Philippa Kaur

Coupling of SM and VPD confounds ecosystem dryness stress
The difficulty to disentangle the respective effects of SM and VPD stems from the fact that SM and VPD are strongly coupled through land–atmosphere interactions7,20. In addition, field experiments that manipulate atmospheric humidity and temperature at the ecosystem scale are lacking21. Given the strong SM-VPD coupling (Fig. 1c), e.g., on the yearly scale, both lower SM and higher VPD are associated with lower ecosystem gross primary production (GPP), indicated by SIF (Fig. 1a, b). This underlies the use of either SM or VPD alone as proxy for dryness stress on ecosystem production in many current models. Note a global spatially contiguous SIF data set was mainly used in this study, which was generated by using the machine-learning algorithm to train SIF observations from Orbiting Carbon Observatory-2 (OCO-2)22. We display the yearly scale because it is typically used to represent the condition of strong SM-VPD coupling globally11, and the study time period mainly spans from 2001 to 2016. However, as SM and VPD are strongly coupled, it is possible that the correlation between SM and SIF is a byproduct of the correlation between VPD and SIF, or vice versa. As a consequence of SM-VPD coupling, the correlations of yearly SM and VPD with SIF is very similar globally (Fig. 1d). Consequently, the correlation between SM and VPD constitutes a confounding factor that is often overlooked when assessing the role of SM and VPD in determining the impact of dryness stress on ecosystem production. There are still low correlations between SIF and SM or VPD in the northern high latitudes or tropical regions, which suggests possible temperature or radiation effects and requires further investigation.
Fig. 1: Strong coupling of soil moisture and vapor pressure deficit confounds ecosystem dryness stress.

a–c Spatial distribution of Pearson’s correlation coefficient between solar-induced chlorophyll fluorescence (SIF) and soil moisture (SM) (r(SIF, SM)), SIF and vapor pressure deficit (VPD) (r(SIF, VPD)), and SM and VPD (r(SM, VPD)), at the yearly scale. Regions with sparse vegetation and regions without valid data are masked in gray. d Relationship between yearly r(SIF, VPD) and yearly r(SIF,SM) across land vegetated areas. Color shows the relative density of data points, with higher density in black and lower density in yellow.

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Decoupling of SM and VPD globally
At yearly scale, there is a strong negative correlation between SM and VPD, indicating that low SM is always accompanied by high VPD (Fig. 1c), which is consistent with previous findings7,20. From yearly to monthly, weekly, and daily scale, the correlations between SM and VPD are generally decreasing (Fig. 2d), but remain large across extensive areas, such as central South America, Sub-Saharan Africa, India, and Southeast Asia (Fig. 2a and Supplementary Fig. 1). However, when binning the data into 10 bins according to percentiles of either SM or VPD per pixel, we find that the correlation coefficient between SM and VPD in each bin becomes approximately zero (Fig. 2b–d and Supplementary Figs. 2 and 3). This shows that SM and VPD are generally decoupled at daily scale in both SM and VPD bins.
Fig. 2: Decoupling of soil moisture and vapor pressure deficit.

a–c Spatial distribution of Pearson’s correlation coefficient between soil moisture (SM) and vapor pressure deficit (VPD) at daily scale, averaged over daily SM bins, and averaged over daily VPD. Regions with sparse vegetation and regions without valid data are masked in gray. d Violin plots of correlations between SM and VPD from yearly to daily bins across land vegetated areas. White dots indicate the median values, gray boxes cover the interquartile range, and thin gray lines reach the 5th and 95th percentiles.

