Life‐history strategy and extinction risk in the warm desert perennial spring ephemeral Astragalus holmgreniorum (Fabaceae)

Abstract This study of Astragalus holmgreniorum examines its adaptations to the warm desert environment and whether these adaptations will enable it to persist. Its spring ephemeral hemicryptophyte life‐history strategy is unusual in warm deserts. We used data from a 22‐year demographic study supplemented with reproductive output, seed bank, and germinant survival studies to examine the population dynamics of this species using discrete‐time stochastic matrix modeling. The model showed that A. holmgreniorum is likely to persist in the warm desert in spite of high dormant‐season mortality. It relies on a stochastically varying environment with high inter‐annual variation in precipitation for persistence, but without a long‐lived seed bank, environmental stochasticity confers no advantage. Episodic high reproductive output and frequent seedling recruitment along with a persistent seed bank are adaptations that facilitate its survival. These adaptations place its life‐history strategy further along the spectrum from “slower” to “faster” relative to other perennial spring ephemerals. The extinction risk for small populations is relatively high even though mean λ s > 1 because of the high variance in year quality. This risk is also strongly dependent on seed bank starting values, creating a moving window of extinction risk that varies with population size through time. Astragalus holmgreniorum life‐history strategy combines the perennial spring ephemeral life form with features more characteristic of desert annuals. These adaptations permit persistence in the warm desert environment. A promising conclusion is that new populations of this endangered species can likely be established through direct seeding.


| INTRODUC TI ON
A major goal of plant evolutionary ecology is to understand patterns of life-history evolution (Gadgil & Solbrig, 1972;Grime, 1977;Pianka, 1970 Population matrix modeling is a valuable approach for integrating demographic data into an analytical framework in order to understand plant life histories (Caswell, 2001;Morris & Doak, 2002).
These models use multiple years of demographic data from the field to calculate transition probabilities for vital rates important in the life history of a species. These are combined into matrix elements that represent transitions among the stage or age classes in the life cycle diagram. The traditional approach to modeling plant life history has used deterministic models that include only mean values for vital rates and thus for elements of the transition matrix. A deterministic model converges to a single asymptotic value, the deterministic population growth rate (λ d ). These models can be useful as heuristic tools, as well as for plant species where environmental drivers and consequently vital rates do not vary widely and where the mean condition permits persistence (Crone et al., 2011). For plant species that grow in stressful environments with high inter-annual variation in year quality, however, deterministic models are not good predictors of population growth.
Stochastic population matrix models incorporate vital rate variances and covariances as well as means into the estimate of population growth rate and are thus better able to predict the consequences of environmental variation on population growth (Morris & Doak, 2002). The result of each iteration of a stochastic model is an estimate of the stochastic population growth rate (λ s ). It is based on the random draw of values from a set of vital rate probability distributions to populate the matrix for each time step (Morris & Doak, 2002).
A considerable body of theoretical and empirical evidence has supported the idea that mean λ s will almost always be lower than λ d , that is, that adding vital rate variation to the model will slow population growth (Lewontin & Cohen, 1969;Tuljapurkar & Orzack, 1980). This is because population growth as specified in these models is a multiplicative process that is more sensitive to bad years than good years. More recent work has shown that this constraint may be relaxed if certain assumptions are not met. These include the assumptions that vital rates and logλ s are linearly related and that the mean condition permits population persistence, that is, λ d ≥ 1 (Boyce et al., 2006;Drake, 2005;Morris & Doak, 2004).
More recently, the review by Lawson et al. (2015) examined both theoretical and empirical evidence to investigate the conditions under which increased environmental variance (stochasticity) could increase the population growth rate. They found that the effect of environmental stochasticity on population growth rate is determined by the shape of the relationships between population growth rate and independent variables that are related to environmental quality. When this relationship is concave, increased stochasticity decreases the mean population growth rate as predicted by classical models (Lawson et al., 2015, Figure 2). When the relationship is linear, environmental stochasticity has no effect. However, when the relationship between population growth rate and environmental quality is convex, stochasticity can increase the population growth rate. Lawson et al. (2015) also discuss the idea that different vital rates could have response curves with different relationships to environmental quality, and that these could interact to increase the positive effect of stochasticity on population growth rate. "Labile" vital rates, (e.g., reproductive output) could potentially increase nonlinearly with environmental quality, accelerating population growth in years when environmental quality is higher. These could be combined with "buffering" vital rates (e.g., seed dormancy loss rate) that are insensitive to changes in environmental quality and thus provide protection against population decline in unfavorable years.
In this study, we examine the life history of the warm desert endemic Astragalus holmgreniorum Barneby ( Figure 1) to ask how well the life history of this species permits it to persist in its warm desert environment and whether it has evolved specific traits that comprise a warm desert-adapted life-history strategy. Warm deserts are stressful environments with high inter-annual variation in quality that is mediated by large and unpredictable variation in precipitation around a generally low mean.
Astragalus holmgreniorum has a hemicryptophyte spring ephemeral life history (Rominger et al., 2019) that is unusual in the warm desert. Warm desert floras worldwide show a bimodal life-history strategy distribution dominated by stress-tolerant shrubs and annual plants (Danin & Orshan, 1990;Pierce et al., 2017;Whittaker & Niering, 1964). Hemicryptophyte spring ephemerals, that is, spring ephemeral herbaceous perennial species that position their dormant meristems at or very near the soil surface, are especially uncommon.
The soil surface in warm deserts can be many degrees hotter in summer than either the air above the surface or the soil at a depth of a few centimeters (Geiger et al., 2009), adding another element of stress and posing a significant risk of low dormant-season survival for these species.
As mentioned above, divergent life-history strategies have evolved in response to the constraints of the warm desert environment. Desert shrubs have evolved to tolerate the stressful summer season by raising their meristems well above the hot soil surface, as well as through numerous other morphological and physiological mechanisms for reducing or tolerating both heat and water stress (Peguero-Pina et al., 2020). They represent the stress-tolerator corner on the Grime life-history triangle (Grime, 1977). This is an essentially K-selected life-history strategy (Gadgil & Solbrig, 1972) that emphasizes a long life span over recruitment through sexual reproduction. Seed production may only occur sporadically, and persistent seed banks are rare.
In contrast, the desert annual life-history strategy is closer to the ruderal corner of the Grime life-history scheme (Grime, 1977). These species are similar to R-selected ruderal species (Gadgil & Solbrig, 1972) in that they establish and produce seeds quickly in favorable environments that are only available for short periods. Ruderal species may find new favorable environments in space, through dispersal, or in time, through persistent seed banks. Warm desert annuals are also able to complete their life cycles quickly during the most favorable season and thereby escape the stresses of the unfavorable season as seeds. Because years show extreme variation in quality in the warm desert, this life-history strategy includes the ability to maximize both recruitment and seed production in favorable years and to survive multiple years of unfavorable conditions in the persistent seed bank (Pake & Venable, 1996).
Perennial spring ephemerals in warm deserts are similar to annuals in that they grow actively only in the most favorable season and escape the stressful season through dormancy. Most of these are long-lived geophytes with deeply buried dormant meristems that are not exposed to the extreme heat of the surface soil. As in desert shrubs, the emphasis for most desert geophytes is on longevity and sometimes clonal reproduction, not sexual reproduction (Fragman & Shmida, 1997 4. Because of the key role of the seed bank in population persistence for this species in its present environment, modeling will show that population persistence is not possible without a long-lived seed bank.
5. Modeling will also predict that artificial seed bank augmentation can substantially decrease extinction probability in at-risk populations and that seed introduction can potentially be used to establish viable new populations of this endangered species in suitable unoccupied habitat.

