Ecology

E.D.S. was supported by the International Mobilities of Researchers of the Biology Centre (grant no. CZ.02.2.69/0.0/0.0/16_027/0008357). L.E. is supported by funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (ERC Starting grant no. 803151). M.W.B. was supported by the United States National Science Foundation Division of Environmental Biology (grant no. 1456054). M.K. was supported by Fellowship Purkyně (Czech Academy of Sciences) and by the project Centre for research of pathogenicity and virulence of parasites r.n.: CZ.02.1.01/0.0/0.0/16_019/0000759. More

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The videos were imported from digital videotapes using iMovie 4 and iMovie HD 6.0.3. They were deinterlaced using JES Deinterlacer 3.8.4. Images are processed here using QuickTime Player 7.3.3, GraphicConverter 8.8.3, and GIMP 2.10. Within these applications, it is possible to interpolate and adjust brightness, contrast, color, and other parameters. The simple processing applied here is effective for some cases. With advanced processing techniques that involve greater control and analysis of parameters, experts in image processing might be able to extract additional information.
The 2006 video
The first video was obtained from a kayak with a Sony DCR-HC36 standard video camera (which captures interlaced video at 720 × 480 pixels) in the Pearl River swamp in Louisiana on February 20, 2006, in an area along English Bayou where there were five sightings that week; the ‘kent’ calls of the Ivory-billed Woodpecker were heard twice during the same period, once coming simultaneously from different directions. The 2006 video shows a large woodpecker perched on a tree, climbing upward, taking a short flight between limbs, and then taking off into a longer flight. Part of the perch tree, which includes two forks that facilitated scaling, was used in the size comparison in Fig. 2; the bird in the video appears to be larger than a Pileated Woodpecker specimen8. According to Julie Zickefoose, whose paintings of the Ivory-billed Woodpecker have appeared on the covers of the January 2006 issue of the Auk and both editions of Ref.3, the “long but fluffy and squared-off crest,” “extremely long, erect head and neck,” “large, long bill,” “bill to head proportions,” “rared-back pose,” “long and thin” wings, “flapping leap” between limbs, and “ponderous and heavy” flight are suggestive of the Ivory-billed Woodpecker but not the Pileated Woodpecker13.
Figure 2

A pileated Woodpecker specimen is mounted on part of the perch tree. Frames from the 2006 video were scaled using forks in the tree (dashed lines). A meter stick is placed at the point where the flight between limbs occurred. The inset shows Pileated Woodpecker and Ivory-billed Woodpecker specimens that were photographed side by side at the National Museum of Natural History. The bird in the video is partially hidden by vegetation in the image on the lower left, but it is fully in view in the images at the top when it took the flight between limbs.

Full size image

The 2008 video
A short distance up the same bayou, another video was obtained with the same camera on March 29, 2008, from 23 m up a tree that was used as an observation platform for keeping watch for Ivory-billed Woodpeckers flying over the treetops in the distance. A large bird that flew along the bayou and passed below was identified as an Ivory-billed Woodpecker on the basis of two white stripes on the back and black leading edges and white trailing edges on the dorsal surfaces of the wings (those definitive field marks were observed from an ideal vantage point at close range and nearly directly above). The appearance in the video of the bird, its reflection from the still surface of the bayou, and reference objects made it possible to determine positions along the flight path and obtain estimates of the flight speed and wingspan. The bird in the 2008 video folded its wings closed during the middle of each upstroke as illustrated in Fig. 3. The two large woodpeckers are the only large birds north of the Rio Grande that have this distinctive wing motion, which is clearly resolved in the video. Using an approach that he had previously developed and applied to other woodpeckers17, Bret Tobalske, an expert on woodpecker flight mechanics, digitized the horizontal and vertical motions of the wingtips and concluded that the bird in the video is a large woodpecker13. The flap rate of the bird in the video is about ten standard deviations greater than the mean flap rate of the Pileated Woodpecker13.
Figure 3

