Power-law productivity and positive regime shift of symbiotic and climate-resilient edible ecosystems

Transformative change in primary food production is urgently needed in the face of climate


Introduction
Many studies have sounded alerts about global ecological deterioration due to the accelerating impacts of human activity in the last century (e.g., ref. 1-4).The 6 th mass extinction is underway in a wide range of biotic communities, including primary forests 5 , vertebrates 6 , and insect fauna 7 .
These impacts are largely due to the primary food production on land and have caused critical environmental shifts in marine ecosystems 8 : Here, the agricultural sector is responsible for 25% of greenhouse gases (GHGs) 9 , and it has disrupted global biochemical flows and biosphere integrity 10 .However, interactive responses to changes in human activities, material cycles, and biodiversity distribution, including effects induced by climate change mitigation and conservation activities, are extremely complex and difficult to simulate.Globally assessed scenarios (e.g., ref. 4, 11) are not capable of predicting actual social emergencies, such as the COVID-19 pandemic, and cannot promptly address the root causes.Moreover, the importance of an integrated approach to the science of climate and biodiversity changes and the development of coherent policies has only recently been realized (e.g., ref. 12).Current economic theory and practice do not sufficiently incorporate a valuation of biodiversity and multiple ecosystem services 13 ; we need to take comprehensive measures interconnecting direct drivers of ecosystem deterioration and underlying economic, social, and technological causes, in order to regenerate the ecologically driven material cycles and substantially reducing agricultural inputs and runoff 3,14 .
Many global-scale simulations have suggested possible scenarios toward sustainable land use aimed at recovery of biodiversity and the carbon cycle (e.g.ref. 15, 16).On the other hand, despite their scale, these studies are based on databases that do not necessarily encompass the whole socialecological complexity required for an actual implementation.The interactions of many parameters and the complexity of community dynamics have largely been ignored (e.g., in a food-system change scenario 15 , the cross-field phosphorus cycle 17 and management breakthrough on the carbon cycle 18 are not included; in a global afforestation scenario 16 , the implausibility of afforestation of naturally maintained grasslands and savannas and thermodynamic trade-off between tree cover increase and consequent diminishment of albedo 19 are not considered), and deep case studies are needed in connection to a realistic driving force.The ground truth is often ignored even in basic statistical studies; this makes the applicability of global scenarios to actual situations quite elusive-while 84% of farms are owned by smallholders producing on less than 2 ha, estimates of the total surface of smallholds vary from 12% to 40% of the global farmland depending on the method of measurement 20 .
In order to convert the majority of food producers (especially resource-, knowledge-, and technology-deprived smallholders) into positive drivers of biodiversity, on-site tailoring and proactive management of agrobiodiversity in a comprehensive social-ecological context are important leverage points 3,21 .An essential pillar of transformative change in food production is to deliver a managementintensive typology of sustainable practices that contains interfaces with the diversity and uniqueness of real-world operations on a scientific basis, which has been studied in the field of open complex systems science 14,22,23 .We need complementarity between a general theory based on averaged statistics and deep analyses of individual cases in order to make progress toward the inclusion of neglected diversity.With the rise of big data, such a paradigm has emerged in the management of living systems, such as in precision medicine (e.g., N-of-1 studies 24 and longitudinal deep phenotyping 25 ).This study aims to provide the pioneering cases of such a paradigm for planetary health with the basis of community ecology perspective, towards the application to the grassroot majority of world food production.

Crop production at ecological optimum
Empirical studies in ecology have revealed the positive contribution of species diversity and the symbiotic relationship between plants to the primary production of ecosystems at the community level (e.g., ref. 26) , especially in relation with surface patterns that follow power-law distribution 27,28 .Although knowledge of selforganized natural vegetation constitutes a better understanding of community dynamics and has been used for planning conservation practices, little of it has been applied to crop production.Synecological farming (synecoculture) takes advantage of the sustainable productivity of self-organized vegetation that occurs when there is an extremely high diversity of crops 14,29,30 .The principle of production in synecoculture is fundamentally different from those of other low-input organic and natural farming methods that are limited in their association and rotation of a few crops (e.g., ref. 31).In contrast to the conventional definition of productivity based on a single crop and a field environment controlled toward its physiological optimum, synecoculture relies on the primary production of a mixed community that comprises tens to hundreds of edible plant species; this sort of production is known as augmentation of the ecological optimum (explained in Box 1).

Symbiosis-dominant ecosystems with crops
To evaluate the self-organization process of a mixed community of crops, a 420-sq.mplot in the temperate zone (Oiso, Japan) was used to measure the species-wise surface The probability density of the species-wise surface in each 2-sq.m measurement section also followed a power law (Figure 5 (top) of the Extended Data).
The relative degree of symbiotic relationship can be compared with the parameter  and showed that naturally occurring spontaneous species (usually considered to be weeds) form vegetation patterns that contain more positive interactions ( closer to zero) than the introduced crop species.This tendency was also observed in another classification of edible and non-edible plants based on past usage in synecoculture practices.Positive diversity responses to climate variability were also dominant in spontaneous species (see Fig 2 of Extended Data).The direct implication is that the coexistence of naturally occurring non-edible species serves as a substantial source of symbiotic gain for the whole community dynamics that promotes ecological succession, and it may contribute to the productivity of crops and other edible plants through an overall increase in resources such as soil organic matter and soil microbial activity 32 .

