Ecology

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Plant materialRice (Oryza sativa L.) plants used for the experiment were from the collection of Taiwan Agricultural Research Institute. The rice variety (Tainan No. 11) used in this study has enhanced resistance to rice blast. The use of plant materials complies with international, national, and/or institutional guidelines and legislation.Field areaThis study was carried out in experimental rice fields under low-external-input and conventional farming in central Taiwan (23.5859 N, 120.4083 E; 8.0 ha). The annual average temperature ranged from 23 to 25 °C, the annual average relative humidity ranged from 75 to 92%, and the annual rainfall ranged from 1020 to 2873 mm year−1 (average data between 2006 and 2016 measured at a nearby weather station; Fig. 1). The experimental paddy plots were defined by considering the typical dimensions of the agricultural fields in Taiwan (0.5 to 1.0 ha). A long-term experiment was conducted from 2006 to 2016 to study the effects of different agronomic management on biodiversity, productivity, and environment, including traceability system, soil fertility, nitrogen leaching, production costs, disease incidence and severity, the abundance of pest and beneficial insects, and weed dynamics.The treatments consisted of conventional farming with high chemical fertilizer input (CF) and low-external-input farming with low fertilizer input (LF). In the CF farming, we followed the fertilizer recommendations that are constructed to meet the nutrient requirements of the crop. In the LF farming, the chemical fertilizers were largely reduced compared to the recommendation (see next paragraph for the details). The experiment was conducted as a randomized complete block design (RCBD) with four replicates. In agricultural experiments, the RCBD is a standard procedure by grouping experimental units into blocks. For example, the design can control variation in the experiments by considering spatial influences and adjusting the effects of target factors in fields. Each experimental unit consisted of a 0.58 ± 0.16 ha of the area of the field. Additional nutrient management in the LF system includes (1) nitrogen-fixing and cover crops, (2) manure and compost applications, (3) plant and soil nutrient analyses for adjusting fertilization, and (4) reduced tillage. Soil-available potassium gradually decreased during the 10-year study period in the area of the LF system. Over the study period, the LF system achieved the similar level of crop production as that of the CF system (Fig. S1).In our study area, there were two growing seasons within a year: one in the first half of the year (from February to June) and one in the second half (from August to December). The ground fertilizers were applied before rice seedlings were transplanted, followed by additional fertilizations during the tillering and boosting stages. The total amount of fertilizers used for the CF system included 140–180 and 120–140 kg N ha−1, 70–72 and 60 kg P2O5 ha−1, and 85 and 60 kg K2O ha−1 for the first and second seasons, respectively. For the LF system, 100 and 80 kg N ha−1, 30 and 30 kg P2O5 ha−1, and 30 and 30 kg K2O ha−1 were applied in the first and second seasons, respectively. The larger amount of fertilizers for the first season was due to its longer duration. For each rice growing season, fungicides were applied to both farming systems once during the boosting stage. During the fungicide application, a 10% mixture of Cartap plus Probenazole or 6% probenazole for rice blast (both 30 kg ha−1) and 1.5% Furametpyr for sheath blight (20 kg ha−1) were used.Rice disease monitoringThe major rice disease (rice blast; Fig. S2) was monitored biweekly in the CF and LF systems over the two growing seasons per year, with each growing season including (in chronological order) the tillering, flowering, and maturing stages. There was a total of 123 occasions during our study. The plants were disease free when planted out. When the lesion of the rice blast began to appear in the fields from the tillering stage to the maturing stage, the effects of the two treatments (CF and LF systems) in the paddy fields on the disease incidence of rice blast (caused by Pyricularia oryzae) were investigated. For each plot (or experimental unit), the incidence of rice disease was randomly examined at 5 points and for 25 plexuses (i.e., each derived from one primary tiller) per point. The disease incidence was quantified as the percentage of infected plexuses that were determined based on the presence of infected leaves.The area under the disease progress curve (AUDPC) was used to quantify disease incidences over time, and the relative AUDPC ((RAUDPC)) was used because of unequal sampling duration in the growing seasons during our study period. For each plot (or experimental unit), we used the (RAUDPC) to summarize the incidences of disease during each growing season as follows:$$RAUDPC=frac{sum_{i=1}^{n-1}frac{{y}_{i}+{y}_{i+1}}{2}times left({t}_{i+1}-{t}_{i}right)}{100 times left({t}_{n}-{t}_{1}right)},$$
(1)
where ({y}_{i}) and ({t}_{i}) are the disease incidence (%) and time (day) at the (i)th observation, respectively, and (n) is the total number of observations.Bayesian modelingWe built a mechanistic model that was applied to assess the interplay within a network of relationships among weather fluctuation, farming system, and disease incidence in the paddy fields. The model describes how (1) temperature and relative humidity together influence the development of primary inoculum, (2) rainfall detaches the fungal spores on the host tissues, and (3) rainfall and wind catch the airborne spores onto the leaf area. These environmental processes determine the disease incidence in the model. In addition, this model considers that farming systems can suppress or accelerate disease incidence. By fitting our model to the observed incidence, Bayesian inference was used as the parameter estimation technique for the models. In addition, we tested the alternative mechanistic hypotheses based on a model-selection criterion and cross vaidation (see subsequent paragraphs).With a linearity assumption, the incidences of disease (RAUDPC) were modeled as an inverse-logit function of the progress rate of the development of primary inoculum ((IP) with values between 0 and 1) and the net catchment of the airborne spores by rainfall and wind ((CT) with values between 0 and 1; when subtracting the detachment of spores by rainfall from the host tissue) as follows:$$RAUDPC=invLogitleft({a}_{f}+{b}_{1}bullet logitleft(avg_IPright)+{b}_{2}bullet logitleft(avg_IPbullet avg_CTright)right),$$
(2)
where ({a}_{f}), ({b}_{1}), and ({b}_{2}) describe the constant baseline for different farming systems ((f) = the CF or LF system), the direct effect size of (avg_IP), and the mediating effect size of (avg_CT) through (IP), respectively. The two parameters ((avg_IP) and (avg_CT)) are averaged (IP) and (CT) during the growing season, respectively (see below for details). The effect sizes ({b}_{1}) and ({b}_{2}) have values more than zero. The constant baseline allows the management-specific acting in the model when they can influence the disease incidence.The process rate (IP) was simulated as a function of the temperature response ((fleft(Tright)) with values between 0 and 1) and hourly air relative humidity ((RH,) %) as follows20:$$IP= left{begin{array}{ll}0& mathrm{if}, RH 0) are the steepness and midpoint parameters to control the portion of spores caught by the wind, respectively.The Bayesian framework ‘Stan’49 and its R interface ‘RStan’50 were used to construct and fit the models. There were two competing models: either considering the difference between the CF and LF systems by not fixed to the same values of the constant baseline ({a}_{f}) in Formula (2) or not. For each model, four Markov Chain Monte Carlo (MCMC) chains (for numerical approximations of Bayesian inference) ran, each with 5,000 iterations, and the first half of the iterations of each chain were discarded as burn-in. The R-hat statistic of each parameter approaches a value of 1, indicating model convergence. With a total of 2,000 samples, collected as one sample for every 5 iterations for each chain, the model parameters and their posterior distribution were estimated. To compare the two competing models, we calculated the widely applicable information criterion (WAIC) using the R package ‘loo’51. The best model was determined based on the lowest WAIC. By using the same R package, we also performed the approximate leave-one-out cross-validation (LOO-CV) to estimate the predictive ability of the two Bayesian models. Here, we used the expected log predictive density (ELPD) to be the predictive performance.Compliance with ethical standardsThe authors declare that they have no conflict of interest. This article does not contain any studies involving animals performed by any of the authors. This article does not contain any studies involving human participants performed by any of the authors. More