Illustrations of large woodpeckers in flight. Left: The Pileated Woodpecker typically swoops upward a short distance before landing on a surface that faces the direction of approach; the Ivory-billed Woodpecker has long vertical ascents that allow time for maneuvering and landing on surfaces that do not face the direction of approach. Center: An Ivory-billed Woodpecker takes off with rapid wingbeats into a horizontal flight that quickly transitions into an upward swooping flight. Right: Illustration of a flight in the Pearl River swamp on March 29, 2008, that was viewed from 23 m up in a cypress tree. When the wings are folded closed in flight, the dorsal stripes and the white triangular patch have the same appearance as they do for the perched birds in Fig. 1. As discussed in Movie S6 of Ref.8, the wings of an Ivory-billed Woodpecker in a historical photo and of the bird in the 2008 video have the swept-back appearance of the wings in the middle image.

Full size image

Additional characteristics of the bird in the video that are consistent with the Ivory-billed Woodpecker but not the Pileated Woodpecker are the high flight speed, narrow wings, swept back wings, and prominent white patches on the dorsal surfaces of the wings8,13. There is one characteristic of the bird in the video that was initially thought to be inconsistent with the Ivory-billed Woodpecker. On the basis of historical accounts of a ‘duck-like’ flight, the Ivory-billed Woodpecker was thought to have a duck-like wing motion in which the wings remain extended throughout the flap cycle. In a series of paintings of the large woodpeckers in flight by Zickefoose18, the wings of the Pileated Woodpecker are correctly shown folding closed during the middle of the upstroke; in a proper representation of conventional wisdom at the time, the wings of the Ivory-billed Woodpecker are shown remaining extended throughout the flap cycle (duck-like flaps). An apparent paradox arose during the initial inspection of the video, which revealed an unexpected wing motion. The paradox was resolved after the discovery that a photo from 1939 shows an Ivory-billed Woodpecker in flight at an instant when the wings are nearly folded closed13.
The 2007 video
The other video was obtained with a Sony HDR-HC3 high-definition video camera (which captures interlaced video at 1,440 × 1,080 pixels) that was mounted on kayak paddles8 in the Choctawhatchee River swamp in Florida on January 19, 2007, in an area where an ornithologist and his colleagues had recently reported a series of sightings7. During an encounter with a pair of birds that were identified as Ivory-billed Woodpeckers on the basis of field marks and remarkable swooping flights, the camera captured a series of events that involve flights, field marks, and other behaviors and characteristics that are consistent with the Ivory-billed Woodpecker but no other species of the region. The analysis of the 2007 video is based in part on the fact that the probability of a series of unlikely events becomes extremely small as the number of events increases12. There is a downward swooping takeoff with a long horizontal glide that is consistent with the following account by Audubon15: “The transit from one tree to another, even should the distance be as much as a hundred yards, is performed by a single sweep, and the bird appears as if merely swinging from the top of the one tree to that of the other, forming an elegantly curved line.” There are upward swooping landings with long vertical ascents that are not consistent with the Pileated Woodpecker but are consistent with an account by Eckleberry of an Ivory-billed Woodpecker that “alighted with one magnificent upward swoop”19.
A long vertical ascent allows time for maneuvering, and the bird appears to rotate about its axis during two of the ascents as illustrated in Fig. 3. In a film of the closely related Magellanic Woodpecker (Campephilus magellanicus)20, there is maneuvering during a landing with a long vertical ascent. During and after one of the ascents, a woodpecker in the 2007 video shows field marks and body proportions that are consistent with the Ivory-billed Woodpecker but no other species of the region. There is a takeoff into horizontal flight with deep and rapid flaps that are not consistent with the Pileated Woodpecker but are similar to the deep and rapid flaps during a takeoff of the closely related Imperial Woodpecker (Campephilus imperialis)21. In another event, a woodpecker climbs upward and engages in a series of behaviors that are consistent with the Ivory-billed Woodpecker but no other species of the region, including delivering a blow that produces an audible double knock and taking off with rapid wingbeats into a flight that immediately transitions into an upward swooping flight that is illustrated in Fig. 3. More