Production experiments
The productivity of synecoculture in temperate and semi-arid tropical zones was tested in two farms, on a 1,000 sq.m farm in Ise, Japan over the course of four years The average profitability (measured as gross profit minus costs) of synecoculture in the Ise farm rose 2.35-to 3.87-fold, which corresponds to an estimated 0.981-to 1.16-fold increase in harvest biomass, compared with the conventional databases of all scales and small scale (<0.5ha) (see the description of the relative biomass ratio BR in Methods).Compared with the median (and 25 th and 75 th percentiles) of conventional market gardening, the profitability of synecoculture in the Mahadaga farm rose 88.0(202/54.4)-fold,which corresponds to an estimated 33.8(49.6/25.1)-foldincrease in harvest biomass, on average over two 18-month periods before and after November 2016 under different social conditions.In particular 121(278/74.9)-foldincrease in profitability corresponding to an estimated 37.8(55.3/28.0)-foldincrease in harvest biomass under high market accessibility, and a 55.0(126/34.0)-foldincrease in profitability corresponding to a 29.9(43.8/22.2)-foldincrease in harvest biomass under low market accessibility (see Methods).The on-site comparison at Mahadaga farm showed that synecoculture excelled in showing 258fold increase profitability in correspondence with an estimated 12.4-fold harvest biomass compared with the five other simultaneously tested alternative methods of sustainable farming.
A most dramatic change was the local reversal of the regime shift in the Mahadaga farm.From an analysis of satellite images taken before the experiment, the vegetation patches that surrounded Mahadaga farm corresponded to spotted vegetation patterns that strongly implied warning signals of imminent desertification 34 .The subsequent intensive introduction of 150 edible plant species, including 40 staples, reestablished a lush ecosystem that maintained high productivity year-round that had positive regeneration effects on neighboring plots (Fig 2 (c1-c3)).

Climate resilience
In all of the experiments conducted at the three sites, a significant positive correlation of plant species diversity with the fluctuation components of major meteorological parameters was observed, which could not be totally reduced to a correlation with the mean components ( Extended Data), seasonality was weaker in the fluctuation than in the mean components, indicating that the observed biodiversity response may be an adaptive diversification of the species composition to climatic variability rather than seasonal patterns in community dynamics 35 .The observed positive correlation between the meteorological variance and plant species diversity in self-organized edible ecosystems implies the presence of evolutionary acquired biodiversity maintenance mechanisms, because increasing diversity to cope with environmental fluctuation generally contributes to sustain ecological community.We believe that it could constitute a fundamental mechanism to augment the climate resilience by mainstreaming biodiversity in food production 36 , which could provide an enhanced portfolio of agrobiodiversity beyond substitution and relocation of major crops 37 , and thereby enlarge the range of options to cope with the inevasible global biodiversity redistribution under climate change 38 and keep the food systems within the planetary limits 15,39 .

