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Literature search and study selectionA systematic literature search was conducted in the ISI Web of Science database for observational and experimental studies published from 1975 to 13 January 2020 using the following search terms: TOPIC: (grassland* OR prairie* OR steppe* OR shrubland* OR scrubland* OR bushland*) AND TOPIC: (drought* OR ‘dry period*’ OR ‘dry condition*’ OR ‘dry year*’ OR ‘dry spell*’) AND TOPIC: (product* OR biomass OR cover OR abundance* OR phytomass). The search was refined to include the subject categories Ecology, Environmental Sciences, Plant Sciences, Biodiversity Conservation, Multidisciplinary Sciences and Biology, and the document types Article, Review and Letter. This yielded a total of 2,187 peer-reviewed papers (Supplementary Fig. 1). At first, these papers were screened by title and abstract, which resulted in 197 potentially relevant full-text articles. We then examined the full text of these papers for eligibility and selected 87 studies (43 experimental, 43 observational and 1 that included both types) on the basis of the following criteria:

(1)

The research was conducted in the field, in natural or semi-natural grasslands or shrublands (for example, artificially constructed (seeded or planted) plant communities or studies using monolith transplants were excluded). We used this restriction because most reports on observational droughts are from intact ecosystems, and experiments in disturbed sites or using artificial communities would thus not be comparable to observational drought studies.

(2)

In the case of observational studies, the drought year or a multi-year drought was clearly specified by the authors (that is, we did not arbitrarily extract dry years from a long-term dataset). Please note that some observational data points are from control plots of experiments (of any kind), where the authors reported that a drought had occurred during the study period. We did not involve gradient studies that compare sites of different climates, which are sometimes referred to as ‘observational studies’.

(3)

The paper reported the amount or proportion of change in annual or growing-season precipitation (GSP) compared with control conditions. We consistently use the term ‘control’ for normal precipitation (non-drought) year or years in observational studies and for ambient precipitation (no treatment) in experimental studies hereafter. Similarly, we use the term ‘drought’ for both drought year or years in observational studies and drought treatment in experimental studies. In the case of multi-factor experiments, where precipitation reduction was combined with any other treatment (for example, warming), data from the plots receiving drought only and data from the control plots were used.

(4)

The paper contained raw data on plant production under both control and drought conditions, expressed in any of the following variables: ANPP, aboveground plant biomass (in grassland studies only) or percentage plant cover. In 79% of the studies that used ANPP as a production variable, ANPP was estimated by harvesting peak or end-of-season AGB. We therefore did not distinguish between ANPP and AGB, which are referred to as ‘biomass’ hereafter. We included the papers that reported the production of the whole plant community, or at least that of the dominant species or functional groups approximating the abundance of the whole community.

(5)

When multiple papers were published on the same experiment or natural drought event at the same study site, the most long-term study including the largest number of drought years was chosen.

In addition to the systematic literature search, we included 27 studies (9 experimental, 17 observational and 1 that included both types) meeting the above criteria from the cited references of the Web of Science records selected for our meta-analyses, and from previous meta-analyses and reviews on the topic. In total, this resulted in 114 studies (52 experimental, 60 observational and 2 that included both types; Supplementary Note 9, Supplementary Fig. 2 and ref. 25).Data compilationData were extracted from the text or tables, or were read from the figures using Web Plot Digitizer26. For each study, we collected the study site, latitude, longitude, mean annual temperature (MAT) and precipitation (MAP), study type (experimental or observational), and drought length (the number of consecutive drought years). When MAT or MAP was not documented in the paper, it was extracted from another published study conducted at the same study site (identified by site names and geographic coordinates) or from an online climate database cited in the respective paper. We also collected vegetation type—that is, grassland when it was dominated by grasses, or shrubland when the dominant species included one or more shrub species (involving communities co-dominated by grasses and shrubs). Data from the same study (that is, paper) but from different geographic locations or environmental conditions (for example, soil types, land uses or multiple levels of experimental drought) were collected as distinct data points (but see ‘Statistical analysis’ for how these points were handled). As a result, the 114 published papers provided 239 data points (112 experimental and 127 observational)25.For the observational studies, normal precipitation year or years specified by the authors was used as the control. If it was not specified in the paper, the year immediately preceding the drought year(s) was chosen as the control. When no data from the pre-drought year were available, the year immediately following the drought year(s) (14 data points) or a multi-year period given in the paper (22 data points) was used as the control. For the experimental studies, we also collected treatment size (that is, rainout shelter area or, if it was not reported in the paper, the experimental plot size).For the calculation of drought severity, we used yearly precipitation (YP), which was reported in a much higher number of studies than GSP. We extracted YP for both control (YPcontrol) and drought (YPdrought). For the observational studies, when a multi-year period was used as the control or the natural drought lasted for more than one year, precipitation values were averaged across the control or drought years, respectively. Consistently, in the case of multi-year drought experiments, YPcontrol and YPdrought were averaged across the treatment years. When only