| Study species description
Astragalus holmgreniorum is restricted in its current distribution to an area within 15 km of a rapidly expanding urban center, St.
George, Utah, USA, at the northeastern edge of the Mojave Desert (US Fish and Wildlife Service, 2006). The species was first described by Barneby (1980) and was listed as federally endangered in 2001 (US Fish and Wildlife Service, 2001). Its narrow geographic range, habitat specialization, and locally low abundance characterize it as a rare species at apparently high risk of extinction (Rabinowitz, 1981 preventing this potential seed loss (Figure 1; Houghton et al., 2020).

F I G U R E 2 Life cycle diagram for Astragalus holmgreniorum.
Square boxes represent life stages between which plants can transition each year. These include three size classes (S1-S3), dormant seeds of different ages in the persistent seed bank (SB1-SB9), and nondormant carryover seeds (NONDORMANT). See Table 1 for vital rates contributing to each transition The species is facultatively autogamous but has much higher reproductive success with pollinator-assisted selfing (geitonogamy) and especially with outcrossing (Tepedino, 2005). The principal pollinators are large ground-nesting native bees of the genus Anthophora; there is no evidence for pollen limitation under field conditions (Tepedino, 2005).
Astragalus holmgreniorum has phenology typical of a spring ephemeral hemicryptophyte (Rominger et al., 2019). Plants emerge as seedlings or returning adults in very early spring, grow actively through the spring, and adults complete flowering and seed production by early summer (Figure 1)

| Long-term demographic study
The demographic data for this analysis were collected over the SISURV*S1S1 Size Class 1 to Size Class 2 S1SURV*S1NOTS1*SIS2NEW Size Class 1 to Size Class 3 SISURV*SINOTS1*S1S3NEW Size Class 2 to Size Class 2 (Stasis) S2SURV*S2S2 Size Class 2 to Size Class 1 (S2SURV*S2NOTS2*S2S1NEW)+ (S2RO*SRS*ND1FRAC0*GFRAC *GSURV) Size Class 2 to Size Class 3 S2SURV*S2NOTS2*S2S3NEW plot each year. We also quantified the survival probability and probability of transitioning from one size class to another from one year to the next for each size class. These data were used to calculate means and variances for each of these vital rates. See Appendix 1 for a detailed explanation.

| Reproductive output study
We also recorded the number of flowers on each plant in the demography study each year, then calculated the proportion of plants of each reproductive size class (S2 and S3) that flowered as well as the mean number of flowers per flowering plant for each size class each year.
To translate flower number data from the demography study into an estimate of seed production, we utilized an independent reproductive output dataset (Searle, 2011). Fruit set was mea- We included an additional vital rate, seed rain survival, in the model to account for possible post-dispersal losses prior to incorporation into the seed bank. We based our estimate of seed rain survival on the study of Houghton et al. (2020), which showed only small losses to seed predators or to removal from suitable habitat via overland flow. In the absence of yearly data, this vital rate was treated as time-invariant. This parameter was included in order to avoid the assumption that all seeds produced can successfully enter and persist in the seed bank.