Discussion
One of the greatest challenges in this study that seems contradictory to conventional monoculture methods is the stabilization of yield that relies on ecological niche formation.The rationale of synecoculture lies in productivity at the community level with a hyper-diverse portfolio of products and reduced input costs, which is compatible with the primary production of self-organized plant communities in natural environment 29 .In Figure 5 of the Extended Data, although the fitted Pareto distributions for all experiments are situated in the parameter range where analytical mean converges to a finite value (i.e.,  > 1), a large deviation is inherent even at the annual scale (the 12-month gross profit ranged between 56 and 141% of the total average for the Mahadaga farm and was between 27 and 214% of the total average for the Ise farm).Therefore, productivity in terms of arithmetic means is not a stable indicator for management.Still, the cumulative cost-benefit ratio converged to a higher level of performance compared with the conventional and other alternative methods (Figure 6 (a2) and (b2) of the Extended Data), which conforms to the theoretical prediction of power-law productivity and stability of harmonic means in our previous study 30 .This is due to the positive correlation of productivity with introduced species diversity that develops over time, which is particularly enhanced in the ecological optimum production and performs increasingly better in marginal environments for both gains in gross profit and cost reductions (see total overyielding in Fig 1 of the main text and Figure 1 of Extended Data for the theoretical predictions, and Figure 6 (a1) and (b1) of the Extended Data for the measured data).
Not only the higher productivity of the Mahadaga farm, but also the ecological optimization with synecoculture could rebuild the power-law distribution of patch patterns and may help to prevent state shifts in the farm plots near the living area in a semi-arid environment 34,40 .The recovery and enhancement of diverse vegetation in farm plot represents a major shift from negative to positive externality on biodiversity in crop production 14 , which is also compatible with massive greening initiatives to reestablish a viable environment against desertification (e.g., ref. 41).It also sets a new baseline of increased crop diversity and yield against the declining trend in dryland 11 , which can minimize land clearing and protect habitats of threatened large mammals especially in sub-Saharan Africa 42 , where animal-source foods are nutritionally valuable in food-deficient settings 43 .Given the importance of sustainability of smallhold farms and the positive social-ecological impacts that synecoculture could have, international initiatives in ECOWAS are being formed to better utilize the capacity of ecological optimum production, with a short-term goal to provide healthy and balanced diets to 3.5 million people impacted by COVID-19 44 .
Asia and sub-Saharan Africa will see the largest growth of agricultural emissions and will account for two-thirds of the increase in overall food demand by 2050 45 .In the face of climate change and current pandemics, food systems that support these regions and other nations harboring smallholders need to be scaled bottom up and should realize synergy between provisioning and regulating services (including pathogen suppression) that have been historically put in massive trade-off in agricultural land use 1,3 .In accordance with the biodiversity maintenance mechanisms that have been progressively revealed in the field of community ecology, our in-depth operational case studies imply that there exist fundamental principles that bring about such synergy through the leveraging of self-organized edible plant communities.It will lead to a novel typology for transformative change from resource-to management-intensive farming capable of creating essential biodiversity and ecosystem services in highly resilient form without resorting to fertilizers and agrochemicals.With appropriate development of supportive information technologies 23,46 and sustainable distribution networks for various farm products 47 and neglected and underutilized plant genetic resources 48 , ecological optimum production could be applicable to small-scale farms less than 5 ha that make up 94% of agricultural holdings 49 and which combined with middle-scale farms less than 50 ha produce up to 77% of the major commodities and nutrients in the world 50 .Taken as a whole, the expansion and site-specific tailoring of human-augmented farming ecosystems has the potential to uplift the baseline of multiple ecosystem services globally and provide fundamental measures to cope with growing food demand and for proactive adaptation of various crop portfolio to climate change, which will introduce a human-driven form of resilience in biosphere integrity along with the expansion of essential human activities, by involving increasing population as a positive driver of biodiversity in Anthropocene 14,29 .

Box 1. Integrated model of physiological and ecological optima (IMPEO) 29 .
The physiological optimum is the basis of monoculture optimization in agronomy, which is generally expressed as a unimodal distribution along the environmental gradient (Fig 1 (a)).In actual ecological situations, however, isolated growth is not fully attained and mixed communities are prevalent, which results in diverse shifting, division, and modification of the growth curve leading to the emergence of ecological The contribution of symbiotic gain to the total overyielding in mixed polyculture could become increasingly significant as the mean environment shifts from a physiologically favorable condition (yellow background) to the marginal ranges (orange background), by creating new stretches of arable land in harsh conditions where little monoculture growth can be expected (red background).(regularized productivity is daily and species-wise productivity in terms of Japanese yen (JPY) multiplied by the number of harvest events per year for synecoculture or yearly reported profit for conventional methods), both with an offset of total costs in order to compare the yearly mean profits (vertical solid lines) and costs (vertical dashed lines) summed as positive and negative values, respectively (see Methods).

See the Supplementary Information and
The dotted lines on the y-axis represent the estimated probability distributions for each production category based on the data shown as the rug plots along the x-axis.In (b) left, the 78 academic names of total synecoculture products are shown as a list with a color gradient, and the associated numbers define the value of the y-axis in (b) right, in which the sales for each product according to date on the x-axis is represented as the diameter of the circle with the same color gradient as the list.
The correlational analysis in (c) shows significant positive correlations between the number of produce types from synecoculture and meteorological variances for each 30-day interval.There was no significant correlation with the mean of the meteorological parameters.
Harvested crop diversity versus mean or variance of three meteorological parameters is plotted as circles following the color gradient of the date.Black solid line: linear regression with less than 5% significance; dashed line: linear regression with 95% confidence; dotted line: linear regression with prediction intervals.The three-year production experiment in Mahadaga, Burkina Faso shows a power-law distribution of product sales with (b in the red rectangle) asynchronous harvests of 37 kinds of crop.The x-axis of (a) represents sales of each product for synecoculture and for five alternative farming methods that were simultaneously tested on 500 sq.m (regularized productivity is daily and species-wise productivity in terms of West African CFA franc (XOF) multiplied by the number of harvest events per year for synecoculture and five alternative farming methods or yearly reported profit for the conventional methods), both with an offset of total costs in order to compare the yearly mean profits (vertical solid lines) and costs (vertical dashed lines) summed as positive and negative values, respectively (see Methods).The dotted lines represent the estimated probability distributions for each production category on the y-axis based on the data shown by the rug plots along the x-axis.The total productivity of synecoculture (red line and distribution) is shown on a monthly aggregated scale (orange distribution) and in the two periods before (cyan line and distribution) and after (magenta line and distribution) November 2016, which was the turning point of market accessibility (see Methods).In (b) left, the 37 academic names of total synecoculture products are shown as a list with a color gradient, and the associated numbers define the value of the y-axis in (b) right, in which the sales of each product according to date on the x-axis is represented as the diameter of the circle with the same color gradient as the list.
The correlational analysis in (c) shows significant positive correlations between the number of produce types from synecoculture and meteorological variances for each 14-day interval.There are also significant negative correlations with the means of the meteorological parameters.Harvested crop diversity versus mean or variance of three meteorological parameters is plotted as circles following the color gradient of the year's date.Black solid line: linear regression with less than 5% significance; dashed line: linear regression with 95% confidence; dotted line: linear regression with prediction intervals.