Sentinel-1 data selectionThe Copernicus Sentinel-1 mission was launched by the European Space Agency (ESA) in 2014 with the Sentinel-1A satellite, complemented with the second Sentinel-1B satellite in 2016. Each satellite has a 12-days repeat cycle. Continuity of the Sentinel-1 mission has been approved by ESA until 2030 and replacement satellites will be launched. The satellites operate in different acquisition modes over different parts of the globe. Land masses are covered primarily by the Interferometric Wide-Swath Mode (IW) with a 250 km swath width across-track. Single-look-complex (SLC) Level 1.1 data are required for interferometric processing. Along-track, Sentinel-1 data are sliced into consecutive frames (slices) of about 250 km length. Data are distributed via ESA’s Scientific Sentinel-1 Hub, which is mirrored at NASA’s Alaska Satellite Facility DAAC (ASF-DAAC). During production, Sentinel-1 SLC data were accessed on the ASF-DAAC data repository which resides on Amazon’s AWS S3 bucket in region us-west-2.Sentinel-1 satellites cover various parts of Earth in ascending and descending flight direction in a total of 175 relative orbits. ESA’s flight plan has some areas covered every six days and in both flight directions, predominantly over Europe. For the production of this data set, Sentinel-1 SLC frames were selected from all available scenes acquired between December 1st 2019 and November 30th 2020. Over the one-year timeframe, a maximum of 30 to 31 acquisitions at 12-days repeat, and 60 to 61 acquisitions at 6-days repeat intervals can be expected. The following selection criteria were applied consecutively to achieve global coverage with uniform distribution of acquisitions across seasons (Fig. 1):