| Seed bank persistence study
We used a 6-year retrieval study with seeds of known age to examine seed bank persistence (Searle, 2011;A. Searle, unpublished data Size Class 2 to Size Class 2 (Stasis): S2SURV survival probability for Size Class 2, S2S2 probability that a plant of Size Class 2 will remain in Size Class 2.
Size Class 2 to Size Class 3: S2SURV Survival probability for Size Class 2, S2NOTS2 probability than an S2 will not remain in Size Class 2, S2S3NEW probability that an S2 that does not remain in Size Class 2 will transition to Size Class 3.
Transitions from Size Class 2 or Size Class 3 to Size Class 1 include transitions among existing plants as above and also recruitment (transition to S1) from nondormant seeds produced in the current year: S3RO = seed production of Size Class 3 plants; SRS = seed rain survival; NDFRAC0 fraction of current year seeds that are nondormant; GFRAC = fraction of nondormant seeds that germinate; GSURV fraction of germinants that survive to census as S1 recruits.
The fraction of seeds produced by S2 and S3 plants that are dormant transition directly into the dormant carryover seedbank (SB1): S3RO*SRS*DFRAC0. Nondormant seeds produced in the current year can germinate and have the potential to transition to S1 plants as above, or they can fail to germinate and enter the carryover nondormant seed pool: S3RO*SRS*ND1FRAC0*(1-GFRAC).
Dormant seeds in the seed bank that carry over each year from seeds of one age to seeds of an age 1 year older: DFRAC1 through DFRAC8 Seeds of each age in the seed bank that become nondormant can (1) germinate and potentially transition to S1 recruits or (2) enter the carryover nondormant seed pool as in: (1) ND1FRAC1*GFRAC*GSURV (2) ND1FRAC1*(1-GFRAC) Seeds in the nondormant carryover seed pool can either germinate (GFRAC) and potentially lead to S1 recruits (GSURV) or remain in the pool for another year (1-GFRAC).

TA B L E 1 (Continued)
nondormant each year, that is, the remaining dormant seed fraction decreased linearly through time (11.1% per year) to a value of zero at 9 years. The dormancy loss rate was insensitive to environmental variation. The fraction of nonviable ungerminated seeds in a cohort in this retrieval experiment was also insensitive to environmental variation. It increased linearly with time (3% per year) to an estimated maximum of 27% viability loss at 9 years.
We used this dataset as a basis for the calculation of vital rates for seed bank transitions in the model. The seeds that become nondormant each year can either germinate within the year and potentially transition to S1 the following year, or they can remain ungerminated and enter the nondormant stage the following year (see Appendix 1 for details of study and calculations).

| Germinant survival to census study
Germinant survival to census is an important vital rate in the model that cannot be obtained from observational data. To estimate this parameter, we used data from a recent experimental seeding . Seeds were acid-scarified to render them nondormant and sown by broadcasting and raking in the field in late fall 2018 at ten introduction sites. Winter precipitation was almost twice the long-term average in 2019-2020, and the seedings overall were highly successful, with an average return on seed (recruits/seeds planted) of 24%. All scarified seeds germinated in retrieval bags that year, making return on seed equivalent to germinant survival to census. We extrapolated from this data point to construct a regression for predicting germinant survival to census from winter precipitation and used the equation to estimate this parameter for each year of the demographic study. We could then calculate the estimated germinant survival mean and variance (see Appendix 1 for details).

| Correlation analysis
We performed correlation analysis based on the demographic dataset using yearly precipitation data for the prior-year summer-fall dormant season (June-October) and for the current-year growing season (winter: November-January, early spring: February-March, late spring: April-May) from the PRISM Data Explorer for our study site as predictors. Response variables were yearly measurements of plant survival, size class transition, and flowering. Seed output and germinant survival variables could not be included, as precipitation data were used in their estimation. We also examined whether response variables (vital rates) were correlated with each other.

| Population matrix modeling
We adopted a stochastic discrete-time matrix modeling approach to simulate population trends through time for A. holmgreniorum using the program MATLAB to perform simulations based on minor modifications of MATLAB code published by Morris and Doak (2002; see Appendix 2 for details of model development). This approach to stochastic population modeling uses measured or estimated variation in vital rates to introduce stochasticity.

| Defining the population matrix
We  (Table 1).

| Modeling procedure
The model starts with values for each time-invariant vital rate and probability distributions for each variable vital rate, generated from measured means, variances, and within-and between-year correlations among vital rates. It then picks vital rate values at random based on the constraints of their distributions to generate a matrix for each yearly time step in each iteration. The first time step uses an initial vector of stage values as input, and output from each matrix calculation is used as input for the next yearly time step. The model initiates each time step in May, soon after census, when seeds are mature but not yet dispersed (pre-birth-pulse). This is also the beginning of the dormant season, just before the last point in time when actively growing plants were present. We used our best estimates for each vital rate to create the base population matrix model. We could then simulate the effect of changes in specific vital rates on both mean stochastic growth rate (λ s or stochastic lambda) and extinction probability.

| Extinction risk estimates
We used a cumulative distribution function to generate an extinction risk time series for each model run (Morris & Doak, 2002). Extinction risk for a given point in time into the future is defined as the proportion of model iterations in which the population falls below a quasiextinction threshold by that time. The quasi-extinction threshold was conservatively but arbitrarily set at 200, that is, when numbers fell below 200 plants + seeds, the local population was considered extinct. As ca. 99% of the genets on average are present as seeds, this threshold represents a seed bank reduced to a dangerously low level. Extinction risk was positively correlated with quasi-extinction threshold, but this relationship showed an exponential rise to maximum, with little change in extinction risk at quasi-extinction thresholds >200 (data not shown). At very small thresholds, extinction risk was greatly reduced, but these low thresholds are not ecologically realistic.

| Hypothesis 1: Population fluctuations and precipitation drivers
We used the results of the demographic study directly to evaluate our hypothesis that the effect of extreme inter-annual variation in seasonal precipitation would be evident as extreme population fluctuations in density and seed production through time. Proportion of individuals surviving from one yearly census to the next on the demography plot for adults and recruits was less than 3 years, and no individual in the study was known to survive for more than 6 years. The mean survivorship curve for this species was close to a Deevey Type II curve (Deevey, 1947), that is, a log-linear relationship of survivorship with age, resulting in relatively constant mortality risk through time (See Appendix 3 for full discussion).