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Disentangling the relative role of SM of VPD
We now disentangle the respective effects of SM and VPD in limiting ecosystem production by exploiting the fact that SM and VPD are decoupled in binned daily SM or VPD data (Fig. 2). SM and VPD are also largely decoupled in 4-day bins, which is the temporal resolution of the mainly used SIF data set (Supplementary Figs. 4 and 5). The analysis is guided by the assumption that if SM dominates dryness stress, low SM will limit ecosystem production regardless of VPD variations (Supplementary Fig. 6a, c). In the same way, if VPD dominates dryness stress, high VPD will limit ecosystem production regardless of SM variations (Supplementary Fig. 6b, d).
To illustrate this further, we select an example pixel located in Mali (West Africa). Without decoupling SM and VPD, it is difficult to conclude whether the decrease in SIF is caused by low SM, high VPD, or both in conjunction (Fig. 3a, b). However, when looking at the variation of SIF across VPD gradients in SM bins (without SM-VPD coupling), high VPD does not reduce SIF but even increase SIF a bit under moderate SM conditions (Fig. 3c). In contrast, low SM reduces SIF noticeably in VPD bins (Fig. 3d). This shows that high VPD does not limit SIF in the absence of the SM-VPD coupling at the example pixel, whereas low SM can still limit SIF. In other words, the apparent VPD limitation on SIF is largely the byproduct of SM-VPD coupling. The respective effects of SM and VPD on SIF is also illustrated in Fig. 3e. The changes in SIF from low VPD to high VPD without SM-VPD coupling (termed ΔSIF(VPD|SM)) can quantify the VPD stress on SIF. Likewise, changes in SIF from high SM to low SM without SM-VPD coupling (termed ΔSIF(SM|VPD)) quantify the SM stress on SIF. The effect of SM and VPD on SIF is estimated using two approaches: (i) SIF in the maximum VPD bin minus SIF in the minimum VPD bin or SIF in the minimum SM bin minus SIF in the maximum SM bin; (ii) using linear regression to derive changes in SIF caused by high VPD or low SM. The two approaches lead to similar results (Methods and Supplementary Fig. 16). As shown in Fig. 3f, the SM effect is strong at the example location (ΔSIF(SM|VPD) = −0.17 mW m−2 nm−2 sr−1), in contrast to the VPD effect (ΔSIF(VPD|SM) = −0.03 mW m−2 nm−2 sr−1). Thus, the comparison of (ΔSIF(SM|VPD) and ΔSIF(VPD|SM) enables the disentangling of their relative role in governing dryness stress.
Fig. 3: Disentangling soil moisture and vapor pressure deficit limitation effects.

a Daily solar-induced chlorophyll fluorescence (SIF) versus daily vapor pressure deficit (VPD). b Daily SIF versus daily soil moisture (SM). c Daily SIF versus daily VPD, binned by SM. d Daily SIF versus daily SM, binned by VPD. c, d circles denote the averaged SIF within each bin of VPD and SM. e Average SIF in each percentile bin of SM and VPD. The cyan arrows indicate the VPD limitations on SIF without SM-VPD coupling (ΔSIF(VPD|SM)), and the orange arrows indicate the SM limitations on SIF without SM-VPD coupling (ΔSIF(SM|VPD)). For better readability, only four arrows are shown. f Distribution of ΔSIF(VPD|SM) and ΔSIF(SM|VPD). Circles denote the ΔSIF(VPD|SM) and ΔSIF(SM|VPD) in each bin. Squares denote the corresponding mean. The example pixel is located in Mali, West Africa at 14.25°N, −4.75°E. See Methods for more details.