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Study area and nest-box occupancy
The study area is located in the Kauhava region, western Finland (62° 54′–63° 16′ N, 22° 54′–23°47′ E; ca. 1,300 km2 area; altitude 42 m), where the landscape is mainly characterized by a mosaic of commercially managed coniferous forests, agricultural land and peatland bogs23,49. Some mixed and old-growth forests as well as many clear-cuts and sapling areas are also found within the area. The area is sparsely populated, and settlement mainly consists of one-family houses and farmhouses.
The flying squirrel is dependent on natural cavities, which have become scarce in Finnish managed forests, including our study area40. In the study area, flying squirrels used nest boxes built for Pygmy owls (Glaucidium passerinum) that are set up for research purposes (e.g.23,40,50). This nest-box type resembles cavities made by the great spotted woodpecker (Dendrocopos major) with the thickness of the front wall  > 50 mm and the diameter of the entrance-hole of 45 mm. Nest boxes were grouped so that there are 2 boxes 80–100 m apart within a forest site, the sites being at least 0.8–1.0 km apart23,40. The 2 boxes per site were within an average flying squirrel female territory (8 ha51), and the data for these boxes were combined, that is, if one of the boxes was occupied the site was classified as occupied. In other words, the site was used as a sampling unit (on average 364 ± 121 nest boxes in 208 ± 61 sites yearly).
The occupancy of the nest boxes by flying squirrels was checked every spring and autumn in 2002–2018. Sites were often visited more than once in both spring and autumn, and we control for the number of visits in our analysis (on average 4 ± 1 visits per year on a site). Occupancy was determined by the presence of flying squirrel nesting material within the nest box (ball-shaped nest made of lichen, moss and other soft material, distinct from the nests built by any other animal in the area). If the nest was not used, the nest material was lacking or was a flat layer in the bottom of the nest box, often covered with bird nest materials. The flying squirrel occupied about 9% of the available nest-boxes23. The density of nest boxes was low (0.3 boxes per 1 km2) suggesting that nest boxes had only a minor role in the spatial distribution of the flying squirrel population within the area. The nest boxes were in various forest types, but the detection probability in different forest types does not differ substantially in our data23.
The occupancy patterns were expected to reflect the seasonal mortality and dispersal patterns described in the introduction of this study (seasonal models: (i) dispersal model, (ii) summer survival model, and (iii) winter survival model). Individuals do have more than one nest during year in nest-boxes, dreys and natural cavities21. We could not observe individuals if they did not use nest boxes. Natural cavities were, however, rare near the nest boxes40 and the nest boxes were made to resemble natural cavities by using the trunk of spruce (Picea abies) or aspen (Populus tremula). Communal nesting behavior or reproductive success do not differ for flying squirrels living in these nest boxes and natural cavities in Finland21. The lack of cavities means that flying squirrels present in the area had a reason to build nests to nest boxes, because cavities or nest boxes are preferred nesting places over dreys21. Thus, there should not be much individuals not using our nest boxes, although it is clear that such a cases do occur (see “Discussion”). The data includes, for example, cases where the residents died during summer, but we did not detect them, because dispersers recolonised the nest box. In practise, the number of such cases remains low in our data. It would simultaneously require that in an occupied nest box (occupancy rate of available nest boxes was on average 9%) the resident adult dies during summer (adult summer mortality is not high) and a disperser arrives to the site, which likelihood for a specific nest box remains low. Finally, we are unaware of species that might prevent flying squirrels from using the nest boxes, except for the Pygmy owl. In spring, 5 to 10% of nest-box sites (3–6% of nest boxes) were occupied by breeding pygmy owls and in autumn 17% of nest-boxes included food-stores of Pygmy owls50. Pygmy owls do not prey on flying squirrels but may affect the availability of nest boxes. However, one nest box per forest site was available for flying squirrels even in the sites used by a Pygmy owl, thanks to the study design of two nest boxes per site.
Winter food
Birch catkins are the main food for the flying squirrel in winter21, likely, because the birch is the most abundant deciduous tree in Finnish forests. However, alder catkins are preferred over birch catkins21, and recent studies indicate that the availability of alder catkins in the winter and spring preceding reproduction is an important determinant of breeding success22,25. Temperature in summer determines catkin production30, that is, catkins mature during summer, are available for flying squirrels starting in autumn and stay dormant over winter. Thus, in the current analyses temperature measured in summer is related to next winters’ catkin availability. Catkins flower in spring but flying squirrels may extend the period of catkin usage by storing them21.
For birch catkin availability, we used estimates from an annual birch catkin survey conducted by the Natural Resources Institute Finland (www.luke.fi). These data are collected to describe nation-wide pollen conditions in Finland. Catkin production of deciduous trees is spatially auto-correlated at scales of up to a few hundred kilometres in Finland30,51, and we used the estimate for central-western Finland, where our study area is located. The birch catkin data for central-western Finland is collected annually at approximately six different locations from 304 trees within the region. We did not have an estimate for alder catkin production, but following earlier studies22,25,26,52, we used aerial pollen estimates for central-western Finland as a proxy for alder catkin production (https://www.norkko.fi/). Pollen data were collected by the aerobiology unit of the University of Turku from 10 locations in Finland using EU standard methods and Burkard samplers. The data consisted of accumulated sums of average daily counts of airborne pollen in 1 m3 of air during spring30. Thus, winter food data used in this study describes yearly changes in catkin availability in the region.