| Seed production
Most S3  When we calculated seed production on an area basis, we obtained the striking result that significant seed production was highly episodic (Figure 4d). Most of the seeds produced over the 22-year study period were produced in only a handful of years ( Figure 4e). An estimated 150,000 seeds were produced in the study plots over the 22-year period, of which 115,000 or approximately 77% were produced by S3 plants. Estimated total annual production exceeded 15,000 seeds in only 6 years, and these 6 years accounted for almost 85% of total seed production.
Another 12% was accounted for by two additional years that approached production of 10,000 seeds. The remaining 14 years accounted for <3% of total production.

| Precipitation as a driver of vital rate variance
Results of the correlation analysis generally supported Hypothesis 1, namely that high variance in vital rates was driven by stochastic variation in environmental quality as reflected in inter-annual variation in seasonal precipitation. Vital rates were generally positively correlated with winter-spring precipitation. However, these correlations were often not very strong, even though the inter-annual variation in precipitation was extreme (Table 2, Figure 3a). Very dry years (2002,2007) were associated with failure to emerge successfully from dormancy and a complete lack of actively growing plants, but exceptionally high-precipitation years (e.g., 2005) were not necessarily associated with high survival or high reproductive output (Figures 3 and 4). The pattern of positive correlation with precipitation was generally consistent even when correlation coefficients were not significant, however.
Precipitation correlations associated with the growth of smaller size classes or stasis in the largest size class were all positive and mostly significant, as were all correlations associated with flowering. In contrast, correlation coefficients associated with size regression or stasis in the smaller size classes were all negatively correlated with winter-spring precipitation, though none of these were statistically significant.
Precipitation during the summer-fall dormant period had no strong or consistent effect on any vital rates except those associated with adult survival over the dormant season (Table 2).
Surprisingly, the significant effect of dormant season precipitation on adult survival was negative, that is, plants suffered higher mortality with increased warm-season precipitation. The reason for this is not known.
Correlations among vital rates confirmed the patterns seen for winter-spring precipitation. Over-dormant season survival was positively correlated among size classes and these correlation coefficients were significant or marginally significant (Table 3).  Note: Probabilities shaded yellow are significant at p < .05; probabilities shaded peach are marginally significant at p > .05 but p < .10.
TA B L E 2 Correlations between vital rates and precipitation drivers for the previous summer-fall dormant period (Jun-Oct) and the winter-spring active period (Nov-May), based on the 22-year demographic dataset (Figures 3 and 4) and PRISM Data Explorer precipitation estimates ( Figure 3a) TA B L E 3 Correlations among Astragalus holmgreniorum vital rates calculated from 22 years of demographic data (S1SURV, S2SURV, S3SURV=over-dormant season survival for size classes S1, S2, and S3; S2FLPROP, S3FLPROP, S2FLPLT, S3 FLPLT = proportion of plants flowering and flowers per flowering plant for S2 and S3; remaining nine columns represent size class transitions among S1, S2, and S3) S1 SURV

| Hypothesis 2: The effect of environmental stochasticity
Best estimates for vital rates resulted in a base model for A. holmgreniorum with a mean stochastic growth rate (λ s ) of 1.0942 ± 1.242 (mean ± standard deviation; Table 4). This indicates that, on average, this model population of A. holmgreniorum can be expected to grow through time. However, mean λ s > 1 was associated with a very large standard deviation due to high variance in vital rates, which in turn is a result of extreme inter-annual variation in year quality (Figure 3a). The population represented by this version of the model has a 47% chance of extinction (i.e., dropping below the 200-individual quasi-extinction threshold) within 50 years, assuming that the conditions that generated its vital rates in the past continue into the future. This extinction risk is based on the likelihood that a chance series of unfavorable years will prevent seed bank replenishment and lead to steep population decline some time during the 50-year period. Even though mean λ s is >1, there is still a relatively high probability of extinction for this small population.
The large standard deviation around mean λ s makes it difficult to project the model population into the future with any certainty and thus to predict its probability of extinction in concrete terms.
However, the deterministic version of the model , which is based on vital rate mean values (standard deviation zero), resulted in a λ d of 0.9435 and virtually certain extinction (Table 4). This indicates that the level of stochasticity reflected in measured vital rate variances resulted in a dramatically higher probability of population persistence than in the deterministic version, even though this stochasticity creates uncertainty in predicting whether or when the population will become extinct. This result supports Hypothesis 2: Given the low quality of an average year in its warm desert habitat, this population requires a stochastically varying environment to persist.

| Hypothesis 3: Compensating for low dormant season survival
Examining the effect of changing vital rate means on model outcomes is a form of sensitivity analysis for stochastic population modeling that helps to identify which vital rates are most important in determining population trajectories. We proposed in Hypothesis 3 that low dormant-season survival is a given in the life history of this species, and that other stages of the life cycle would be able to compensate for this low survival and thus reduce its importance. To test this hypothesis, we reduced or increased survival vital rates of germinants, S1, S2, or S3 plants by 10% or 50%.
As predicted, these changes in survival for S2 and S3 adult plants had negligible impact on mean λ s (Table 4, Figure 5f,h) and also had minimal effect on cumulative extinction risk (Figure 5e,g). When S1 survival, which includes dormant season survival of S1 recruits, was changed, mean λ s and extinction risks showed more sizeable changes (Figure 5c,d). These changes were even more evident when germinant survival to census was manipulated (Figure 5a,b).
Germinant survival was the only case where reducing survival by 50% resulted in λ s < 1. These results support the hypothesis that the population growth rate is insensitive to reductions in survival for adult plants, but is much more sensitive to survival reductions for germinants and new recruits.
We hypothesized that high reproductive output would be a key adaptation for the survival of this species in a warm desert environment and that seed production per plant would thus be important in determining model outcomes. The sensitivity analysis also supported this hypothesis (Table 5, Figure 6a,b), with a strong effect of 50% reduction similar to the effect of 50% reduction in germinant survival.
The sensitivity analysis showed parallel changes in λ d when survival and reproductive vital rates were manipulated, but even with 50% increases, λ d only approached or barely exceeded 1.00 for any of these vital rates (Tables 4 and 5).