Methods Summary
We developed a theory that connects the differing definitions of productivity of monoculture-based optimization in agronomy and mixed community-based growth in ecology, which defines the protocol of synecological farming (synecoculture) as an extreme typology of plant food production based on self-organized ecological niches of a highly diverse community of crops and other spontaneous vegetation.
Three small-scale plots representative of the basic smallest surface for smallholders were prepared in Japan and Burkina Faso following the protocol of synecoculture, and maintained without the use of tillage, fertilizers, or agrochemicals.
We measured the species-wise surface in a small harvest-free surface in Japan and analyzed whether the vegetation patch pattern followed a power law that reflects symbiotic interaction between plants or an exponential distribution based merely on the competition of resources.
Two production experiments in Japan and Burkina Faso were performed in collaboration with commercial farms with market access.A wide variety of specieswise product sales was recorded and the statistical properties of the time series were analyzed in comparison with official statistics on productivity and the cost of conventional market gardening and other parallelly tested farming methods.
In all experiments, we compared the mean and variance parameters of meteorological records of the finest satellite open data with the observed plant diversity and analyzed statistical correlation that represents the biodiversity response to a changing environment during the growth period.Let us describe the diverse ecological niches as   =   (;   ,   ) for centrally competent species and   =   (;   ,   ) for marginally competent species under the following assumptions,   =   =   and   <   , where   and   stand for the growth rates,  is an environmental parameter, and   ,   and   ,   are the means and standard deviations of  for centrally and marginally competent species, respectively.For simplicity, we set the same surface ratio between centrally and marginally competent species, but the model is valid for any arbitrary ratio of mixed polyculture.

Methods
Random harvesting from all environments in those niches (i.e., random sampling from the growth rate distributions   and   ) results in a normal distribution of mean productivity through the central limit theorem, such that   ~([];   ,   ) and   ~([];   ,   ), where ( • ; , ) is a normal distribution with mean  and standard deviation ,   and   respectively represent the harvest rate of centrally and marginally competent species of the mean environmental parameter [] over the sampling.We can also obtain the mean monoculture productivity  ′ ~�[];   ,   � by using the same sampling method, which results in   <   <   .In

Implementation of synecological farming (synecoculture) in Oiso and Ise, Japan and Mahadaga, Burkina Faso (Fig 2).
Following the protocol of synecoculture farming method, the following three ecosystems were started from bare ground 23,52,53 :

Surface distribution analysis and correlation analysis between species diversity and meteorological parameters at the synecoculture field in Oiso, Japan (Fig 3).
The covering surface of each plant species at low ground level in field A was measured with the 2-step visual analog scale method 33 on 80 sections measuring 2 sq.m each, 22 times at an interval of 1 week to 1.5 months (about once every 2.3 weeks on average) at a frequency depending on the degree of growth during January -December 2011 [Supplementary Data 1].The observed plant species were categorized into 1) introduced crop species and 2) naturally occurring spontaneous species, which were also parallelly labeled as 3) edible species that were utilized and 4) non-edible species that were not yet utilized as synecoculture products.
In Fig 3 (a), the inverse cumulative distribution of the number of different species is plotted with respect to the minimum threshold of yearly averaged covering surface ratio.Theoretical models show that the size distribution of self-organized vegetation surface tends to an exponential distribution that reflects competition between plants for resources, but that it tends to a power-law distribution when there is locally symbiotic relationship 27,28 .This assumption applies to the analysis of both the inverse-cumulative and non-cumulative distributions, since power-law and exponential functions are conserved under the transformation from a probability density to its cumulative distribution.The experiment in Oiso focused on measuring the relative degree of contribution between local symbiotic interactions and resource competition at the inter-species level (i.e., symbiotic gain and competitive loss in IMPEO) through an analysis of the species-wise averaged surface distribution.We devised an integrative model to evaluate the goodness of fit between the power-law and exponential distributions:

Productivity analysis and correlation analysis of species diversity and meteorological parameters of synecoculture field in Ise, Japan (Fig 4).
78 kinds of vegetable and fruit products were harvested from field B and sold as delivery boxes from January 2011 to February 2014 at a price rate of 315 JPY per 100 g, which is approximately equivalent to the rate for certified organic products (about 1.5 times higher than the price of conventional farm products) in the same region [Supplementary Data 2].From June 2010 to May 2014, other edible plant products, seeds and seedlings were also occasionally harvested and sold on-site, including as ingredients for a local restaurant; the data are summarized for each month [Supplementary Data 3].The principal cost was comparable to that of the conventional methods and comprised the cost of seeds and seedlings [Supplementary Data 4].
Yearly average data of productivity (gross profit in JPY) and material costs Products from 37 plant species in field C were harvested and sold at a local market from June 2015 to May 2018 53,57,58 .The price rate was set to those of organic products (about two times higher than conventional products) from June 2015 to May 2017, and to the prices of conventional products from June 2017 to May 2018, because of deterioration of local security situation and consequent loss of customers.
Five alternative methods that aim for sustainable farming were also tested alongside the synecoculture production during the same period, namely 1: a system of rice intensification and trees, 2: conservation agriculture, 3: permaculture, 4: biointensive market gardening, and 5: traditional market gardening.We obtained the gross profit of synecoculture sales at a daily resolution [Supplementary Data 5] and those of the five alternative methods in terms of the monthly aggregated sum

Estimation of harvest biomass from product sales
Although the land equivalent ratio (LER) 51 is used to evaluate polyculture productivity, it is not suitable for evaluating highly diverse mixed polycultures for two reasons: 1.For any probability distribution with the mean  and standard deviation , the effect of fluctuations expressed as a ratio Therefore, even if the monoculture and polyculture productivities are equal, the effect of fluctuation in LER gives a positive bias to polyculture.

2.
Actual monoculture productivity data is a weighted sum of many monoculture crops 56,59 , which is equivalent to a polyculture based on a mosaic of different monoculture surfaces.Therefore, the proportion of each crop surface within a given social-ecological context affects the overall productivity, which is not considered to be a realistic constraint in LER.
To overcome this pitfall, we defined the relative biomass ratio (BR) that represents the community-based land equivalent ratio as follows: Where   is the mixed polyculture yield ( > 1 crops are mixed together on the same surface) of the  th crop, and   is the mosaic polyculture yield (a combination of separate monocultures with  > 1 different crops on the same surface area) of the  th crop.Note that BR coincides with  ∶= ∑    ′  =1 in the IMPEO of one or more crops with the same physiological growth curve  ′ .
In the case that  crops for   are included in the  crops of   , which is the case for field B, it is possible to calculate the BR of the mixed polyculture products using the sales data weighted with the per-price weight of each crop: Where If  crops for   are not totally included in the  crops of   , which is the case of field C, we considered the possible variable range of conventional productivity based on the median and 25 th and 75 th percentiles of productivity in  crops (see also       Japanese yen (JPY) multiplied by the number of harvest events per year for synecoculture or yearly reported pro t for conventional methods), both with an offset of total costs in order to compare the yearly mean pro ts (vertical solid lines) and costs (vertical dashed lines) summed as positive and negative values, respectively (see Methods).The dotted lines on the y-axis represent the estimated probability distributions for each production category based on the data shown as the rug plots along the x-axis.In (b) left, the 78 academic names of total synecoculture products are shown as a list with a color gradient, and the associated numbers de ne the value of the y-axis in (b) right, in which the sales for each product according to date on the x-axis is represented as the diameter of the circle with the same color gradient as the list.The correlational analysis in (c) shows signi cant positive correlations between the number of produce types from synecoculture and meteorological variances for each 30-day interval.There was no signi cant correlation with the mean of the meteorological parameters.Harvested crop diversity versus mean or variance of three meteorological parameters is plotted as circles following the color gradient of the date.Black solid line: linear regression with less than 5% signi cance; dashed line: linear regression with 95% con dence; dotted line: linear regression with prediction intervals.
Productivity of synecoculture experiment in the tropical semi-arid zone.The three-year production experiment in Mahadaga, Burkina Faso shows a power-law distribution of product sales with (b in the red rectangle) asynchronous harvests of 37 kinds of crop.The x-axis of (a) represents sales of each product for synecoculture and for ve alternative farming methods that were simultaneously tested on 500 sq.m (regularized productivity is daily and species-wise productivity in terms of West African CFA franc (XOF) multiplied by the number of harvest events per year for synecoculture and ve alternative farming methods or yearly reported pro t for the conventional methods), both with an offset of total costs in order to compare the yearly mean pro ts (vertical solid lines) and costs (vertical dashed lines) summed as positive and negative values, respectively (see Methods).The dotted lines represent the estimated probability distributions for each production category on the y-axis based on the data shown by the rug plots along the x-axis.The total productivity of synecoculture (red line and distribution) is shown on a monthly aggregated scale (orange distribution) and in the two periods before (cyan line and distribution) and after (magenta line and distribution) November 2016, which was the turning point of market accessibility (see Methods).In (b) left, the 37 academic names of total synecoculture products are shown as a list with a color gradient, and the associated numbers de ne the value of the y-axis in (b) right, in which the sales of each product according to date on the x-axis is represented as the diameter of the circle with the same color gradient as the list.The correlational analysis in (c) shows signi cant positive correlations between the number of produce types from synecoculture and meteorological variances for each 14-day interval.There are also signi cant negative correlations with the means of the meteorological parameters.Harvested crop diversity versus mean or variance of three meteorological parameters is plotted as circles following the color gradient of the year's date.Black solid line: linear regression with less than 5% signi cance; dashed line: linear regression with 95% con dence; dotted line: linear regression with prediction intervals.
at the early stage of synecoculture introduction (Fig 2 (a1-a4)).The inverse cumulative distribution of the species diversity on the surface was closer to a power-law distribution than an exponential distribution, implying that the symbiotic interactions between plants are inherent besides the competition for resources (Fig 3 (a), see Methods).