Weather data
We used mean monthly weather information from the weather station maintained by the Finnish Meteorological Institute in Kauhava53. The weather recording station was in the middle of the study area and at the same altitude as the rest of the area. There is minimal spatial variation in mean monthly weather measures within our flat study area. We counted mean temperature and precipitation from monthly means for the following periods: winter (December–February), spring (April–May), summer (June–August) and autumn (October–November). March and September were excluded, as they could not be unequivocally assigned to a specific season and, thus, to life stages of flying squirrels (spring: reproduction; summer: rising juveniles; autumn: dispersal period; winter: surviving from the elements). Including these months to analysis did not change the current results or conclusions.
During the study period of 2002–2018, the temperature had an increasing trend in winter and autumn, and a negative trend in summer (effect of continuous variable year on temperature: in winter positive relationship r2 = 0.09; in spring positive r2 = 0.01; in summer negative r2 = 0.09; in autumn positive r2 = 0.1). For precipitation, the trends were positive or non-existing (effect of year on precipitation: in winter positive r2 = 0.04, in spring positive r2 = 0.02, in summer positive r2 = 0.07, in autumn r2 = 0).
Predation pressure
Flying squirrels are negatively affected by the presence of the Ural owl in our study area23. Other predators play a lesser role without having major impacts on flying squirrels (the goshawk Accipiter gentilis23), or are not very common in the area (the pine marten Martes martes and the eagle owl Bubo bubo48). The Ural owl prefers mature mixed and spruce-dominated forest54, just like the flying squirrel. Data on Ural owls was collected by surveys on natural cavities and nest boxes and by searching for new nest sites annually in 2002–2018. Long-term studies of birds of prey have been carried out in the Kauhava region (e.g.40,48,49), so the locations of Ural owl nests are known. The density of Ural owls was approximately 2 pairs per 10 km2 (48; M. Hänninen & E. Korpimäki, unpublished data).
Using the data for Ural owl nests located during the field surveys, the predator presence at flying squirrel nest-box sites was described by calculating flat-top bivariate Gaussian kernels around each nest (see23,55). Following our earlier analysis23, we calculated the kernels with a flat top distance of 500 m, SD of 4 and cut off distance of 5 km. The flat-top part represents the area where the impact of the avian predator is strongest, beyond which it declines, following the Gaussian distribution. The height of the kernel (0–1) at flying squirrel nest box was used as a proxy for predation pressure (referred to as Ural owl index). The kernels were calculated using ArcGIS 10.1 software by Esri and R 3.2.555. The Gaussian kernels were used because the location of nests was known, but we do not know the exact hunting area of individuals. The kernels were based, however, on expert knowledge on likely hunting distance of the species56. That is, the hunting effort was assumed to be highest close to the bird’s nest and to remain at a high level within a given distance and then decrease symmetrically in all directions when moving further from the nest.
Habitat data
The areas of different land use classes within a buffer of 200 m were calculated for each nest box in ArcGIS and R. The buffer corresponds roughly to the estimated home-range size of female flying squirrels50. Thus, the selected spatial scale captured habitat composition at the level central for reproductive success. Landscape maps were based on SLICE dataset57, two forest classifications from 1997 and 2009 (METLA, https://www.maanmittauslaitos.fi/en/opendata), and Landsat images (https://landsat.usgs.gov/), so that yearly changes in forest cover (e.g. clear-cutting of forest) were taken into account. For a detailed description of map processing, see40. We compared which forest composition best describes the squirrel presence and selected the one best fitted to the data based on an Akaike Information Criterion (AIC). That is, model combinations with different forest types and age classes were tested and the one with lowest AIC-values was selected to final models being best fitted for the analysis. The habitat best explaining flying squirrel occurrence included all mature and old spruce and mixed coniferous–deciduous forests. Pure pine forests, which are not preferred by the species21, were excluded.
The available habitat data ended in the year 2015 because we had no information for changes in the forest cover after 2015. We updated the habitat data until 2018 with the values for 2015, but in the end decided to use only the habitat data until 2015 and omitted it from the final models, because it had no effect (see “Results”). Thus, we gained full power to analyse the effects of weather, winter food and predator pressure on flying squirrel occupancy patterns.
Analyses—dispersal model, summer survival model, and winter survival model
We built three binary models (using GLIMMIX in SAS 9.4. software) with nest-box occupancy in different seasons as a response variable. In each model, the nest-box site was a repeated factor (using generalized estimation equations, GLIMMIX SAS) and the year and average number of nest-box visits per year were continuous explanatory variables. To simplify the models, we used an AIC comparison to select the weather variables that were best fitted to the model (AIC  More