GSP was published in the paper (63 of 239 data points), we used this to obtain YP data as follows: we regarded MAP as YPcontrol, and YPdrought was calculated as YPdrought = MAP − (GSPcontrol − GSPdrought). From YPcontrol and YPdrought data, we calculated drought severity as follows: (YPdrought − YPcontrol)/YPcontrol × 100.For production, we compiled the mean, replication (N) and, if the study reported it, a variance estimate (s.d., s.e.m. or 95% CI) for both control and drought. In the case of multi-year droughts, data only from the last drought year were extracted, except in five studies (17 data points) where production data were given as an average for the drought years. When both biomass and cover data were presented in the paper, we chose biomass. For each study, we consistently considered replication as the number of the smallest independent study unit. When only the range of replications was reported in a study, we chose the smallest number.To quantify climatic aridity for each study site, we used an aridity index (AI), calculated as the ratio of MAP and mean annual PET (AI = MAP/PET). This is a frequently used index in recent climate change research27,28. AI values were extracted from the Global Aridity Index and Potential Evapotranspiration (ET0) Climate Database v.2 for the period of 1970–2000 (aggregated on annual basis)29.Because we wanted to prevent our analysis from being distorted by a strongly unequal distribution of studies between the two study types regarding some potentially important explanatory variables, we left out studies from our focal meta-analysis in three steps. First, we left out studies that were conducted at wet sites—that is, where site AI exceeded 1. The value of 1 was chosen for two reasons: above this value, the distribution of studies between the two study types was extremely uneven (22 experimental versus 2 observational data points with AI  > 1)25, and the AI value of 1 is a bioclimatically meaningful threshold, where MAP equals PET. Second, we left out shrublands, because we had only 14 shrubland studies (out of 105 studies with AI  More

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Seasonality of human–bear conflictOn the Deccan Plateau and Gujarat, most sloth bear attacks occurred in winter, which differs significantly from the seasonality of attacks reported by other studies. Unlike other study areas, people on the Deccan Plateau and in Gujarat are more active in the forest in winter when monsoons and crop harvests have ended. The higher incidence of attacks during monsoons in central India correlates with the increased presence of people farming and protecting crops from cattle depredation, as well as from bears and other wildlife species grazing in nearby forested areas5, 16,17,18. The Kanha–Pench Corridor study was the only one which documented an increase in sloth bear attacks during summer. This increase is concurrent with an increase of people in the forest that collect mahua flower (Madhuca spp) and tendu leaf (Diospyros spp)19. In Sri Lanka, most attacks occurred in the dry season, coincident with the highest levels of human activity in forested areas. People in Sri Lanka enter forests for alternative sources of income as agriculture activity declines during the dry season4.Across all studies, the majority of sloth bear attacks are correlated with the time of year when human activity is greatest in bear habitat. However, the time of year that the peak of human activity occurs in sloth bear habitat varies by region. We conclude that the seasonal activity of bears plays a much smaller role on attack rates than the seasonal activity of humans. Consistent with findings in other studies, human incursion into bear habitat is the primary factor responsible for precipitating conflict21.Time of day influences on human–bear conflictMost studies attributed the time of day that attacks occurred to when most humans were active in the forest4, 17,18,19,20. However, the Deccan plateau differed in that the majority of attacks occurred after dark when fewer people were active in or near the forest. Working in agricultural areas after dark is a more common practice on the Deccan Plateau than for the other study areas due to the availability of electricity and artificial lighting, though even with artificial lighting human activity after dark on the Deccan Plateau is still substantially less than during daytime. While a contributing factor, we do not feel that the increase in nighttime activity on the Deccan Plateau fully explains the significant increase in attacks during that time period as compared to other areas. We suspect that sloth bear activity patterns on the Deccan Plateau, and how bears use their environment, accounts for the shift in attack timing.Sloth bears, though potentially active throughout the day, are predominately crepuscular and nocturnal17, 22,23,24. During daytime, sloth bears seek shelter in naturally occurring caves, crevices between big boulders, the spaces between tree roots, beneath fallen trees, or under bushes1, 25,26,27,28. On the Deccan Plateau, however, sloth bears utilize rocky caves almost exclusively for daytime denning29. A cave reduces chance encounters with people and predators while providing a modicum of security, hence the lower incident rate for areas with naturally occurring caves.Conversely, studies conducted in Sri Lanka, Maharashtra and the Kanha-Pench corridor documented more attacks during daytime when people are more active but sloth bears are less active4, 5, 19. Large areas where sloth bears are located in Sri Lanka do not have caves for resting, though they do have dense vegetation and tree cavities (S. Ratnayeke, personal communication July 28, 2020). The Dnyanganga Wildlife Sanctuary, in the state of Maharastra, is mostly lower plains forest without rocky caves (N. Dharaiya, personal communication June 25, 2020). The Kanha-Pench corridor landscape is largely comprised of sal (Shorea spp) and teak (Tectona spp) forests largely devoid of caves30. The role of caves in minimizing daylight sloth bear attacks may be best exemplified by an attack in Sri Lanka as quoted in Ratnayeke et al.4:
“I was following two of my companions and saw a black form lying at the foot of a clump bushes, about 10 m from me. I called out to my companions. Before I knew it, the impact of the charging bear knocked me off my feet. It happened so fast, I didn’t see the bear coming… just dust, flying leaves, and the screams and roars of the bear.”