(
Fig 2 (b1-b2)) and on a 500 sq.m farm in Mahadaga, Burkina Faso over the course of three years (Fig 2 (c1-c5)).The probability density of product-sales data based on asynchronous thinning of highly diverse mixed polyculture showed a long-tail distribution that largely deviated from a conventional normal distribution (Fig 4 (a, b) and Fig 5 (a,b)), and it followed power law (See Figure 5 (middle and bottom) of the Extended Data, and examples of harvests in Fig 2 (b2) of the main text), regardless of the differences in climate region and species composition.Despite the no-input practice except water and introduction of seeds and seedlings, on-site observation implied overall and multiple increases in ecosystem functions along with the ecological succession in the fields, such as improvement in crop yield, the establishment of a complex food chain that supported ecological regulation of pests, thick development of porous soil structure, increased humus and soil organic matter, improved water retention and permeability, and the resulting activation of soil microbiota (see e.g., Fig 2 (c4-c5), Figure 6 (a1) and (b1) of the Extended Data, and ref. 23, 29, 33).
Fig 3 (b), Fig 4 (c), and Fig 5 (c) of the main text and Figs 2-4 of the Extended Data).Because of the non-linear relationships between the mean and standard deviation of meteorological parameters (bottom line of Figs 2-4 of the niches (Fig 1 (b)).Random harvesting from various environments asymptotically converges the mean productivity to a normal distribution under the mean environmental conditions of the samples (Fig 1 (c)).According to the nature of competition with other species, the plants can qualitatively be classified as those with central or marginal competence (orange and blue distributions, respectively, in Fig 1).Such differences generally produce competitive loss and symbiotic gain of productivity, and both contribute to the total overyielding in mixed communities (green distribution in Fig 1 (c)).
Figure 1 of the Extended Data for the multi-dimensional version of IMPEO.

Fig 1 .
Fig 1. Relationship between physiological and ecological optima and the total effect of overyielding.(a) y-axis: examples of physiologically optimum isolated growth rate versus x-axis: environmental parameters such as temperature, precipitation, sunlight, etc.(b) y-axis: primary productivity of various ecological niches in the same environment (x-axis) but mixed communities.(c) Top: random sampling from various niches in (a) (blue and orange dashed lines) and (b) (blue and orange solid lines) converges to normal distributions via the central limit theorem, their frequencies correspond to mean productivity measures such as harvest rate (yaxis) under averaged environmental conditions (x-axis).The overall productivity (green line) includes the productivities of plants of both growth-rate types.(c) Bottom: Effects of symbiotic gain (blue line and arrows) and competitive loss (orange line and arrows) of plants with marginal and central competence, respectively, measured as the land equivalent ratio (LER) on the scale of LER ′ ≔ log(log(LER) + 1).The main components of the total overyielding (green line) transit from centrally to marginally competent species as the environment shifts from the physiological optimum (yellow background) to marginal (orange background) and monoculture intolerant ranges (red background).

Fig 2 .
Fig 2. Synecoculture experimental plots.(a1-a4) Initial vegetation stages during the second year of crop species introduction from bare land in the temperate zone, in Oiso, Japan.After the construction of furrows in January, pictures show the transition of vegetation in (a1) early February, (a2) early May, (a3) late August, and (a4) late October.(b1) Pilot farm production experiment in the temperate zone, in Ise, Japan.Typical mixed polyculture state that augments diversity and productivity of vegetables in November is shown, with (b2) an example of the products packed in a delivery box.(c1-c5) Reversal of the regime shift in the semi-arid tropics, in Mahadaga, Burkina Faso.(c1) The control plot with no intervention remained bare for three years, while (c2) the introduction of 150 edible species established vigorous ecosystems including (c3) a strategic combination of crops with high density and vertical diversity.Partial regeneration of grass is observed in the background of (c1), which appears to be a positive effect from the neighboring synecoculture field (c2-c3).(c4) Little organic

Fig 3 .
Fig 3. Spatial distribution and positive correlation with environmental variances in the initial stage of ecologically optimum crop growth in the temperate zone.The initial-stage experiment in Oiso, Japan (Fig 2 (a1-a4)) shows that (a) the estimated inverse cumulative distribution of the number of different plant species versus the percentage of the surface they occupy is closer to a power-law distribution that reflects symbiotic interactions  = 0 than to an exponential distribution that merely reflects competition for resources  = 1.(b) There exist positive correlations between the mean number of observed species and the variance of meteorological parameters over the 30 days preceding the daily plot observation.There is no observable correlation with the means of the meteorological parameters.Mean plant species diversity versus mean and variance of three meteorological parameters are plotted with circles following the color gradient depicting the date.Black solid line: linear regression with less than 5% significance; dashed line: linear regression with 95% confidence; dotted line: linear regression with prediction intervals.