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Next, we examine the respective SM and VPD effects on SIF globally. To ensure comparability in space, the SIF time series at each pixel are normalized by the average SIF exceeding the 90th percentile. Temperature and radiation can also limit ecosystem production, therefore, we have filtered out days when other meteorological drivers were likely to be more important than SM or VPD in limiting ecosystem carbon and water fluxes throughout the analyses, following previous studies12,23. We find that ΔSIF(SM|VPD) is negative across most vegetated land areas, robustly indicating the limiting role of low SM to SIF (Fig. 4a, b) and consistent with plant physiological understanding and previous studies4,7. The units refer to the fractions relative to average SIF exceeding the 90th percentile in each grid cell. Large ΔSIF(SM|VPD) are identified in mid-latitudes, including southern North America, central Eurasia, southern Africa, and Australia. In contrast, ΔSIF(VPD|SM) is small and close to 0 across large areas, but it was larger than ΔSIF(SM|VPD) in tropical Africa surrounding the equator (Fig. 4c, d). Globally, a change from the wettest SM to the driest SM under constant VPD reduces SIF by up to 14.9% on average, whereas a change in VPD from lowest to highest quantiles under constant SM has little effect on SIF (−3.8%) on average. Locally, the areas where the strength of SM effects on SIF (|ΔSIF(SM|VPD)|) exceeds that of VPD effects (|ΔSIF(VPD|SM)|) are widespread, which is also visible along the latitudinal gradient (Fig. 4e, f). In total, |ΔSIF(SM|VPD)| is larger than |ΔSIF(VPD|SM)| across 71.3% of land vegetated areas with valid data, by contrast, VPD is more important than SM in 26.7% of corresponding areas. Furthermore, our findings suggest that many previous estimates of the role of VPD on ecosystem production are likely exaggerated16,24 as they did not account for the strong SM-VPD coupling as a confounding factor. In boreal and tropical regions, both SM and VPD have little effect on SIF, which is controlled by radiation and temperature7,25. The spatial patterns of ΔSIF(SM|VPD)—ΔSIF(VPD|SM) are robust to the choice of the particular forcing data set (Supplementary Figs. 7–11). However, when using the GOME-2 SIF and SCIAMACHY SIF with the local overpass time at 9:30 am and 10:00 am, the VPD effects are weaker than that in CSIF (reducing SIF by 0.1% and 0.02% on average globally), including most of Africa (excluding the Sahara) as well as large areas of central South America, southern Asia, and Australia (Supplementary Figs. 9–11). This raise a caveat that using SIF retrieved in the morning would underestimate the VPD effects. To further test the robustness of our result, we standardized the SIF by photosynthetically active radiation (PAR) to remove possible radiation effects26, limited the data to a narrow temperature range to remove possible temperature effects and aggregated data to a coarser time resolution or using 20 percentile bins, yielding similar results (Supplementary Figs. 12–15). Thus, we demonstrate that SM is the dominant factor in driving the response of ecosystem production to dryness at the ecosystem scale across most land vegetated areas, except for tropical and boreal areas.
Fig. 4: Effect of soil moisture and vapor pressure deficit on ecosystem production globally.

a, c, e Spatial distribution of the changes in solar-induced chlorophyll fluorescence (SIF) caused by low soil moisture (SM) (ΔSIF(SM|VPD)) and high vapor pressure deficit (VPD) (ΔSIF(VPD|SM)), and their differences in absolute values (i.e., |ΔSIF(SM|VPD)|−|ΔSIF(VPD|SM)|). b, d, f Zonal means of SM and VPD effects on SIF and their differences in absolute values. The units refer to the fractions relative to average SIF exceeding the 90th percentile in each grid cell. Black lines indicate the mean values, and gray shaded bands show the standard deviation. Regions with sparse vegetation and regions without valid data are masked in white.

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Different from a recent global assessment of SM stress on ecosystem production that estimates the relation between SM stress and background climate from a small sample of flux sites18, our results build on data with global coverage and hence provide spatially explicit information of SM stress. Further converting the SIF decrease to the actual carbon loss would largely help quantify changes in terrestrial carbon fluxes under drought. Furthermore, our conclusions contradict many laboratory experiments that show strong VPD effects on stomatal conductance at the leaf scale27,28. This again indicates that the stomatal sensitivity to VPD do not definitely determine the same VPD response of plant water and carbon fluxes at the ecosystem scale29,30, but some ecosystem scale measurements reveal that stomatal sensitivity to VPD can matter in some cases11,12. Key processes driving the weak plant photosynthesis response to VPD at the ecosystem scale need to be addressed in future work, such as the role of ecosystem water use efficiency, water storage and hydraulic strategies29.
Dependence of SM stress on climate and vegetation gradients
We find that SM limitation effects (ΔSIF(SM|VPD) are largest in semi-arid ecosystems (Fig. 5a), including shrubland, grassland, and savannah ecosystems. These are the ecosystems that are the main drivers of the interannual variability in global terrestrial CO2 flux31,32. In contrast, VPD effects are much weaker in these regions (Fig. 4c). This suggests that SM could be more important than VPD in driving interannual variability of global terrestrial carbon uptake. As SM stress is strongest in drylands, the projected expansion of drylands33 is likely to increase the influence of SM on the future global carbon cycle. In addition, we find that regions with lower tree fraction exhibit a larger response to SM stress globally (Fig. 5b). This is in line with recent findings34, and further verifies the robustness of our results. Our findings also highlights the differential dryness response of ecosystems along a tree cover gradient.
Fig. 5: Dependence of soil moisture dryness stress on climate and vegetation gradients.

