively open areas preferred by A. ilicifolia, allowing some
428
patches to escape burning.
429
430
A.7 One-stage demographic model
19
431
We created a one-stage (one-equation) compaction of the two-stage (two-equation)
432
demographic model to test the effect of uncertainty in model structure on results. Eq. A.1
433
became
434
435
nplant(t+1) = c(t) nplant(t).
436
437
In a non-fire year, the single vital rate c(t) was assumed to be an independent draw from a
438
lognormal distribution with mean equal to the dominant eigenvalue of M and standard deviation
439
equal to the dominant eigenvalue of S. In a fire year, a lognormal vital rate was drawn as usual,
440
but then multiplied by the eigenvalue of F from Eq. 5, which was 0.0766.
441
442
A.8 Impact of invasive plants interacting with fire
443
There are no quantitative data describing the impact of invasives on A. ilicifolia.
444
However, vegetation managers have noted the spread of Brachipodium distachyon, Bromus sp.,
445
and Avena sp. in A. ilicifolia habitat, often following a fire (P. Gordon-Reedy and J. Vinje,
446
unpublished data). Thus, we created two scenarios to suggest the possible effect of invasive
447
species, interacting with fire, on A. ilicifolia.
448
To parameterize the invasives scenarios, we used A. ilicifolia data describing the impact
449
of removing invasives on seedling survival and fecundity in the field (Bauder and Sakrison
450
1997). Reanalyzing the data (Bauder, unpublished data), we found that seedling survival
451
increased modestly with invasive weeding (Figure A4). A regression using data from three years
452
showed that unweeded plots had roughly 90% of the seedling survival of weeded plots. Seed
453
production (ratio of number of seeds to number of germinants) in plots that were not weeded nor
454
had weeding in an adjacent plot was 87% as large as in plots in which weeding occurred within
20
455
the plot or within an adjacent plot. In view of these data, we supposed that invasives reduce the
456
fecundity rate to 85% of its no-invasives value, and reduced the survival rate to 90% of its no-
457
invasives value. That is, invasives reduce the two rows of the mean vital rates matrix M to 85%
458
and 90%, respectively, of their benchmark values in Eq. A.2.
459
To model the interaction of invasives with fire, we created two particular scenarios. In
460
the first scenario, we assume that fire, by burning invasives (analogous to weeding), allows the
461
fecundity and survival rates of A. ilicifolia, immediately following the fire, to achieve 100% of
462
their no-invasives values. That is, the mean vital rates matrix M is at its benchmark setting in
463
Eq. A.2. Then, as time passes, the invasives reestablish; and the first and second rows of M
464
(representing fecundity and survival) gradually decline to 85% and 90%, respectively, of their
465
benchmark values, remaining there until the next fire. The trajectory of the gradual percentage
466
decline is assumed to be as shown by the dashed lines on Fig. A.5. These changes in vital rates
467
only occur in the patch that experienced fire and not in all patches in the metapopulation.
468
In the second scenario, we assume that a fire, by burning A. ilicifolia, allows invasives to
469
become well-established. In particular, immediately following a fire, the vital rates matrix M
470
changes to its invasives value, with the first and second rows of M at 85% and 90% of their no-
471
invasives values in Eq. A.2. However, as time passes, it is assumed that A. ilicifolia fights off
472
the invasives and M returns to its benchmark value of Eq. A.2, until the next fire. The trajectory
473
of the return is assumed to be as shown by the undashed lines on Fig. A.5. Again, these changes
474
occur only in the patch that experienced fire.
475
21
476
477
478
479
480
481
482
Figure A.4. Seedling survival as a function of six weeding categories: (i) no weeding in the
plot or in adjacent plots, (ii) weeding in one adjacent plot but not the plot itself, (iii) weeding in
two or more adjacent plots but not the plot itself, (iv) weeding in the plot but not in any
adjacent plots, (v) weeding in the plot and one adjacent plot, and (vi) weeding in the plot and in
all adjacent plots.
Multiplier
1
Fecundity, scenario 1
Survival , scenario 1
Fecundity, scenario 2
Survival , scenario 2
0.95
0.9
0.85
0
483
484
485
486
487
20
40
60
80
Time since last fire
100
120
Figure A.5. Vital rates multipliers as a function of time since last fire. In scenario one, fire
removes invasives allowing A. ilicifolia to re-establish. In scenario two, invasives re-establish
immediately following a fire and A. ilicifolia recovers to out-compete invasives with time.
488
Keeping the impact of invasives the same (15% reduction in fecundity and 10% in
489
survival), we also tried additional functional forms for the vital rates multiplier from those in Fig.
22
490
A.5. For example, we employed a step-wise function that occurred 15 years after a fire. These
491
alternative functional forms did not impact the population results. Results of simulations under
492
the two invasives scenarios are reported in Fig. 5 of the main text.