Had this bear been in a cave rather than the shade of a bush, it likely would not have felt threatened and reacted defensively. We speculate that during daylight on the Deccan Plateau, sloth bears rest securely within a cave and are not threatened by humans passing nearby. We know that farmers and livestock herders work in relatively close proximity to known den locations without fear of being attacked (S. Shanmugavelu, pers. observation). Clearly, caves afford a level of protection and separation that benefits both bears and humans. Consequently, we suggest this is the most likely explanation as to why there are relatively few attacks on the Deccan Plateau during daytime.Season and sloth bear safety messagingBear attack research and safety messaging often recognizes a seasonal component17,18,19,20, 31 (e.g., more sloth bear attacks occur during the monsoon season than during other seasons). Sloth bears are active year-round, and the rate of attacks is strongly correlated with the level of human activity in the forest. Similarly, in Alaska, Smith and Herrero32 reported that human-brown bear conflicts were strongly seasonal in their occurrence. Additionally, they reported that attacks occurred most often when both people and bears vied for the same resource, such as salmon or ungulates. Farther north, human-polar bear conflict peaks when bears are on land awaiting freeze up in the fall33. Not infrequently, sloth bear safety messaging amounts to little more than general statements such as “when in the forest or in sloth bear country be aware”. In other words, an individual’s odds of being attacked by a sloth bear while in the woods may not significantly vary regardless of season. But, where it has been found to vary by season, this information should be conveyed to the public.Time of day and sloth bear safety messagingSloth bear research and safety messaging often reports and warns of the “most dangerous” time or times of the day to be active in the forest17,18,19,20, 31, 34. Sloth bear attacks, like grizzly bear or American black bear attacks33, can occur anytime, day or night6. However, due to an abundance of naturally occurring caves on the Deccan Plateau, stumbling across a sleeping sloth bear mid-day is much less likely to occur than it is in Sri Lanka or in the Kanha-Pench corridor. Therefore, regional sloth bear safety messaging should acknowledge this significant difference which will promote bear safety.The Corbett Foundation31 and Dharaiya et al.34 do an admirable job of focusing their safety messaging to a specific regional group of people in their respective publications. This type of regional messaging is necessary for optimizing sloth bear safety messaging efficacy. However, there is also value to non-site-specific sloth bear safety messaging. The short film “Living with Sloth Bears”35 intentionally addresses general safety messaging that applies to sloth bears across their entire range. Consequently, in the making of this film, we purposely avoided referring to the timing of attacks, seasons or time of day, or other aspects of human-bear conflict because we were aware of significant differences with respect to these variables between locations.Yet another aspect of bear safety messaging is to keep it simple so that a person, under duress, will remember what to do in the event of a bear encounter Attempting to recall the details of an extended message, especially when being threatened by a bear, can be difficult, if not impossible. Therefore, the trend has been to keep bear messaging as simple as possible and we agree with it. However, teaching people that work in bear habitat the most likely times of day encounters occur can be beneficial. In summary, there is a time and place to provide detailed information that is regionally specific, and other situations in which to keep messaging simple.Sloth bear denning ecology on the Deccan Plateau and its role in human–bear conflictThe Deccan Plateau is known as high quality sloth bear habitat, as evidenced by the relatively high density of bears in this area (S. Shanmugavelu, pers. observation). While there is ample food on the Deccan Plateau, the abundance of caves there sets it apart from other areas within the specie’s range. Sloth bears use only caves or cave-like structures on the Deccan Plateau for resting (Shanmugavelu et al. In Print). Caves provide protection from the elements, such as the heat of the day or severe storms, as well as protection from potential predators. Sloth bears do not have many predators and while a cub or very young bear may be at risk from leopards (Panthera pardus) or wolves (Canis lupes pallipes), the only natural predator of adult sloth bears is the Bengal tiger (Panthera tigris tigris). Tiger scat studies revealed that sloth bears can comprise up to 2% of a their diet36,37,38,39. Tigers no longer occur on the Deccan Plateau, but the abundance of caves in the area undoubtedly historically benefited sloth bears, perhaps facilitating a higher density than would have been otherwise attainable. Presently, however, an increase in human population and habitat loss represents greater threat to the species. More