Experimental design
Raw data used in the manuscript were collected by the ARC-SGI of South Africa in open field test plots. The raw data include observed dryland wheat yields matched by location with daily minimum and maximum temperatures and total precipitation recorded during the growing season at near-by weather stations. Weather station data were downloaded from NASA GSOD using GSODR59. From the raw data, we only include wheat trial locations that have a weather station within 75 km and at least five years of wheat field trials, and wheat cultivars must appear in at least two trial years. It is possible that there are slight differences in the weather station observations and the actual weather at wheat trial locations, particularly with respect to precipitation. However, the weather station observations in the study region appear representative based on climatic norms60 and are the best available data for capturing daily extremes. This results in 18,881 wheat yield observations from ARC-SGI spanning 17 locations and 71 cultivars from 1998 to 2014.
The yield and weather data vary substantially in-sample, which supports robust estimation of wheat yield responses to extreme and average weather conditions (Supplementary Tables 1, 2; Supplementary Figs. 1 and 2). The growing season for each location-year-cultivar is defined by the planting and harvest dates, and typically span mid-May to late October. Planting and flowering dates are observed, not estimated. Flowering is defined as the day at which 50 percent flowering occurs. Harvest dates are not observed and thus inferred using a rule of 30 days after the observed flowering date, which provided consistent results with other alternatives discussed in the robustness checks below. Phenological information were collected by both ARC staff and wheat producers based on a field observation conducted once per weekday for ARC run stations and daily for producer fields. The temperature bins are calculated from maximum and minimum temperatures using a sinusoidal interpolation of temperature exposure within each day and span 5 °C intervals. Total days (24 h) spent within intervals for the entire season are summed into eight temperature exposure bins. All negative temperatures are summed into a single bin, as well as all temperatures above 30 °C. Notably, exposures greater than 30 °C occur substantially more in the Free State compared to the Western Cape.
Statistical analysis
The preferred regression model specifies log wheat yield as a function of location, cultivar, and year fixed effects, as well as a quadratic polynomial for cumulative precipitation and the eight temperature bins mentioned above61. The weather variables are seasonal aggregates from the observed planting date to the inferred harvest date. The highest temperature bin of >30 °C represents exposures known to negatively affect wheat yields62,63,64. Average exposures across bins are provided for the two main dryland wheat-growing provinces, the Free State and Western Cape, in Supplementary Fig. 1. We considered simplified models that include linear and quadratic trends instead of year fixed effects, or (alternatively) omitting the temperature bins in favor of average temperatures, and found that they substantially reduced model performance (Supplementary Tables 3, 4). In addition, we considered extensions of the model that added pre-season precipitation (30 days before planting), or (alternatively) a cubic polynomial for in-season precipitation instead of a quadratic, and found that they also did not improve model performance. The preferred model is specified in Eq. (1):