| Hypothesis 4: Importance of the seed bank for population persistence
The most dramatic effects on model behavior were obtained by manipulating the vital rates that determine maximum seed longevity (Table 5, Figure 6c,d). As outlined in Hypothesis 4, we expected that a long-lived seed bank would be necessary for population persistence in the stochastic warm desert environment, where harsh conditions in some years preclude the presence of actively growing plants ( Figure 3). To test this hypothesis, we modified the fraction of seeds expected to remain dormant from 1 year to the next (as well as the corresponding fraction expected to become nondormant) in the transitions from stages SB1 through SB9 to reflect shortened seed persistence periods of seven, five, three, one, and zero years.
We did not change viability loss rates, but instead limited seed bank persistence by increasing the fraction transitioning from dormant to nondormant each year (see Appendix 1 for full explanation).
When seed bank maximum persistence was shortened from 9 years to 7 years, mean λ s dropped to 1.0413 and the extinction probability at 50 years increased to 0.619 (Table 5, Figure 6d). For 5-year maximum persistence, mean λ s was reduced to 0.9253 and 50-year extinction risk increased to 0.774, while for 3-year maximum persistence, mean λ s reached a low of 0.7337, with 0.922 probability of extinction within 50 years. With a maximum persistence of one year, mean λ s dropped to 0.4020 and 50-year extinction became virtually certain, while a model with no seed bank carryover, that is, with all seeds becoming nondormant the year of their production, resulted in essentially immediate extinction. In contrast, reductions in seed bank longevity increased λ d , although these increases were small, and λ d never exceeded 1.0. These increases are consistent with theory that predicts that delaying reproduction should reduce λ d , that is, in a constant environment, early reproduction is always favored. With a value of 5 years or less for seed bank persistence, λ d > λ s , also as predicted by theory (Table 5; Tuljapurkar & Orzack, 1980). These results strongly support the hypothesis that a persistent seed bank is essential to population persistence in A. holmgreniorum, and also that the advantages of stochasticity for population persistence cannot be realized without the presence of a persistent seed bank.

| Hypothesis 5: Seed addition effects on extinction risk
We performed the base model using the initial stage vector from 2001, a year with a somewhat above the average total number of individuals, including actively growing plants. We first tested whether a change in initial stage vector values would substantially change extinction risk relative to the base model. We substituted the initial stage vector for 2007, a year with a below-average total number of individuals and no actively growing plants. This change had a substantial effect, raising 50-year extinction risk from 47% to 60% (Table 6).
The importance of the seed bank in population persistence coupled with the result above suggested that population augmentation through the addition of seeds could substantially lower extinction risk. Changing the vector of initial stages in the 2001 base model by adding 2000 or 5000 seeds to SB1 had a small effect (47% to 45% and 44% 50-year extinction risk, respectively), while similar changes to the vector of initial stages for the 2007 model lowered the 50- year extinction risk from 60% to 56% and 52% (Table 6). This supported the hypothesis that population augmentation through seed addition could potentially reduce extinction risk.
We also tested the hypothesis that new populations in unoccupied suitable habitat could be initiated through seed introduction.
Adding 2000 seeds to SB1 with other stages set at zero generated TA B L E 5 Values for the model parameters λ d (deterministic growth rate) and mean and standard deviation of λ s (stochastic growth rate) for the base population viability model for A. holmgreniorum (2001 initial vector of stage values) and for models testing the sensitivity of model parameters to variation in vital rates that measure seed production and seed bank longevity

| Environmental drivers of inter-annual vital rate variation
Our first hypothesis was that stochastic inter-annual variability in seasonal precipitation is a principal driver of high variance in vital rates. This was supported by a strong trend overall for positive correlations of vital rates related to survival, growth, and flowering with winter-spring precipitation over the 22 years of the study, even though many individual correlation coefficients were not statistically significant ( Table 2). The relatively weak correlation with seasonal precipitation suggests that more subtle differences in rainfall periodicity, combined with other factors that influence vital rates, must act to obscure the more obvious effects of inter-annual variation in precipitation. Efforts to correlate specific vital rates with precipitation during key time periods within the season did not result in any appreciable improvement in correlation coefficients, however. Very small sample size in some years resulted in imprecise vital rate estimates that could negatively impact correlation analysis. Also, vital rates do not necessarily vary linearly with precipitation, which would also tend to lower correlation coefficients. Consistent positive correlations among vital rates associated with survival, growth, and flowering lent support to the hypothesis that they were responding to the same precipitation drivers. We found no evidence for significant temporal autocorrelation in either precipitation drivers or vital rates, indicating that correlations among them are more likely to reflect direct or indirect causal relationships.

| The role of environmental stochasticity
Our second hypothesis was that, given the mean current conditions that result in λ d < 1, environmental stochasticity is necessary for the survival of this species. This hypothesis was supported. The deterministic version of the base model, with vital rates fixed at mean values, resulted in eventual near-certain extinction (λ d = 0.9485).
Values for mean λ s were almost always much higher than those for λ d (Tables 4 and 5). The value of mean λ s approached and became smaller than λ d only when seed longevity was drastically reduced, showing that the persistent seed bank is necessary for the advantages of stochasticity for this species to be realized (Table 5).