Global descending data (Fig. 1a) were selected where the one-year stack size had at least 25 acquisitions.

Spatial gaps were filled with ascending data (Fig. 1a) where the one-year stack size had at least 25 acquisitions.

For spatial consistency, over conterminous North America north of Panama, preference was given to ascending data where both ascending and descending data existed with stack sizes over 25 acquisitions.

For stack sizes less than 25 acquisitions, preference was given to the flight direction with the larger number of acquisitions.

Remaining gaps were filled with data from the flight direction available.

Fig. 1Flight direction, polarization mode, and InSAR stack sizes of 6- and 12-days repeat coverage of Sentinel-1 data acquired between December 1st 2019 and November 30th 2020 selected for processing.Full size imageArctic and Antarctic regions are typically covered with polarization modes of horizontal transmit (HH single- or HH/HV dual-polarization). Figure 1b shows the global distribution of the processed data in horizontal and vertical polarization transmit modes, respectively. Table 1 summarizes the number of selected scenes in the two flight directions and various polarization modes. The total number of processed Sentinel-1 SLC frames came to ~205,000 scenes with a total raw input data volume of about 850 Terabytes. Figure 1c,d show the spatial distribution of the final scene selection with the number of 6- and 12-days repeat-pass image pairs. Consistent 6-days repeat coverage with about sixty image pairs from either ascending or descending orbits could be processed over Europe, the coastal areas of Greenland and Antarctica, and some smaller areas around the world; note that in some regions (e.g., India, interior Greenland, Northern Canada, Eastern China) 6-days repeat coverage was available in certain seasons only (Fig. 1c). A consistent coverage with 12-days repeat-pass imagery, instead, could be processed almost globally with the nominal maximum of about thirty repeat-pass pairs in areas where only one satellite, Sentinel-1A or Sentinel-1B, acquired data in all but few areas above 60° N in Canada, Greenland, or Russia (Fig. 1d). In some small areas in the Midwestern United States, the Khabarovsk region in Far-Eastern Russia, or in the Northern Sahara, neither Sentinel-1A nor Sentinel-1B acquire data in IW mode, leading to small gaps in the final data set.Table 1 Number of Sentinel-1 Single Look Complex scenes processed.Full size tableProcessing approachThe overall processing workflow was developed based on the interferometric processing software developed by GAMMA Remote Sensing and geared towards efficient processing in the Amazon Web Services (AWS) cloud environment utilizing Earth Big Data LLC’s cloud scaling solutions. The workflow is divided into three main blocks as illustrated in Fig. 2. Sentinel-1A and -1B acquire data along 175 relative orbits/orbital tracks. Blocks 1 and 2 were processed on a per relative orbit basis; block 3 was initiated after blocks 1 and 2 had been completed for all relative orbits.Fig. 2Implementation of the Sentinel-1 interferometric processor in the AWS cloud environment.Full size imageProcessing Block 1For each SLC of a given relative orbit, processing block 1 entailed:

1.

Conversion of SLC image files to a GAMMA specific format. Each Sentinel-1 SLC, covering an area of ~250 × 250 km, consists of six SLC image files (one SLC image file for each of the three sub-swaths in co- (VV or HH) and cross-polarizations (VH or HV).

2.

Compensation of the SLC amplitudes for the noise equivalent sigma zero (NESZ).

3.

The orbit state vectors provided with the original Sentinel-1 SLCs were updated with the precision state vectors (AUX_POEORB) distributed by the Sentinel-1 payload data ground segment 20 days after data take with a precision (3σ) generally of the order of 1 cm (target requirement  More

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Daily change in primary pollutantsTo elucidate the change trend of primary pollutants under the 13th Five-Year Plan, we calculated the daily primary pollutants in 2015 and 2019 based on formula (1) and formula (2). Such diurnal comparisons can reduce the effects of seasonal weather to some extent. From the 19 UAs (224 prefecture-level cities), the heat diagram of the daily change transfer matrix of primary pollutants from 2015 to 2019 is shown in Fig. 2, including six primary pollutants and clean day conditions.Figure 2Transfer change matrix heatmap of primary pollutants from 2015 to 2019.Full size imageFrom the sum of the diagonal numbers, 37% of the primary pollutants had no shift during the 13th Five-Year Plan period. PM2.5, PM10 and O3 were the main primary pollutants, especially PM2.5. More primary pollutants were diverted to ozone pollution, indicating that the proportion of O3 as the primary pollutant is gradually increasing. In addition, the proportion of clean air has increased significantly, which shows that pollution control has been effectively reflected during the 13th Five-Year Plan period. However, the proportion of NO2 before and after metastasis was approximately the same, with approximately 5% NO2 pollution. This may imply that the governance of NO2 pollution was rendered nonsignificant. It is noteworthy that ozone pollution in China has become an increasingly prominent task in recent years. Similar to Xiao’s16 research on ozone pollution, they argue that present-day ozone levels in major Chinese cities are comparable to or even higher than the 1980 levels in the United States. Taken together, ozone and PM2.5 have become the top two air pollution pollutants in China.Monthly distribution of primary pollutantsTo further explore the spatiotemporal distribution of the primary pollutants across the UAs, we obtained the most primary pollutants per month by dividing the number of days with the most pollutants by the number of cities in each UA from the 2019 data. In Fig. 3, the UAs location was plotted on the abscissa, and the monthly variance of the primary pollutant was plotted on the ordinate. As shown in Fig. 3, PM2.5 appeared as dark green, PM10 appeared as light green, O3 appeared as orange, NO2 appeared as yellow, and clean days appear as dark blue. The main pollutants in the 19 UAs are PM2.5, PM10 and O3. NO2, as the primary pollutant, only appeared in the HBOY UA in January. Ordos, located in HBOY, possess rich oil and coal resources, with coal mining as its leading industry38. According to the China Energy Statistical Yearbook 2019, nearly 250 million tons of raw coal were used for thermal power generation in Inner Mongolia Autonomous Region, making it the region with the largest amount of raw coal for thermal power generation in China39. To a certain extent, the increase of heating40 and the imperfect denitration technology41 are both contributing to the increase of NO2 pollution in the atmosphere. CO and SO2 did not become major pollutants. Clean days (where AQI  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