Violin plots of soil moisture (SM) limitation effects (ΔSIF(SM|VPD)) across a aridity gradients and b tree cover gradients. c Violin plots of the sensitivity of solar-induced chlorophyll fluorescence (SIF) to SM (i.e., (frac{{delta SIF}}{{delta SM}}|_{VPD})) within different plant functional types: SHR(S), shrubland (south of 45° N); GRA, grassland; CRO, cropland; WSA(S), woody savanna (south of 45° N); SAV, savanna. White dots indicate the median values, gray boxes cover the interquartile range, and thin gray lines reach the 5th and 95th percentiles.

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The representation of dryness stress on plant photosynthetic CO2 assimilation can differ largely between TEMs and is considered one of the largest uncertainties in predicting future land carbon uptake and climate35,36,37. Their representations in TEMs often uses an empirical function that only varies by plant functional type (PFT)38, which have generally not been validated against observational empirical data. Therefore, we explored the observed standardized sensitivity of SIF to SM. We find that the sensitivity of ecosystem production to changes in SM can vary largely even in the same PFT with strong observed dryness effects (Fig. 5c). This is consistent with recent findings that the grassland’s sensitivity to dryness can vary greatly39. The differences of dryness response in the same PFT are, e.g., related to plant species, plant height and plant hydraulic processes, such as plasticity variations in xylem and mesophyll conductance, embolism resistance, or water storage40. At present, evaluating and incorporating more plant hydraulic processes into the next generation of terrestrial ecosystems is on the way41. Our results of dryness effects on ecosystem production thus enables an evaluation of further TEM evolution.
In summary, we provide global results of SM and VPD stress on SIF and demonstrate that SM, rather than VPD, is the dominant driver leading to drought limitation on vegetation productivity at the ecosystem level across most vegetated land areas. VPD stress on ecosystem production is almost lost across large areas without SM-VPD coupling. We thus make the case for revisiting the role of VPD in previous studies that neglected the strong SM-VPD coupling. Furthermore, models that do not correctly disentangle the respective VPD and SM limitations cannot adequately predict the dryness stress on ecosystems and associated rough risks to human well-being. The next challenge is to incorporate the observations to constrain the representation of dryness stress on plants in models, which would also reduce uncertainties in the projection of terrestrial CO2 fluxes and associated climate projections. More

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Study areas
Time series of population survey data were used from nonmigratory elk populations in three different locations along the West Coast of the USA (Fig. 4). Five of the populations were in the Prairie Creek drainage (Davison), the Lower Redwood Creek drainage (Levee Soc), the Stone Lagoon area, the Gold Bluffs region, and the Bald Hills region of Redwood National and State Parks (RNSP), Humboldt County, California (41.2132° N, 124.0046° W). These populations occupy an area of about 380 km2. The climate in this region was mild, with cool summers and rainy winters. Annual precipitation was usually between 120 and 180 cm and most of the precipitation fell between October and April. Snow was rare since average winter temperatures rarely dropped below freezing and ranged from 3 to 5 °C. Average summer temperatures ranged from 10 to 27 °C, depending on the distance inland. Elk in RNSP were not legally hunted, and displayed strong social bonding between females, juveniles, and sub-adult males7.
Figure 4

Map of study areas in Arid Lands Ecology (ALE) Reserve, southern part of Redwood National and State Parks, and Tomales Point Elk Reserve in Point Reyes National Seashore. This map was created in ArcMap (Version 10.6; https://desktop.arcgis.com/en/arcmap/).