| Importance of the persistent seed bank
Our fourth hypothesis, that the persistent seed bank would be essential for population persistence in this species, was strongly supported. Reductions in the number of years that seeds could persist in the seed bank had a dramatic negative effect on mean λ s as well as on extinction risk (Table 5, Figure 5c,d). This shows that without a persistent seed bank, the chances of long-term survival in the warm desert for a species with the spring-ephemeral hemicryptophyte life form would be near zero. Episodic high seed production, high recruitment success, and the capacity to form a long-lived seed bank are key features of the warm desert-adapted life-history strategy that enables long-term persistence of A. holmgreniorum in spite of its short life span.
The coupling of higher reproductive output with a shorter life span in comparison to the life histories of perennial spring ephemerals of more mesic habitats is an expression of a more r-selected or "fast" life-history strategy (Salguero-Gómez, 2017). Longer-lived spring ephemerals include deciduous forest understory species (Augspurger & Salk, 2017;Lapointe, 2001) as well as some species of steppe habitats (e.g., Astragalus scaphoides; Gremer & Sala, 2013).
The emphasis on a persistent seed bank in A. holmgreniorum makes its life-history strategy even more similar to that of desert annual species, even though it does not produce seeds in the first year of life. Its short life span is presumably imposed by the stresses it experiences as an adult, rather than being genetically programmed into its life cycle as it is for annual plants (Figure 4b).

| Seed addition as a conservation strategy
Our fifth hypothesis was that the model would demonstrate that seed addition could potentially be an effective conservation strategy for this endangered species. This hypothesis was supported.
The model demonstrated high sensitivity of extinction risk to the vector of initial stage values, including increases to the seed bank.

TA B L E 6
Effects of changes in the initial vector of stage values on 50-year quasi-extinction risk, including the effect of: (1) using initial vectors of stage values for different years from the field data, (2) augmenting populations by adding seeds, (3)  It makes intuitive sense that a larger starting population would have reduced extinction risk in a given time period. For a short-lived perennial in the highly stochastic warm desert environment, where plant numbers vary widely between years, this led to widely divergent predictions of extinction risk depending on which year was chosen to provide starting values. There appears to be a moving window of extinction risk through time, with the population passing through periods of lower and higher extinction risk as population numbers and stage distributions fluctuate. We used the initial vector of stages in the base model as a standard for detecting the effect of variation in other model parameters on both mean λ s and extinction risk, but this does not imply that the base model extinction risk time course is more likely than an extinction curve generated using a different initial vector of stages. The value of mean λ s was largely insensitive to the vector of initial stage values as predicted by theory (Caswell, 2001).
Our model has shown that seed addition can substantially reduce extinction risk, supporting the idea of augmentation of at-risk populations through seeding. Introducing seeds into unoccupied habitat was shown to have the potential to create populations that could be persistent, creating the possibility of establishing new populations through seed introduction.
The model has also provided some useful insights into other ways that extinction risk for this species might be reduced through management. The importance of high reproductive output for population persistence has drawn attention to the need to protect and enhance habitat for the ground-nesting bees that are its principal pollinators (Tepedino, 2005). The importance of high recruitment success highlights the need to manage the increasing threat from invasive winter annual grasses that can outcompete A. holmgreniorum seedlings in the spring. And the importance of dormant season survival of recruits prompted a recent study documenting the negative impact of cattle trampling on recruit survival (Searle & Meyer, 2020).

| CON CLUS IONS
Our population matrix model for A. holmgreniorum using best estimates of vital rates and an initial vector of stage values based on field data resulted in mean λ s > 1, indicating that this population of A.
holmgreniorum is predicted on average to grow if past environmental conditions persist into the future. However, extreme inter-annual variation in environmental quality resulted in very large variances in vital rates, generating a substantial risk of local extinction even when mean λ s > 1. Our model permitted meaningful tests of hypotheses about life-history strategy. We determined that the species depends on a stochastic environment for population persistence, but that stochasticity only confers this advantage if there is a persistent seed bank to permit survival through unfavorable years. We confirmed that the species compensates for high adult mortality during the dormant season by employing a strategy further along the "fast-slow" continuum that emphasizes high seed production in the first years of adult life and high recruit survival, particularly in more favorable years. We found that large inter-annual variation in year quality related to precipitation generally resulted in very large variances in vital rates. The wide variance associated with mean λ s as well as the impact of the initial vector of stage values on extinction risk showed that extinction risk is not predictable in this highly variable environment.
This led to the concept of a moving window of extinction risk through time, with higher risk during periods of lower abundance.
The strong effect of the initial stage vector on extinction risk also served as a reminder that the extinction risk evaluated in this analysis is for only a small subset of the total number of individuals of this species measured over a very small area. Given the same life-history strategy and the same stochastic environment, the extinction risk to the species as a whole or even to the larger population that was subsampled for this study would be far lower (Table 6).
Simulations showing the possible effectiveness of seed addition or introduction may provide a novel management strategy for reducing extinction risk. Following recommendations based on our modeling effort, augmentation and introduction projects have already been successfully implemented for this species  using salvaged seeds as well as seeds produced in cultivation (Schultz et al., 2021).

ACK N OWLED G M ENTS
The University who helped with fieldwork on this project over the years.

CO N FLI C T O F I NTE R E S T
The authors declare no conflict of interest.