Fig 4 .
Fig 4. Productivity of synecoculture experiment in the temperate zone.The fouryear production experiment in Ise, Japan shows (a) a power-law distribution of product sales with (b in the orange rectangle) asynchronous harvests of 78 kinds of crop.The x-axis of (a) represents sales of each product in synecoculture on 1,000 sq.m

Fig 5 .
Fig 5. Productivity of synecoculture experiment in the tropical semi-arid zone.
Simulation of the integrated model of physiological and ecological optima (IMPEO): Box 1 and Fig 1.
Based on ref.29, we simulated a typical scenario of overyielding with a mixed polyculture of two plant species.First, let us describe the unimodal distribution of physiological growth of two species with the same physiological optimum range (Fig 1 (a)).We define this distribution as (;   ) with an environmental parameter  and its physiologically optimum value   giving the maximum growth rate.The emerging ecological niches through interactions between the two species and the environment have several typologies, such as shifting and division, and other modifications of the growth curve, which are impossible to simulate precisely (Fig 1 (b)).Nevertheless, we will assume that there are qualitatively two different types of niche differentiation dynamics: 1) One plant type shows the superiority in growth of the physiological optimum to the other species (i.e., central competence expressed as the orange distributions in Fig 1 (b)); 2) The other plant type shows superiority in regard to growth in the marginal condition relative to the physiologically favorable range (i.e., marginal competence expressed as the blue distributions in Fig 1 (b)).
Fig 1 (c) top,   is depicted as an orange line,   as a blue line, and   +   as a green line.The parameters   = 20,   = 19.7,and   = 40 were typical values chosen to illustrate the effects of competitive loss (orange arrows) and symbiotic gain (blue arrows).In Fig 1 (c) bottom, the land equivalent ratio (LER) 51 is the value calculated between the mean monoculture productivity  ′ and its polyculture counterparts   and   , as  =   +   ′ (green line), and its species-wise components    ′ (orange line) and    ′ (blue line).These LER components are depicted on a scale of LER ′ ≔ log(log(LER) + 1), where the straight dotted black line is the separatrix LER ′ = 0 between symbiotic gain (upper part, LER ′ > 0) and competitive loss (lower part, LER ′ < 0).
Field A: From January 2010 to December 2011, randomly mixed communities of 52 edible plant species and other naturally occurring species on 420 sq.m without harvesting or watering and little weed maintenance in Oiso, Japan (GPS coordinates in decimal degrees: 35.31675, 139.32515).• Field B: From April 2008, a preliminary observation of ecological niches of various plant species; from June 2010 to May 2014, a strategically mixed association of 133 edible plant species and other naturally occurring species on a commercial farm of 1,000 sq.m with harvesting and occasional watering and weed maintenance in Ise, Japan (GPS coordinates in decimal degrees: 34.53022, 136.6873).• Field C: After the introduction of seeds and seedlings on March 2015, from June 2015 to May 2018, a strategically mixed association of 150 edible plant species on a commercial farm of 500 sq.m with harvesting, watering, and a small amount of weed maintenance in Mahadaga, Tapoa province, Burkina Faso (GPS coordinates in decimal degrees: 11.72328, 1.76136).For all implementations, only seeds and seedlings and necessary water as specified were introduced in the fields.No synthetic and organic fertilizers, no agrochemicals or other phytosanitary products, no ground cover materials, and no other amendments were used.No agricultural machinery was used, except for a small handy mower in the field B. No external financial support was given to the commercial synecoculture farms (field B and C).

1 𝜆𝜆( 1 ≥
log  =  • (, ) +  where (, ) = �   − > 0) log  ( = 0) is the Box-Cox transformation with a continuous parameter 1 ≥  ≥ 0, which converges to an exponential distribution log  =  •  −  +  in the  = 1 case and a power-law distribution log  =  • log  +  in the  = 0 case.The fitting was performed using the bcPower() and nls() functions in R 54 .In Fig 3 (b), mean species diversity in daily observed sections versus the mean and standard deviation of major meteorological parameters during the past 30 days from the observation (substantial growth period of the crops in the field) are plotted.Complete plots are shown in Figure 2 of the Extended Data.Eight parameters representing major environmental factors for plant growth (temperature, humidity, and sunlight) in an area measured at a daily 1-km grid resolution from December 2010 to December 2011 were obtained from the Agro-Meteorological Grid Square Data System, NARO (https://amu.rd.naro.go.jp/) 55 : daily mean air temperature, daily maximum air temperature, daily minimum air temperature, daily precipitation (reanalysis), mean relative humidity, global solar radiation, downward long-wave radiation, and sunshine duration.The correlation analysis was performed using the lm() function in R 54 .