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An elk population in the Point Reyes National Seashore inhabited part of the Point Reyes Peninsula in Marin County, California (38.0723° N, 122.8817° W). The elk were restricted to an area of 10.52 km2 on the northern tip of the peninsula by a 3-m-tall fence. The climate of this study area was Mediterranean, with an average annual precipitation of 87 cm27. Most of the precipitation fell from autumn to early spring. Temperatures averaged about 7 °C in winter and 13 °C in summer27,35.
Another elk population was in the Arid Lands Ecology (ALE) Reserve and occupied a 300 km2 area within the U.S. Department of Energy’s Hanford Site, Washington (46.68778° N, 119.6292° W). The climate in this area was semi-arid with dry, hot summers and wet, moderately-cold winters. Average summer temperatures were around 20 °C and average winter temperatures were around 5 °C with an average annual precipitation of 16 cm, half of which fell in the winter as rain36.
Population surveys
In RNSP, females, juveniles, and subadult males were often in the same group and tended to use open meadow habitat more frequently than adult males37,38. These behavioral patterns likely explain why females, juveniles, and subadult males were sighted more frequently than adult males7. Moreover, in size-dimorphic ungulates such as elk, recruitment was strongly correlated with female abundance and weakly correlated with male abundance7,13,39. In RNSP, the abundance of groups of females, juveniles, and subadult males drove the dynamics of the group and of adult males7. Therefore, for the RNSP populations, we used herd counts where a herd was comprised of females, juveniles, and subadult males. We also used herd counts for the Point Reyes and ALE Reserve populations to remain consistent.
Systematic herd surveys of elk were conducted during January from 1997 to 2019 in RNSP. Surveys in the Davison meadows, the Levee Soc area, the Stone Lagoon area, the Gold Bluffs region, and the Bald Hills region were conducted by driving specified routes 4 to 10 times on different days throughout the month of January. The time series for these five herds ranged from 19 to 23 years of data. The elk were counted and classified by age and sex as adult males, subadult males, females, and juveniles. Females could not be visually differentiated into adult and subadult age categories37. The highest count of females, juveniles, and subadult males from the surveys conducted each year was used as an index of abundance of each herd since the detection probabilities were high both on an absolute basis ( > 0.8) and relative to variation in detection probabilities (CVsighting = 0.05)7,40. For the Bald Hills herd, which is the only herd in RNSP where harvests occurred, we added hunter harvests to the highest count of each year to account for this source of mortality. These harvests occurred only when elk from the Bald Hills herd left RNSP.
Elk population surveys were conducted at the Point Reyes National Seashore from 1982 to 2008. Weekly surveys were conducted after the mating season. Surveys were conducted on foot or horseback of female elk that were ear-tagged or had a collar containing radio telemetry32,35. Individuals counted were classified as females, juveniles, subadult males, and adult males. Data were not available for the years 1984 to 1989 and 1993, so the time series included 20 years of data. We used the highest count of females, juveniles, and subadult males in each year in our analyses. This herd was also not hunted.
Elk population surveys in the ALE Reserve were conducted in winters after hunting and before parturition. From 1982 to 2000, biologists used aerial telemetry studies, in which they located all collared elk during each survey and classified them by sex and age. We used the total counts of females, juveniles and subadult males. For years in which multiple surveys were conducted, we used the highest count in each year as an index of abundance for that year25,41. We omitted population survey data collected in 1982 from our analysis because individuals were not classified by sex and age in this year. Consequently, the time series included 18 years of data. For all years of data used, we added hunter harvests to the highest count of each year to account for this source of mortality. The count in 2000 was much lower than in the previous year, likely due in part to a large wildfire which occurred in the summer of 2000, which probably had an immediate effect of reducing available elk forage in the reserve and caused elk to spend more time outside of the ALE Reserve42,43. In addition, the highest recorded number of elk (about 291) were harvested that year43.
Ricker growth models
We fit linearized Ricker growth models simultaneously to the seven time series to estimate population growth parameters as well as temporal variation in r and β. We estimated K as the x-intercept of the Ricker growth model (i.e., when r = 0). Notably, preliminary analyses showed that not accounting for observer error did not bias our results (see Supplementary Information).
We used a Bayesian Markov Chain Monte Carlo (MCMC) algorithm with 3 chains, 150,000 iterations, a burn-in period of 75,000, an adaptation period of 75,000, and no thinning. We used Bayesian inference and MCMC because these methods offer advantages when fitting hierarchical models to model variation in ecological data44,45. We conducted these analyses in the RJAGS program (JAGS Version 4.0.0; https://sourceforge.net/projects/mcmc-jags/files/JAGS/4.x/Windows/) in RStudio (R Version 3.5.0; https://cran.r-project.org/bin/windows/base/old/3.5.0/). We used uninformative priors for the y-intercept (i.e., rmax) and the slope (i.e., β) in order to allow solely the data to influence posterior estimates of these parameters. Informative priors were not necessary as long as parameter estimates from each chain converged. Convergence among chains was determined when the Gelman-Rubin diagnostic ((hat{R})) was less than 1.01, and through visual checks of trace and density plots46.
The estimate of rmax borrowed information among herds because this parameter should be similar among populations within a species22. Therefore, we modeled rmax for each herd (j) as a random effect following a normal distribution with (mu_{{r_{max} }} sim Normalleft( {0, 0.001} right)) and (sigma_{{r_{max} }} sim Uniformleft( {0, 100} right)). To model temporal variation in r for each herd, we included a zero-centered random effect which was also modeled following the normal distribution (gamma_{t,j} sim Normalleft( {0,sigma_{{gamma_{j} }} } right)), where (sigma_{{gamma_{j} }} sim Uniformleft( {0, 100} right)). The estimate of β did not borrow information among herds because this parameter can vary widely among herds18. The prior for β for each herd (j) followed the normal distribution (beta_{j} sim Normalleft( {0, 0.001} right)). To model temporal variation in β for each herd, we modified how we modeled β by using a normal distribution ({beta_{{delta }_{t,j}}} sim Normalleft( {mu_{{beta_{{delta }_{j} }}}} , {sigma_{{beta_{{delta }_{j}} }} } right)), where ({mu_{{beta_{{delta }_{j}} }}} sim Normalleft( {0, 0.001} right)) and ({sigma_{{beta_{{delta }_{j}} }}} sim Uniformleft( {0, 100} right)). Thus, there were four possible Ricker growth models for each herd; (1) no temporal variation in r and β,