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We use the notation in Table 1.Table 1 Notation used in the description of the optimisation framework.Full size tableCurrent crop production and areas, P
i(x), H
i(x)We used 5-arc-minute maps of the fresh-weight production Pi(x) (Mg year−1) and cropping area Hi(x) (ha) of 25 major crops (Table 2) in the year 201037. These represent the most recent spatially explicit and crop-specific global data75. Separate maps were available for irrigated and rainfed croplands, allowing us to estimate the worldwide proportion of irrigated areas as 21% of all croplands.Table 2 Crops included in the analysis.Full size tableAgro-ecologically attainable yields ({widehat{Y}}_{i}(x))
We used 5-arc-minute maps of the agro-ecologically attainable dry-weight yield (Mg ha −1 year−1) of the same 25 crops on worldwide potential growing areas (Supplementary Movie 3) from the GAEZ v4 model, which incorporates thermal, moisture, agro-climatic, soil, and terrain conditions42. These yield estimates were derived based on the assumption of rainfed water supply (i.e., without additional irrigation) and are available for current climatic conditions and, assuming a CO2 fertilisation effect, for four future (2071–2100 period) climate scenarios corresponding to representative concentration pathways (RCPs) 2.6, 4.5, 6.0, and 8.576 simulated by the HadGEM2-ES model77. Potential rainfed yield estimates for current climatic conditions were available for a low- and a high-input crop management level, representing, respectively, subsistence-based organic farming systems and advanced, fully mechanised production using high-yielding crop varieties and optimum fertiliser and pesticide application42. We additionally considered potential yields representing a medium-input management scenario, given by the mean of the relevant low- and high-input yields. Future potential yields were available only for the high-input management level. Thus, we considered a total of 175 (=25 × 3 present + 25 × 4 future) potential yield maps. Potential dry-weight yields were converted to fresh-weight yields, ({widehat{Y}}_{i}(x)), using crop-specific conversion factors42,78.Both current and future potential rainfed yields from GAEZ v4 were simulated based on daily weather data, and therefore account for short-term events such as frost days, heat waves, and wet and dry spells42. However, the estimates represent averages of annual yields across 30-year periods; thus, whilst the need for irrigation on cropping areas identified in our approach during particularly dry years may in principle be obviated by suitable storage of crop production79, in practice, ad hoc irrigation may be an economically desirable measure to maintain productivity during times of drought, which are projected to increase in different geographic regions due to climate change80,81.Carbon impact C
i(x)Following an earlier approach8, the carbon impact of crop production, Ci(x), in a 5-arc-minute grid cell was estimated as the difference between the potential natural carbon stocks and the cropland-specific carbon stocks, each given by the sum of the relevant vegetation- and soil-specific carbon. The change in vegetation carbon stocks resulting from land conversion is given by the difference between carbon stored in the potential natural vegetation, available as a 5-arc-minute global map8 (Supplementary Fig. 1a), and carbon stored in the crops, for which we used available estimates8,78. Regarding soil, spatially explicit global estimates of soil organic carbon (SOC) changes from land cover change are not available. We therefore chose a simple approach, consistent with estimates across large spatial scales, rather than a complex spatially explicit model for which, given the limited empirical data, robust predictions across and beyond currently cultivated areas would be difficult to achieve. Following an earlier approach8, and supported by empirical meta-analyses82,83,84,85,86, we assumed that the conversion of natural habitat to cropland results in a 25% reduction of the potential natural SOC. For the latter, we used a 5-arc-minute global map of pre-agricultural SOC stocks7 (Supplementary Fig. 1b). Thus, the total local carbon impact (Mg C ha−1) of the production of crop i in the grid cell x was estimated as$${C}_{i}(x)={{C}}_{{{{{{rm{potential}}}}}},{{{{{rm{vegetation}}}}}}}(x)+0.25cdot {C}_{{{{{{rm{potential}}}}}},{{{{{rm{SOC}}}}}}}(x)-{C}_{{{{{{rm{crop}}}}}}}(i)$$