$$y_{ijt} = alpha _i + alpha _j + alpha _t + beta _1p_{ijt} + beta _2p_{ijt}^2 + mathop {sum}limits_{k = 1}^8 {delta _k} Bin_{ijkt} + varepsilon _{ijt},$$
(1)

where yijt is log yield for cultivar i in location j in year t. Fixed effects (α) are included separately for cultivars, locations, and years. The weather variables include a quadratic polynomial effect for cumulative precipitation pijt and the nonlinear effect of weather across temperature bins Binijkt.
There is likely a large amount of spatial correlation among the error terms of the model across cultivars in the same location, as well as across locations more generally. One could cluster standard errors by year to account for all spatial correlations, however there are 17 years in the data which is a questionably small number of clusters65,66. Instead we cluster errors by year-province as there are only two provinces in the data, Western Cape and the Free State, and their boundaries are several hundred kilometers apart. This method accounts for correlations among the regressors which can also bias standard errors. Cameron and Miller (2015)65 report the variance inflation factor in their equation 6 as 1 + ρx ρu (N – 1), where N is the cluster size, ρu is the within-cluster correlation of the regression errors, and ρx is the within-cluster correlation of the regressor. Note that spatial correlation of the regressors can bias regression standard errors downward even if the errors are only slightly correlated. Just under their equation 6, Cameron and Miller (2015)65 cite a study in which the correlation of the errors was small at 0.03 but the inflation factor was 13 because the regressors were highly correlated.
To better investigate the role that spatial correlation is playing in this analysis, Moran’s I was calculated for each year in the data for both the log yield observations and the regression errors from the preferred model above. The averages of the Moran’s I across years is presented in Table 1. The distance of 1 km captures the within-trial correlations, whereas the distances 100, 500, and 1000 km capture broader groupings. Positive correlation exists in the log yield data and it is highest within-trial as expected. The correlation remains positive as distance increases but dilutes to its smallest value at 1000 km. The regression purges much of the correlation from the data as indicated by the Moran’s I for the errors, although some remains. As noted above, the clustered errors may still produce an adjustment by increasing the variances compared with classical Ordinary Least Squares which does not account for correlations. For example, the standard error on our measure of heat (the >30 °C bin) is 0.0196 under clustering but 0.00433 without clustering (i.e., robust standard errors). This suggests an inflation factor of approximately 20 which is quite large and important for adequately representing the statistical uncertainty in our warming impacts.
Table 1 Moran’s I (MI) spatial autocorrelation for log yield and regression errors.
Full size table