DATA AVA I L A B I L I T Y S TAT E M E N T
Field datasets that form the basis of the analyses in this manuscript are deposited in DRYAD at https://doi.org/10.5061/dryad.tdz08 kq12. US Fish and Wildlife Service (2006). Astragalus holmgreniorum (Holmgren milk-vetch)

A PPE N D I X 1 V ITA L R ATE C A LCU L ATI O N S
In this appendix, we present in detail how the values for vital rates needed to parameterize the elements in the transition matrix were calculated. These methods are described as follows:

Plant survival and size class transitions
Plants were classified into three size classes based on field observations indicating that small plants (Size Class 1; <6 cm in diameter) very rarely flowered and did not set significant amounts of seed.
Plants from 6 to 15 cm in diameter (Size Class 2) generally set seed only in favorable years, while plants >15 cm in diameter (Size Class 3) often set some seed even in poor years. This relationship between size class and seed production was confirmed using a dataset from Searle (2011). We did not consider further division of Size Class 2 or 3 because small sample sizes in many years would have made survival and size class transition probabilities less reliable.

Means and variances for plant survival and size class transitions
were calculated directly from the 22-year demography dataset.
For each pair of years in the study, we first calculated survival, that is, the proportion of individuals in each of the three size classes in Year (n) that survived to Year (n+1) . Of these surviving plants of each size class, we then calculated the proportion that remained in the same size class the following year (stasis) and the proportion that transitioned to each of the other two size classes.
One important feature of the beta distribution used to specify transitions in the matrix model from Morris and Doak (2002) that we implemented (see Appendix 2) is that it can only model binary alternatives. As our model included transitions among three size classes, it was necessary to decompose these probabilities into stepwise binary probabilities. For example, the transition from size class S1 to another size class is broken down into: S1 to S1 (stasis) S1 to notS1 S1 to S2NEW S1 to S3NEW To calculate the probability of the transition from S1 to S2 using these binary vital rates requires multiplying the probabilities (S1 to notS1) and (S1 to S2NEW) together. Thus for the matrix elements that quantify the transition from Size Class 1 to another size class each time step, the following survival and size class transition vital rates were combined: Matrix element S1 to S1 (S1 survival) *(S1 to S1) Matrix element S1 to S2 (S2 survival)*(S1 to notS1)*(S1 to S2New) Matrix element S1 to S3 (S3 survival)*(S1 to notS1)*(S1 to S3New) Matrix elements for the transitions from S2 to S3 and from S3 to S2 were calculated by combining survival and size class transitions in the manner described above. Matrix elements for transitions from S2 and S3 to S1 also included vital rates parallel to those discussed here, but because these transitions also included probabilities for recruitment of S1 plants from seeds produced in the current year by S2 and S3 plants, these matrix elements included additional terms as described later.

R EPRO D U C TI V E O UTPUT PA R A M E TER S
Estimates of seed yield per plant for potentially reproductive individuals (size classes S2 and S3) each year were obtained by combining yearly flower count data from the demography study with estimates of seed production per flower from an independent dataset (Searle, 2011). Each year, every tagged S2 and S3 plant on the study site This approach to estimating seeds per flower represented an effort to improve the precision of seed production estimates as compared to using a constant value for seeds per flower for both size classes and for every year of the study. We only had seeds per flower data from 2 years, so that the analysis has no associated error estimate and assumes a linear relationship. However, the fact that the 2 years of the reproductive output study were so different in terms of both spring precipitation and seed production per flower lends credence to this approach. Seed production per flower was a relatively small contributor to seed production calculations, which were much more heavily influenced by the yearly flower count data.
To obtain the long-term mean and variance for seed production per plant of each reproductive size class for use in the model, these yearly means were then averaged. These reproductive output values for S2 and S3 plants are represented by the vital rates vrs(13) and vrs (14) in Table A2.1 and A2.2.

Seed bank transitions
Estimates for seed bank transitions were derived from a 6-year seed retrieval experiment (Searle, 2011;A. Searle, unpublished data). The study was initiated in summer 2009 with freshly collected seeds from three populations. Thirty groups of 20 seeds from each seed population were packaged into 4-cm 2 nylon mesh bags. The bags were buried ca. 2 cm deep at each of the three sites. At each site two replicates, each consisting of one bag from each of the three seed populations, were included for each potential retrieval date for a total of five possible retrieval dates (for each seed population: 3 sites × 2 replicates × 5 retrieval dates = 30 bags).
Seeds were retrieved in early summer in three subsequent years: 2010 ( Seeds that remained in each bag were quantified and tested for viability. Seeds that were not detectable or were present as seed coats only were assumed to have germinated. The remaining seeds were scarified with sandpaper, then imbibed in petri dishes at 30°C for 1 week. Seeds that germinated were quantified and removed, and the remaining ungerminated seeds were tested for viability using tetrazolium staining. Seeds that did not stain were scored as nonviable. Seeds lots used in the retrieval were also subjected to tetrazolium evaluation at the initiation of the experiment to provide a viability estimate. Initial viability averaged 95% in these late-collected seed lots, but viability fruits∕flower * seeds∕fruit = seeds∕flower. Mean flowers∕plant from field counts * mean seeds∕flower from regression = mean seeds per plant F I G U R E A 1 . 2 Results of a seed retrieval experiment showing initial viability, viability loss, and dormancy loss over 6 years, with values extrapolated to 9 years, the projected maximum time for dormant seeds of a cohort to persist in the seed bank (modified from Searle, 2011) in the reproductive output studies described earlier was near 100%.
All viable seeds in the seed bags were assumed to be dormant, based on the fact that adequate precipitation for full germination of nondormant seeds was received during every year of the retrieval study. This means that only dormant seeds would remain as viable seeds.
The retrieval data were first analyzed as a full factorial using analysis of covariance for binomial data in the SAS program GLIMMIX with seed population and site as the class variables and number of years post-burial as the continuous variable. Because neither seed population nor site was significant, data were pooled by retrieval date for analysis using linear regression with number of years postburial as the continuous independent variable. The analyzed response variables were: (1) fraction of seeds remaining viable, that is, germinable after scarification plus viable using tetrazolium staining and (2) fraction of nonviable seeds. Initially viable seeds not in these two categories were assumed to have germinated; these were still present as empty seed coats in most cases. Intact nonviable seeds were also still present in the bags, likely because the bags spent very little time in a wet condition over the course of the study.