(1)
where ({{C}}_{{{{{{rm{potential}}}}}},{{{{{rm{vegetation}}}}}}}(x)) and ({C}_{{{{{{rm{potential}}}}}},{{{{{rm{SOC}}}}}}}(x)) denote the potential natural carbon stocks in the vegetation and the soil in x, respectively, and ({C}_{{{{{{rm{crop}}}}}}}(i)) denotes the carbon stocks of crop i (all in Mg C ha−1). By design, the approach allows us to estimate the carbon impact of the conversion of natural habitat to cropland regardless of whether an area is currently cultivated or not.In our analysis, we did not consider greenhouse gas emissions from sources other than from land use change, including nitrous emissions from fertilised soils and methane emissions from rice paddies87. In contrast to the one-off land use change emissions considered here, those are ongoing emissions that incur continually in the production process. We would assume that the magnitude of these emissions in a scenario of redistribution of agricultural areas, in which the total production of each crop remains constant, is roughly similar to that associated with the current distribution of areas. We also did not consider emissions associated with transport; however, these have been shown to be small compared to other food chain emissions88 and poorly correlated with the distance travelled by agricultural products89.Biodiversity impact B
i(x)Analogous to our approach for carbon, we estimated the biodiversity impact of crop production, Bi(x), in a 5-arc-minute grid cell as the difference between the local biodiversity associated with the natural habitat and that associated with cropland. For our main analysis, we quantified local biodiversity in terms of range rarity (given by the sum of inverse species range sizes; see below) of mammals, birds, and amphibians. Range rarity has been advocated as a biodiversity measure particularly relevant to conservation planning in general39,90,91,92,93 and the protection of endemic species in particular39. In a supplementary analysis, we additionally considered biodiversity in terms of species richness.We used 5-arc-minute global maps of the range rarity and species richness of mammals, birds, and amphibians under potential natural vegetation (Supplementary Fig. 1c, d) and under cropland land cover94. The methodology used to generate these data38 combines species-specific extents of occurrence (spatial envelopes of species’ outermost geographic limits40) and habitat preferences (lists of land cover categories in which species can live95), both available for all mammals, birds, and amphibians96,97, with a global map of potential natural biomes44 in order to estimate which species would be present in a grid cell for natural habitat conditions. Incorporating information on species’ ability to live in croplands, included in the habitat preferences, allows for determining the species that would, and those that would not, tolerate a local conversion of natural habitat to cropland. The species richness impact of crop production in a grid cell is then obtained as the number of species estimated to be locally lost when natural habitat is converted to cropland. Instead of weighing all species equally, the range rarity impact in a grid cell is calculated as the sum of the inverse potential natural range sizes of the species locally lost when natural habitat is converted; thus, increased weight is attributed to range-restricted species, which tend to be at higher extinction risk40,41.As in the case of carbon, the approach allows us to estimate the biodiversity impact of crop production in both currently cultivated and uncultivated areas.Land potentially available for agriculture, V(x)We defined the area V(x) (ha) potentially available for crop production in a given grid cell x, as the area not currently covered by water bodies42, land unsuitable due to soil and terrain constraints42, built-up land (urban areas, infrastructure, roads)1, pasture lands1, crops not considered in our analysis37, or protected areas42 (Supplementary Fig. 1e). In the scenario of a partial relocation of crop production, in which a proportion of existing croplands is not moved, the relevant retained areas are additionally subtracted from the potentially available area, as described further below.Optimal transnational relocationWe first consider the scenario in which all current croplands are relocated across national borders based on current climate (Fig. 3a, dark blue line). For each crop i and each grid cell x, we determined the local (i.e., grid-cell-specific) area ({widehat{H}}_{i}(x)) (ha) on which crop i is grown in cell x so that the total production of each crop i equals the current production and the environmental impact is minimal. Denoting by$${bar{P}}_{i}={sum }_{x}{P}_{i}(x)$$