Heterogeneous cultivar-level temperature effects are investigated to assess the potential for climate change adaptation via cultivar selection. The preferred specification was modified to account for differences in cultivar effects using the following multilevel model66 specified in Eq. (2):

$$y_{ijt} = alpha _i + alpha _j + alpha _t + beta _1p_{ijt} + beta _2p_{ijt}^2 + mathop {sum}limits_{k = 1}^8 {delta _k} Bin_{ijkt} + u_iBin_{ij8t} + varepsilon _{ijt},$$
(2)

where we extend the preferred model to include a random slope (ui) across cultivars for the highest temperature bin (30 °C+). Note that the fixed effects from the preferred model are include here as dummy variables in the fixed portion of the multilevel model. The only random effect in the multilevel model is for the effect of the >30 °C bin.
Warming impacts are based on uniform changes in the daily temperature data. For example, we use the observed (historical) daily minimum and maximum temperatures and increase them by 1 °C and then re-calculate the growing season bins for all locations and years3,43,45. Averaging these across years and locations then provides a shifted climate to simulate yield change based on the initial regression model parameters and yield estimates. The impacts are calculated as (100left[ {e^{left( {{boldsymbol{Bin}},1 – {boldsymbol{Bin}},0} right)delta } – 1} right])where Bin is a vector of the temperature bins for shifted (1) and baseline (0) climate. The same steps are repeated for the 2 and 3 °C warming scenarios as well. Estimates from the regression in Supplementary Table 3 are used for δ. The point estimation for warming scenarios relies on the Delta Method of asymptotic approximation for large samples as implemented via the nlcom command in Stata version 16.
Robustness checks
The first robustness check we consider is replacing the temperature bins with a quadratic specification of seasonal average temperatures. Interestingly, a two-tailed joint test under this model implies that temperatures do not have a statistically significant effect on yields (F(2,30) = 0.57, p = 0.5716), thereby suggesting that seasonal averages cannot capture yield reductions associated with heat above 30 °C as in our preferred model. The seasonal average model generates misleadingly small warming impacts (Supplementary Fig. 4).
Next we investigate the appropriateness of the equally spaced five degree exposure bins by examining three alternatives: (i) bins of length three degrees, (ii) bins of length five degrees but with a threshold of 29 °C and, separately, (iii) a threshold of 31 °C. We find that all three alternatives produce similar marginal effects of temperatures and warming impacts as our preferred model (Supplementary Figs. 5 and 6).
Under our preferred model the parameters for precipitation and precipitation squared are statistically significant for a two-tailed joint test (F(2,30) = 9.43, p = 0.0007). We find that the yield effects of precipitation are not trivial as a one standard deviation reduction in cumulative rainfall below the average level is associated with a 9.6% yield reduction. To more directly investigate the differentiated impacts of drought and heat, the precipitation component was modified to include the quadratic function (as in the preferred model) along with an indicator variable that takes on a value of “1” when cumulative precipitation is below the 10th percentile of all observed rainfall data. This indicator captures low rainfall conditions likely associated with droughts, and findings suggest the effect of 10th percentile rainfall is an 18% yield reduction (Delta Method = −2.95, p = 0.003). The inclusion of the additional low-rainfall control variable produced similar marginal effects of temperatures and warming impacts as our preferred model (Supplementary Figs. 7 and 8). In addition, we consider controlling for the seasonal variation of precipitation as in Rowhani et al. (2011)67, but found a similar pattern of results for the temperature and warming effects (Supplementary Figs. 8 and 9). Thus, the high temperature effect and precipitation effect seem well differentiated from each other, likely due to the location and year fixed effects that control for (among other things) locations with a more drought-prone climate and widespread droughts across locations within years.
It is essential that cultivars in the data experience sufficient heat exposure to capture the temperature effects, especially when we estimate the cultivar-specific heat effects. Within the sample, every cultivar was exposed to temperatures above 30 °C ranging from 4 to 115 h. Not every cultivar was exposed to temperatures above 30 °C at every location, but cultivars with no exposure above 30 °C at every location account for less than 10 percent of observations. Nonetheless, as a robustness check for the warming impact estimates we drop cultivar-years not experiencing exposures above 30 °C and re-estimate the model. We find similar marginal effects of temperatures and warming impacts as our preferred model (Supplementary Figs. 9 and 10).