Germination and nondormant seed carry-over
Field retrieval experiments have demonstrated that most nondormant seeds germinate, at least in shallowly buried retrieval bags, except in years of very low winter-spring precipitation (<50 mm). To include the possibility that nondormant seeds could carry over to the following year, we included the stage Nondormant in the model.
Lacking detailed information on germination fraction as a function of precipitation, we adopted a simplified version. We set germination at 1.00 for years with >50 mm of winter-spring precipitation and at 0 for years with <50 mm. Using these coarse estimates along with yearly precipitation data resulted in a mean germination fraction of 0.90 and a germination fraction variance of 0.085. This gives a reasonably realistic estimate of what might happen in the field, namely, most years have full or near-full germination, but a few have very low or no germination. Each yearly time step, the model uses this probability distribution to calculate the fraction of nondormant seeds (NDORM0 through NDORM9) that will germinate (GFRAC) and the fraction that will carry over into the Nondormant stage (1-GFRAC).

Germinant survival to census
Calculation of germinant survival to census requires knowledge of the number of seeds that germinate as well the number of seedlings that successfully recruit and are present as S1 plants at census in April. This is not the same as the number of seedlings that emerge, as many seeds may germinate but not produce emerged seedlings.
This fraction (germinated seeds/recruited seedlings) cannot be obtained from observational data without accurate knowledge of numbers of nondormant seeds in the soil seed bank. We used an experimental dataset to measure germinant survival in an exceptionally favorable year. We carried out introduction seedings in fall 2018 at 10 sites in suitable unoccupied habitat . We used both unscarified seeds (expected to persist for 9 years and create yearly recruitment episodes) and scarified seeds Age-specific transitions for seeds in the persistent seed bank as well as transitions to the nondormant seed fraction were also included. Transitions involving actively growing plants varied each year in response to environmental variation, while transitions in the dormant seed bank were treated as invariant, that is, not sensitive to environmental variation.
We calculated means and variances for the vital rates that were variable, and estimated or calculated values for invariant vital rates as described in Appendix 1. We then chose probability distributions for the vital rates to be used in the model. The table of vital rate means, variances, and statistical distributions that follows this section provide these values (Table A2.1).
As per Morris and Doak p. 275-282, we used the beta distribution rather than the normal distribution to model survival and transition probabilities, and the stretched beta distribution rather than the log-normal distribution to model seed production. These distributions have the advantage of being bounded between 0 and 1 for the beta distribution and bounded between user-defined minimum and maximum values for the stretched beta distribution. These limits prevent the model from choosing biologically unrealistic or even impossible values from the probability distribution for each time step. The mathematical calculations for the beta distribution can be cumbersome, but the book provides useful computational shortcuts and also MATLAB code for making these computations that can be embedded in the model program (betaval, Box 8.3; stretchbetaval, Box 8.5).
Another key component of a realistic population matrix model is the inclusion of covariances among vital rates. To accomplish this, ) provided a method for using within- year and across-year correlation matrices for variable vital rates to construct corrected matrices that can be used to pick random values with the correct degree of correlation from the beta or stretch beta distribution for each variable and that will include consideration of the degree of within-year correlation, autocorrelation, and crosscorrelation among variables.
The first step for constructing the derived matrices is to check whether each original correlation matrix meets the matrix algebra assumptions that make its use possible in the model. Missing data, small numbers of observations and other irregularities can cause the matrix to contain unacceptable internal logical inconsistencies that must be corrected before the matrix is usable. Box 8.8 in Morris and Doak provides the MATLAB code to do this. These corrected matrices are then used in combination with the vital rate means, variances, and statistical distributions in a rather complex set of calculations to generate sets of within year, auto-, and cross-correlated vital rates (Box 8.9). The correlated vital rates calculation subroutine is then embedded into the program file that simulates population growth and extinction risk (Box 8.10).
In order to run the program in Box 8.10 for Astragalus holmgreniorum, we listed means and variances for vital rates in our model, specified statistical distributions for each vital rate, and referenced the transition matrix m file (Table A2.2) that defines each matrix element in terms of vital rates. We also loaded the within-year and across-year corrected correlation matrices. We then specified the vector of initial stage values, the quasi-extinction threshold, the number of years to simulate, the number of correlated vital rates in the correlation matrix, the number of uncorrelated vital rates, the dimensions of the population transition matrix, and the number of iterations to perform.
The sections of the program calculate the mean population matrix and deterministic lambda, then create beta distributions that the program can use to select beta-distributed random variables. This is followed by the complex set of calculations used to create correlated beta or stretch beta random variables. Finally, the program proceeds to the simulation of stochastic population growth and the calculation of extinction risk cumulative distribution functions. The program ends by specifying desired outputs of the model run. Morris, W. F., & Doak, D. F. (2002). Quantitative conservation biology:

R E FE R E N C E
The theory and practice of population viability analysis. Sinauer Associates.

A PPE N D I X 3 S U RV I VO R S H I P A N A LYS I S
Our population matrix model for Astragalus holmgreniorum is a stagebased model, but our understanding the life-history strategy of this species hinges on the relationship between environmental quality and the relationship of adult survival with age. Mean survivorship was log-linear, that is, mortality risk was constant on average, especially after the first year (Deevey, 1947; Figure A3.1a). This suggests that mortality is environmentally mediated and not a consequence of senescence in older plants, which would tend to result in increased mortality risk with age.