(2)
the current global production of crop i, any solution ({widehat{H}}_{i}(x)) must satisfy the equality constraints$${sum }_{x}{widehat{H}}_{i}(x)cdot {widehat{Y}}_{i}(x)={bar{P}}_{{{{{{rm{i}}}}}}},{{{{{rm{for}}}}}}quad{{{{{rm{each}}}}}},{{{{{rm{crop}}}}}},i$$
(3)
requiring the total production of each individual crop after relocation to be equal to the current one. A solution must also satisfy the inequality constraints$${sum }_{i}{widehat{H}}_{i}(x)le V(x),{{{{{rm{for}}}}}}quad{{{{{rm{each}}}}}},{{{{{rm{grid}}}}}},{{{{{rm{cell}}}}}},x,,$$
(4)
ensuring that the local sum of cropping areas is not larger than the locally available area V(x) (see above). Given these constraints, we can identify the global configuration of croplands that minimises the associated total carbon or biodiversity impact by minimising the objective function$${sum }_{x}{widehat{H}}_{i}(x)cdot {C}_{i}(x)to ,{{min }}quad{{{{{rm{or}}}}}}quad{sum }_{x}{widehat{H}}_{i}(x)cdot {B}_{i}(x)to ,{{min }}$$
(5)
respectively. More generally, we can minimise a combined carbon and biodiversity impact measure, and examine potential trade-offs between minimising each of the two impacts, by considering the weighted objective function$${sum }_{x}{widehat{H}}_{i}(x)cdot (alpha cdot {C}_{i}(x)+(1-alpha )cdot {B}_{i}(x))to ,{{min }}$$
(6)
where the weighting parameter α ranges between 0 and 1.Considering all crops across all grid cells, we denote by$$bar{C}={sum }_{i}{sum }_{x}{H}_{i}(x)cdot {C}_{i}(x)$$
(7)
the global carbon impact associated with the current distribution of croplands, and by$$hat{C}(alpha )={sum }_{i}{sum }_{x}{hat{H}}_{i}(x)cdot {C}_{i}(x)$$
(8)
the global carbon impact associated with the optimal distribution ({{{widehat{H}}_{i}(x)}}_{i,x}(={{{widehat{H}}_{i}^{alpha }(x)}}_{i,x})) of croplands for some carbon-biodiversity weighting (alpha in [0,1]). The relative change between the current and the optimal carbon impact is then given by$$hat{c}(alpha )=100 % cdot frac{hat{C}(alpha )-bar{C}}{bar{C}}$$
(9)
Using analogous notation, the relative change between the current and the optimal global biodiversity impact across all crops and grid cells is given by$$widehat{b}(alpha )=100 % cdot frac{widehat{B}(alpha )-bar{B}}{bar{B}}$$
(10)
The dark blue line in Fig. 3a visualises (widehat{c}(alpha )) and (widehat{b}(alpha )) for the full range of carbon-biodiversity weightings (alpha in [0,1]), each of which corresponds to a specific optimal distribution ({{{widehat{H}}_{i}(x)}}_{i,x}) of croplands. We defined an optimal weighting ({alpha }_{{{{{{rm{opt}}}}}}}), meant to represent a scenario in which the trade-off between minimising the total carbon impact and minimising the total biodiversity impact is as small as possible. Such a weighting is necessarily subjective; here, we defined it as$${alpha }_{{{{{{rm{opt}}}}}}}={{arg }},{{{min }}}_{alpha in [0,1]}left|begin{array}{ll}frac{frac{partial {hat{c}}(alpha)} {partial {hat{b}}(alpha)}}{hat{c}(alpha)} cdot frac{frac{partial {hat{b}}(alpha)} {partial {hat{c}}(alpha)}}{hat{b}(alpha)}end{array}right|$$
(11)
Each of the two factors on the right-hand side represents the relative rate of change in the reduction of one impact type with respect to the change in the reduction of the other one as α varies. Thus, αopt represents the weighting at which neither impact type can be further reduced by varying α without increasing the relative impact of the other by at least the same amount. Scenarios based on this optimal weighting are shown in Figs. 1,  2, and Supplementary Figs. 3–6, and are represented by the black markers in Fig. 3.Our approach does not account for multiple cropping; i.e., part of a grid cell is not allocated to more than one crop, and the assumed annual yield is based on a single harvest. Allowing for multiple crops to be successively planted in the same location during a growing period would increase the dimensionality of the optimisation problem substantially. However, given that only 5% of current global rainfed areas are under multiple cropping98, this is likely not a strong limitation of our rainfed-based analysis. As a result of this approach, our results may even slightly underestimate