$$ r_{t} = r_{max} + beta N_{t} + varepsilon , $$
(3)

(2) temporal variation in r,

$$ r_{t} = r_{max} + beta N_{t,} + gamma_{t} + varepsilon , $$
(4)

(3) temporal variation in β,

$$ r_{t} = r_{max} + {beta_{{delta }_{t}}} N_{t} + varepsilon , $$
(5)

and
(4) temporal variation in both rmax and β

$$ r_{t} = r_{max} + {beta_{{delta }_{t}}} N_{t} + gamma_{t} + varepsilon . $$
(6)

The residual variance was modeled as (varepsilon sim Uniformleft( {0,100} right)). We fit the model with no temporal variation (Eq. (3)) in either parameter to all seven time series simultaneously. All parameters except for rmax were estimated independently for each herd. For each time series of population survey data, we determined whether models with more parameters provided a better fit. We did so by fitting each possible growth model (Eqs. (4)–(6)) to each time series one at a time, while modeling all other time series with no temporal variation in rmax or β (Eq. (3)). The model with the lowest mean deviance from RJAGS by more than 2 was selected for that herd47.
Environmental and demographic stochasticity
We estimated fluctuation in abundance which can be attributed to demographic and environmental stochasticity for herds with different K for each herd. The stochasticity model was outlined by Ferguson and Ponciano9;

$$ Varleft( {N_{t – 1} } right) = Var_{dem} left( {N_{t – 1} } right) + Var_{r} left( {N_{t – 1} } right) + Var_{{upbeta }} left( {N_{t – 1} } right), $$
(7)

where (Varleft( {N_{t – 1} } right)) was total population stochasticity, (Var_{dem} left( {N_{t – 1,} } right)) was population abundance fluctuation due to demographic stochasticity, (Var_{r} left( {N_{t – 1} } right)) was population abundance fluctuation due to changes in r (i.e., density-independent environmental stochasticity), and (Var_{beta } left( {N_{t – 1} } right)) was population abundance fluctuation due to changes in β. The model assumes density-dependent survival following the Ricker model. Demographic stochasticity was calculated as follows;

$$ Var_{dem} left( {N_{t – 1} } right) = alpha N_{t – 1} e^{{ – beta_{Delta } left( {N_{t – 1} } right)}} left( {1 – e^{{ – beta_{Delta } left( {N_{t – 1} } right)}} } right) + sigma_{dem}^{2} N_{t – 1} e^{{ – 2beta_{Delta } left( {N_{t – 1} } right)}} $$
(8)

where (sigma_{dem}^{2}) was assumed to be equal to α9. Environmental stochasticity that is expressed as changes in r, otherwise known as density-independent or additive stochasticity, was calculated as follows;

$$ Var_{r} left( {N_{t – 1} } right) = sigma_{{beta_{Delta } }}^{2} alpha^{2} N_{t – 1}^{2} e^{{ – 2beta_{Delta } left( {N_{t – 1} } right)}} , $$
(9)

and environmental stochasticity that is expressed as changes in β, otherwise known as density-dependent or multiplicative stochasticity, was calculated as follows;

$$ Var_{{upbeta }} left( {N_{t – 1} } right) = sigma_{{beta_{Delta } }}^{2} alpha^{2} N_{t – 1}^{2} left( {N_{t – 1} } right)^{2} e^{{ – 2beta_{Delta } left( {N_{t – 1} } right)}} . $$
(10)