We also consider whether allowing the temperature and precipitation effects to vary within season affects the warming impacts. We separate the growing season into three stages: (i) planting to 20 days before flowering to capture the vegetative stage, (ii) 20 days before to 10 days after the flowering date to capture the flowering stage, and (iii) 10 days after flowering to the end of season to capture the grain-filling stage. We then re-estimate the model including stage-specific measures of the precipitation and temperature variables, and find that warming impacts are very similar to those from our preferred model approach (Supplementary Fig. 11).
Next, we analyze whether cultivars developed from specific breeders provide differential heat effects by interacting the temperature bin variable for exposures above 30 °C with dummy variables for each of the three breeders represented in our data: Pannar, Sensako, and the South African ARC-SGI. A two-sided joint test of these interactions suggest that the heat effects do differ across breeders for n = 18,629 yield observations with breeder information (F(2,30) = 6.68, p = 0.004), however the magnitude of the differences are small and the warming impacts are similar across all three breeders (Supplementary Figs. 12 and 13). We also consider whether heat effects differ across the spring, facultative, and winter wheat cultivars represented in the data using the same dummy variable approach. A two-sided joint test suggests a lack of statistical significance for these differences (F(2,30) = 2.20, p = 0.128), and the temperature and warming effects are similar across all three types (Supplementary Figs. 13 and 14).
Another robustness check interacts the temperature bin variable for exposures above 30 °C with a continuous variable for the year that each cultivar was publicly released. The in-sample release years span 1984–2012 and we again find a lack of statistical significance for the interaction with a two-tailed test (t(30) = 0.53, p = 0.471) coupled with similar temperature and warming effects (Supplementary Figs. 13 and 15).
The robustness of weather station data was tested by including all available weather stations within 200 km (regardless of missing data) for every wheat trial location using distance-weighting (1/distance2) of the weather observations at the location-year-day level. This increased the number of field trial sites to 32 (some were dropped before because of missing weather data) and the number of unique weather stations to 107. The number of stations matched to a particular site ranged from 12 to 30. We then re-estimate the model using these alternative data and find that the temperature and warming effects are similar to the preferred model (Supplementary Figs. 16 and 17). It is also possible that this distance-weighted interpolation approach is overly simplistic, thereby introducing measurement error that can bias estimates. This type of error would likely affect precipitation more than temperature due to its more localized nature, so we replace our measure of rainfall with that of the gridded Climate Hazards group Infrared Precipitation with Stations (CHIRPS) dataset68. We re-estimate the model and find that the temperature and warming effects are again similar to the preferred model (Supplementary Figs. 16 and 17).
Some studies have shown that wheat maturity occurs more quickly under heat stress62,69. Thus, to test our assumption of a flowering-to-harvest time of 30 days at each location-year, we use this expanded weather station data and re-calculate the temperature bins for a shorter 20 day maturity period. We define the optimal maturity length by running separate regressions of log yield on the weather covariates for each location-year in the data. Each iteration produces two measures of R-squared, one for each of the two maturity lengths, and the higher one is used for that location-year. We find that 30 days is optimal for approximately 2/3 of the location-years (Supplementary Fig. 18). A regression of the improvement in R-squared from varying the maturity length on the occurrence of temperatures above 30 °C suggests that a one percent increase in heat occurrence only improves model fit by approximately 0.001 percent. In addition, we find that optimizing the maturity lengths by location-year produces similar temperature and warming effects as the preferred model (Supplementary Figs. 16 and 17).
Expanding the weather data also provides an opportunity to consider the potential effects of shifting planting dates. Producers may adapt to increasing heat stress by planting earlier to avoid critical periods of heat exposure. To test the implications of this adaptation, warming impacts were simulated based on the initial temperature impacts with different weather variables created by planting date shifts at 7 and 14 days earlier with fixed (days-to-flowering and days-to-harvest) season lengths. For +1 °C, shifting planting dates to 14 days earlier provides approximately one percent reduction in the warming impact on yields, while for +3 °C a 14 day earlier planting date may reduce impacts by about four percent (Supplementary Fig. 19).
Reporting summary
Further information on research design is available in the Nature Research Reporting Summary linked to this article. More