local crop production potential and therefore global impact reduction potentials.Optimal national relocationIn the case of areas being relocated within national borders, the mathematical framework is identical with the exception that the sum over relevant grid cells x in Eqs. (2) and (4) is taken over the cells that define the given country of interest, instead of the whole world. In this way, the total production of each crop within each country for optimally distributed croplands is the same as for current areas. The optimisation problem is then solved independently for each country.Optimal partial relocationWhen (either for national or transnational relocation) only a certain proportion (lambda in [0,1]) of the production of each crop (of a country or the world) is being relocated rather than the total production, Eq. (3) changes to$$mathop{sum}limits_{x}{widehat{H}}_{i}(x)cdot {widehat{Y}}_{i}(x)=lambda cdot {bar{P}}_{i},{{{{{rm{for}}}}}},{{{{{rm{each}}}}}},{{{{{rm{crop}}}}}},i,.$$
(12)
In addition, the area potentially available for new croplands, V(x), (see above) is reduced by the area that remains occupied by current croplands accounting for the proportion ((1-lambda )) of production that is not being relocated. We denote by ({H}_{i}^{lambda }(x)) the area that continues to be used for the production of crop i in grid cell x in the scenario where the proportion λ of the production is being optimally redistributed. In particular, ({H}_{i}^{0}(x)={H}_{i}(x)) and ({H}_{i}^{1}(x)=0) for all i and x. For a given carbon-biodiversity weighting (alpha in [0,1]) in Eq. (6), ({H}_{i}^{lambda }(x)) is calculated as follows. First, all grid cells in which crop i is currently grown are ordered according to their agro-environmental efficiency, i.e., the grid-cell-specific ratio between the environmental impact attributed to the production of the crop and the local production,$${E}_{i}^{alpha }(x)=frac{{H}_{i}(x)cdot (alpha cdot {C}_{i}(x)+(1-alpha )cdot {B}_{i}(x))}{{P}_{i}(x)}.$$
(13)
Let ({x}_{1}(={x}_{1}(i,alpha ))) denote the index of the grid cell in which crop i is currently grow for which ({E}_{i}^{alpha }) is smallest among all grid cells in which the crop is grown. Then let x2 be the index for which ({E}_{i}^{alpha }) is second smallest (or equal to the smallest), and so on. Thus, the vector (({x}_{1},{x}_{2},{x}_{3},ldots )) contains all indices of grid cells where crop i is currently grown in descending order of agro-environmental efficiency. The area ({H}_{i}^{lambda }({x}_{n})) retained in some grid cell ({x}_{n}) is then given by$${H}_{i}^{lambda }({x}_{n})=left{begin{array}{ll}{H}_{i}({x}_{n}) & {{{{{rm{if}}}}}};mathop{sum }limits_{m=1}^{n}{P}_{i}({x}_{m})le (1-lambda )cdot {bar{P}}_{i}\ 0, & hskip-7.5pc{{{{{rm{else}}}}}}end{array}right.$$
(14)
Thus, cropping areas in a grid cell ({x}_{n}) are retained if they are amongst the most agro-environmentally efficient ones of crop i on which the combined production does not exceed ((1-lambda )cdot {bar{P}}_{i}) (which is not being relocated). Growing areas in the remaining, less agro-environmental efficient grid cells are abandoned and become potentially available for other relocated crops. Note that ({H}_{i}^{lambda }) depends on the weighting α of carbon against biodiversity impacts. Finally, instead of Eq. (4), we have, in the case of the partial relocation of the proportion λ of the total production,$$mathop{sum}limits_{i}{widehat{H}}_{i}(x)le V(x)-{H}_{i}^{lambda }(x)quad{{{{{rm{for}}}}}},{{{{{rm{each}}}}}},{{{{{rm{grid}}}}}},{{{{{rm{cell}}}}}},x,.$$
(15)
Solving the optimisation problemAll datasets needed in the optimisation (i.e., (A(x)), ({P}_{i}(x)), ({H}_{i}(x)), ({C}_{i}(x)), ({B}_{i}(x)), ({widehat{Y}}_{i}(x)), (V(x))) are available at a 5 arc-minute (0.083°) resolution; however, computational constraints required us to upscale these to a 20-arc-minute grid (0.33°) spatial grid. At this resolution, Eq. (6) defines a 1.12 × 106-dimensional linear optimisation problem in the scenario of across-border relocation. The high dimensionality of the problem is in part due to the requirement in Eq. (3) that the individual production level of each crop is maintained. Requiring instead that, for example, only the total caloric production is maintained31,99 reduces Eq. (6) to a 1-dimensional problem. However, in such a scenario, the production of individual crops, and therefore of macro- and micronutrients, would generally be very different from current levels, implicitly assuming potentially drastic dietary shifts that may not be nutritionally or culturally realistic.The optimisation problem in Eq. (6) was solved