Population growth parameters from the selected Ricker growth model for each herd were used in these equations to estimate each of these sources of stochasticity for each herd across abundances ranging from five to above K. The relative total population stochasticity was expressed as the total population stochasticity at K for each herd divided by that herd’s K. More

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General
This study was carried out in the Lijiang Forest Ecosystem Research Station, Yunnan Province, China during the period 13 July to 3 August 2019. This field station, which is operated by the Kunming Institute of Botany, Chinese Academy of Sciences, is located to the north of Lijiang on Yulong Snow Mountain at an elevation of 3200 m. Canopy vegetation is dominated by Pinus yunnanensis and Quercus variabilis, and Rhododendron decorum is a highly conspicuous understory shrub species, when in flower. Flowers occur in inflorescences with each plant typically having many inflorescences.
We chose, numbered and bagged one inflorescence on each of 25 plants of Rhododendron decorum, followed the state of individual flowers, and sampled nectar according to a protocol explained below. We selected plants, as encountered, that were flowering and within about 10 m of our walking path, which was along a road and foot track near the field station. We selected inflorescences, one per plant, with at least five unopened buds, and marked five of these buds with small lengths of differently coloured plastic drinking straws22. The different colours enabled us to distinguish flowers during nectar sampling and subsequent measurements of flower colour. All marked flowers were then checked daily to record flower state as bud, beginning to open, open-non-abscised, and open-abscised. Flowers were considered buds if there was no sign of petals unfolding, beginning to open if petals had begun to unfold, and open if petals had unfolded completely. Abscised flowers were clearly indicated by separation between the base of the petals and the rest of a flower, which was along a distinct abscission line (Fig. 1c). Inflorescences were bagged, using green mesh organza bags, to prevent any flower visitation and nectar removal. This species is self-incompatible19, so no pollination occurred.
We carried out two experiments, involving a total of 25 plants. One (Experiment A) involved 10 plants (numbered A-1 to A-10) and was carried out between 13 and 23 July 2019. Experiment B involved 15 plants (numbered B-1 to B-15) and was carried out between 24 July and 3 August 2019. Experiment B was carried out to increase sample sizes for flowers of all ages, and to provide information for relatively young flowers that was not provided by Experiment A (explained further below). Plants were numbered as encountered.
Collection of inflorescences
Inflorescences from Experiment A and Experiment B were collected for sampling of nectar according to the following protocol.
For all inflorescences in Experiment A (i.e., 10 inflorescences) and all in Experiment B, except numbers B-3, 6, 9, 12 & 15 (i.e., 10 inflorescences), each inflorescence was removed from its plant on the first day that abscission of any marked flower was observed. If any marked flower was observed to have abscised, its inflorescence was removed from its plant by breaking its subtending stem and taken to a nearby sheltered ‘nectar sampling station’ where nectar measurements were made. This occurred for flowers between 3 and 9 days of age, counting the first day that a flower was either open or beginning to open as age 1. In a small number of cases, flower abscission occurred when marked flowers were gently touched just prior to nectar sampling. Such flowers were also considered to have abscised.
In addition, five inflorescences from Experiment B (i.e., numbers B-3, 6, 9, 12 & 15) were similarly collected when they were four days old, regardless of whether any flowers had abscised. This provided nectar measurements for relatively young flowers (i.e., ages 1 to 4 days).
Nectar sampling
For collected inflorescences, almost all the marked flowers were open, and we sampled accumulated nectar in each marked and open flower as follows. Nectar was removed using micro-capillary tubes (Hirschmann microcapillary pipettes; 5 µl in Experiment A; 10 µl in Experiment B; both 32 mm long), with volume measured on the basis of nectar length along tube and subsequently converted to µl. When about 0.5 µl of nectar was obtained, this was expelled to a hand-held refractometer (i.e., Bellingham & Stanley, 0 to 50% brix, adjusted for small volumes) for measurement of sugar concentration as % wt/wt sucrose equivalents. These measurements were adjusted for ambient temperature (see Supplementary Information) using a formula developed from information supplied by the manufacturers of the refractometers we use23 and converted to wt/vol using the following formula24: Y = 0.00226 + 0.00937X + 0.0000585X2 where Y is sugar mass per unit volume (mg/µl) and X is % concentration wt/wt. The amount of sugar for a flower (in mg) was then calculated by multiplying nectar volume (µl) by sugar mass per unit volume (mg/ µl).