493
494
495
APPENDIX B. Additional Figures
In Figure B.1, the ratios of average final abundances for each climate and land-use
496
change scenario (numerator) to a null scenario involving no climate or land-use change
497
(denominator) are plotted against FRI for every combination of SDM and population model type.
498
Ratios were substantially lower for the MaxEnt SDM (Figs. B.1a,b) than for the Random Forest
499
SDM (Figs. B.1c,d). That is, MaxEnt predicted a worse outcome for A. ilicifolia than did
500
Random Forest, as discussed further below.
501
For the MaxEnt GFDL scenarios, the final abundance ratio declined steeply with
502
increasing FRI, especially from 20 to 40 years (Figs. B.1a,b). This resulted from the greater
503
predicted habitat fragmentation in the MaxEnt GFDL scenario as compared to the MaxEnt “no
504
change” scenario. A similar phenomenon was observed by Regan et al. (2010). Fragmented
505
landscapes spread fire risk across multiple small patches, rather than having fires restricted to
506
fewer large patches. Because modeled fire events drastically reduced A. ilicifolia's vital rates,
507
preventing a few large fires can avoid extinction, raising the average final abundance for short
508
FRIs in fragmented landscapes as compared to unfragmented landscapes.
509
Figure B.2 shows the coefficient of variation, or the standard deviation divided by the
510
mean, for the 1000-run sets of each model scenario. High coefficients of variation suggest high
511
variability between model runs. Thus, for many runs, A. ilicifolia abundance went to zero.
23
Ratio of Ave. Final Abundance
to "No Change"
Ratio of Ave. Final Abundance
to "No Change"
(a) Maxent, Two-Stage Model
1.8
1.6
1.4
1.2
1.0
0.8
0.6
0.4
0.2
00
40
60
80
Fire Return Interval (years)
100 No Fire
120
0
(c) Random Forest, Two-Stage Model
12
10
8
6
4
2
00
20
40
60
80
Fire Return Interval (years)
100 No 120
Fire
20
40
60
80
Fire Return Interval (years)
100 No Fire
120
(d) Random Forest, One-stage Model
PCM climate change
PCM climate & land use change
GFDL climate change
GFDL climate & lands use change
14
Ratio of Ave. Final Abundance
to "No Change"
14
Ratio of Ave. Final Abundance
to "No Change"
20
(b) Maxent, One-stage Model
1.8
1.6
1.4
1.2
1.0
0.8
0.6
0.4
0.2
0-
12
10
8
6
4
2
0-
0
20
40
60
80
Fire Return Interval (years)
100
No 120
Fire
Figure B.2. Ratio of the average, over 1000 runs, of the final abundance of a given scenario to the final abundance of the no change scenario with
the same fire return interval. This ratio is on the vertical axis, and the fire return interval is on the horizontal axis. The four panels correspond to
four combinations of SDMs and population models: (a) MaxEnt SDM and two-stage population model, (b) MaxEnt SDM and one-stage
population model, (c) random forest SDM and two-stage population model, and (d) random forest SDM and one-stage population model. Within
each panel, the differently labeled points correspond to different climate and land use change scenarios, as indicated in the legend box
superimposed on panel (d). For fire return intervals of 20 and 30 years, the mean of 1000 model runs varies by less than 10% between different
1000-run simulations. For longer fire return intervals, variation is around 5% between different 1000-run simulations.
24
(a) Maxent, Two-Stage Model
(b) Maxent, Scalar Model
3.5
3.0
PCM climate change
PCM climate & land use change
GFDL climate change
GFDL climate & lands use change
Land use change
3.0
Coefficient of Variation
Coefficient of Variation
3.5
2.5
2.0
1.5
1.0
0.5
00
20
40
60
80
100
2.5
2.0
1.5
1.0
0.5
0-
120
No Fire
0
20
Fire Return Interval (years)
(c) Random Forest, Two-Stage Model
4.0
3.0
2.0
1.0
0-
0
20
40
60
80
Fire Return Interval (years)
100
120
No Fire
100
No120
Fire
4.0
3.0
2.0
1.0
0-
120
No
No Fire
Fire
100
(d) Random Forest, Scalar Model
5.0
Coefficient of Variation
Coefficient of Variation
5.0
40
60
80
Fire Return Interval (years)
0
20
40
60
80
Fire Return Interval (years)
Figure B.2. Coefficient of variation of final abundance (standard deviation / mean). This ratio is on the vertical axis of each panel, and the fire
return interval is on the horizontal axis for each panel. The four panels correspond to four combinations of SDMs and population models: (a)
MaxEnt SDM and two-stage population model, (b) MaxEnt SDM and scalar population model, (c) random forest SDM and two stage population
model, and (d) random forest SDM and scalar population model.
25
1
2
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