using the dual-simplex algorithm in the function linprog of the Matlab R2021b Optimization Toolbox100 for a termination tolerance on the dual feasibility of 10−7 and a feasibility tolerance for constraints of 10−4.In the case of a transnational relocation of crop production, the algorithm always converged to the optimal solution, i.e., for all crop management levels, climate scenarios, and proportions of production that were being relocated. For the relocation within national borders, this was not always the case. This is because some countries produce small quantities of crops which, according to the GAEZ v4 potential yield estimates, could not be grown in the relevant quantities anywhere in the country under natural climatic conditions and for rainfed water supply; these crops likely require greenhouse cultivation or irrigation can therefore not be successfully relocated within our framework. Across all countries, this was the case for production occurring on 0.6% of all croplands. When this was the case for a certain country and crop, we excluded the crop from the optimisation routine, and a country’s total carbon and biodiversity impacts were calculated as the sum of the impacts of optimally relocated crops plus the current impacts of non-relocatable crops.This issue is linked to why determining the optimal distribution of croplands within national borders is not a well-defined problem for future climatic conditions. Under current climatic conditions, if a crop cannot be relocated within our framework, then its current distribution offers a fall-back solution that provides the current production level and allows us to quantify environmental impacts. Different climatic conditions in the future mean that the production of a crop across current growing locations will not be the same as it is today, and therefore the fall-back solution available for the present is no longer available, so that a consistent quantification of the environmental impacts of a non-relocatable crop is not possible.Carbon and biodiversity recovery trajectoriesOur analysis in Supplementary Fig. 6 requires spatially explicit estimates of the carbon recovery trajectory on abandoned croplands. Whilst carbon and biodiversity regeneration have been shown to follow certain general patterns, recovery is context-specific (Supplementary Note 1) in that, depending on local conditions, the regeneration in a specific location can take place at slower or faster speeds than would typically be the case in the broader ecoregion. Here, we assumed that these caveats can be accommodated by using conservative estimates of recovery times and by assuming that local factors will average out at the spatial resolution of our analysis. The carbon recovery times assumed here are based on ecosystem-specific estimates of the time required for abandoned agricultural areas to retain pre-disturbance carbon stocks82. Aiming for a conservative approach, we assumed carbon recovery times equal to at least three times these estimates, rounded up to the nearest quarter century (Table 3). Independent empirical estimates from specific sites and from meta-analyses are well within these time scales (Supplementary Note 1).Table 3 Assumed times required for carbon stocks on abandoned cropland to reach pre-disturbance levels.Full size tableApplying the values in Table 3 to a global map of potential natural biomes44 provides a map of carbon recovery times. We assumed a square root-shaped carbon recovery trajectory across these regeneration periods101; similar trajectories, sometimes modelled by faster-converging exponential functions, have been identified in other studies25,27,30,102,103,104,105. Thus, the carbon stocks in an area of a grid cell x previously used to grow crop i were assumed to regenerate according to the function$$left{begin{array}{ll}{{C}}_{{{{{{rm{agricultural}}}}}}}(x)+sqrt{frac{t}{{{T}}_{{{{{{rm{carbon}}}}}}}(x)}}cdot ({{C}}_{{{{{{rm{potential}}}}}}}(x)-{{C}}_{{{{{{rm{agricultural}}}}}}}(x)) & {{{{{rm{if}}}}}},t ; < ; {{T}}_{{{{{{rm{carbon}}}}}}}\ hskip14.7pc{{C}}_{{{{{{rm{potential}}}}}}}(x) & {{{{{rm{if}}}}}},tge {{T}}_{{{{{{rm{carbon}}}}}}}end{array}right.$$ (16) where, using the same notation as further above$${{C}}_{{{{{{rm{potential}}}}}}}(x) ={{C}}_{{{{{{rm{potential}}}}}},{{{{{rm{vegetation}}}}}}}(x)+{{C}}_{{{{{{rm{potential}}}}}},{{{{{rm{SOC}}}}}}}(x)\ {{C}}_{{{{{{rm{agricultural}}}}}}}(x) ={{C}}_{i}(x)+0.75cdot {{C}}_{{{{{{rm{potential}}}}}},{{{{{rm{SOC}}}}}}}(x)$$ (17) Reporting summaryFurther information on research design is available in the Nature Research Reporting Summary linked to this article. More