Nectar was sampled, for both abscised and non-abscised flowers, from where it accumulates after secretion (Fig. 1c). Nectar was sampled for non-abscised flowers from the base of the corolla between the ring of about 10–15 nectaries, around the base of the ovary, and adjacent flower petals. For flowers that had abscised, nectar was separately sampled from both the ring of nectaries and the inside lowest 5 mm of the flower petals, where some nectar becomes attached.
Some flowers were judged to have been affected by rain and their nectar concentration measurements were excluded from analyses. There were periods of rain during our study and occasionally nectar concentration readings of lower than 1.5% wt/wt were obtained (n = 6), and the nectar assumed to have been diluted by rainwater. These records were excluded from analyses. Fortunately, most flowers pointed downwards and were thus not affected by rain.
Flower colour and pigment
Flower colours were measured by means of a modified Panasonic GH-1 camera. The low-pass filter of the camera had been removed in order to increase the sensitivity for ultraviolet light. The camera body was combined to an Ultra-Achromatic-Takumar 1:4.5/85 lens made of fused quartz that transmits UV light. Since the modified camera is sensitive to ultraviolet and infrared light, a UV-/IR-Cut filter transmitting light between 400 nm and 700 only nm was used to capture a normal reference picture. In addition, a UV-picture was captured from the identical position using a Baader UV-filter that transmits near ultraviolet light only. A white Teflon disc reflecting equal amounts of light in a range of wavelength from 300 to 700 nm was used for manual white balance before taking pictures. Using Image J both pictures were split into the RGB color channels, and then a false color photo was merged using the green channel of the color picture as red, the blue channel of the color picture as green, and the blue channel of the UV picture as blue (see Supplementary Information). For more details see article by Verhoeven et al.25. Using IrfanView image’s histogram a uniform non-decomposed area (number of pixels  > 10,000) of the adaxial corolla on the false color picture was selected. The average intensity for the red, green and blue channel of the false color photos with values between 0 and 255 was used for color evaluation. Abscised and non-abscised flowers were photographed together enabling direct comparison of the colours of the flowers.
Pigment content was deduced from the sum of the values of the red, green and blue channel of the false color photos. Since abscised and non-abscised flowers both appear white to the human eye, the possible change in the content of a UV-absorbing pigment was checked by comparing the value for the blue channel in relation to the sum for the values of the green and red channels.
Recordings of the spectral reflectance were done with an abscised and an open, non-abscised flower from each of five inflorescences. Reflectance measurements were performed with an USB2000 + spectrophotometer (Ocean Optics) and illumination was provided by a DH-2000-BAL light-source (Ocean Optics), both connected via a coaxial fibre cable. All measurements were taken in an angle of 90° to the measuring spot with a pellet of barium sulphate used as white standard and a black piece of plastic used as black standard.
Analyses
We used the General Linear Model approach to determine relationships for all flowers between nectar attributes (i.e., volume—µl, concentration—wt/vol, sugar weight—µg) as dependent variables and flower age, whether abscised, experiment (i.e., A vs. B), and Plant ID as independent variables. We also treated Plant ID as an independent categorical variable, but nested within experiment.
We used ANOVA to evaluate relationships between reflectance intensity and whether flower abscised, across different false colours, with Kolmogorov–Smirnov test for normality and Tukey post-hoc comparisons between means. We took log intensity as the dependent variable in order to meet the normality assumption.
We compared spectral reflectance for abscised and open, non-abscised flowers on the basis of the average reflectance across all wavelengths. Here we assumed that the 1140 reflectance values for each flower could be combined into a single average measure and that this average measure adequately represented each flower. We compared the two groups of flowers with a Kolmogorov–Smirnov Two Sample Test.
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