(
seeds and seedlings, fertilizers and other amendments, materials such as plastic mulch, and machinery such as a tractor) of open-field conventional market gardening during 2010-2014 were obtained from the online database provided by the Ministry of Agriculture, Forestry and Fisheries in Japan 56 .These datasets were converted into amounts per 1,000 sq.m.The probability density functions shown in Fig 4 (a) were numerically estimated using the density() function in R 54 .To compare the yearly summed productivity of the conventional methods and with the daily recorded productivity of synecoculture, the scale of the x-axis of Fig 4 (a) is each unit sale multiplied by the number of harvest events per year.The conventional data consists of the yearly mean gross profit   = ∑    =1 that comprise those of  harvest events {  }, which are not explicitly shown in the dataset. is usually small (a few times per year for each crop), and {  } follows a normal distribution because it is based on a large sum of simultaneous harvests of monoculture crops; therefore,   is a good representative value of {  }.One can compare   with the yearly summed gross profit of synecoculture   = ∑    =1 based on the record of  harvest events {  } in daily and species-wise resolution, which is shown as vertical solid lines and rug plots in Fig 4 (a).In synecoculture,  is large (yearly average,  = 285 for the Ise farm and  = 3619 for the Mahadaga farm), and {  } follow a power-law distribution (also plotted in Figure 5 of the Extended Data).Therefore, {  } contains a large deviation from   .In order to plot {  } on a compatible scale with   and   , we need to define the regularized productivity   =  •   (daily and species-wise productivity   multiplied by the number of harvest events  on a yearly scale), because in that way the mean value of {  } coincides with   , i.e.,   = ∑    of the frequency of harvest events.The same scale applies to the yearly costs that are expressed as a negative offset to gross profit, which is depicted with the vertical dashed lines in Fig 4 (a).The correlation between the number of produce types (product diversity measured by the number of different species) sold as delivery box and the mean and standard deviation of eight major meteorological parameters 55 (same as in the Oiso experiment) for each 30-day interval was analyzed.Typical results are shown in Fig 4 (c); complete plots are shown in Figure 3 of the Extended Data.Productivity analysis and correlation analysis between species diversity and meteorological parameters of synecoculture field in Mahadaga, Burkina Faso (Fig 5).

[
Supplementary Data 6], together with the monthly installation, materials and working costs [SupplementaryData 7].Conventional market gardening data based on the estimation of ten crops in Burkina Faso was obtained from a Food and Agriculture Organization of the United Nations (FAO) document59 on standards of gross profit and costs, which included only installation and water costs and excluded other operation costs such as seeds and seedlings, fertilizer and phytosanitary products, and materials and working costs.Datasets of gross profit and costs of the five alternative and conventional methods were converted into amounts per 500 sq.m.The probability density functions in Fig 5 (a) were numerically estimated using the density() function in R 54 .The x-axis in Fig 5 (a) conforms to that of Fig 4 (a).In regard to Fig 5 (c), satellite meteorological data corresponding to the Mahadaga farm at a daily 19.2-km grid resolution was obtained from (http://climengine.appspot.com/) 60.From which, 19 major parameters related to plant growth were taken from the Climate Forecast System (CFS) Reanalysis dataset of the National Centers for Environmental Prediction (NCEP): maximum temperature, mean temperature, minimum temperature, potential evaporation, precipitation, specific humidity, maximum specific humidity, minimum specific humidity, 5-cm soil moisture, 25-cm soil moisture, 70-cm soil moisture, 150-cm soil moisture, net radiation, downward shortwave radiation, upward shortwave radiation, downward longwave radiation, upward longwave radiation, latent heat flux, and sensible heat flux.The correlation between the number of produce types (product diversity measured by the number of different species) and the means and standard deviations of the meteorological parameters for each 14-day interval (a substantial period of growth of crops in the field) were analyzed.Typical results are illustrated in Fig 5 (c); the complete plots are shown in Figure 4 of the Extended Data.

= 1 ,
which results in the LER having a positive bias; e.g.,

Figure 6 (
Figure 6 (b2) of the Extended Data).This estimated biomass does not include the biomass of the established ecosystem permanently present in the synecoculture field, such as trees and seedlings, naturally occurring non-edible plants, fallen leaves, stems after harvest, and highly developed root systems that are sources of soil organic matter.

Field B :
Crop-wise daily sales data of the delivery box from the Ise farm [SupplementaryData 2].Sales data above 1,000 JPY and the estimated probability density above 1.0e-7 were used for the fitting.

Field C :
Crop-wise daily sales data of the Mahadaga farm [Supplementary Data 5].Sales data above 1,000 XOF and the estimated probability density above 1.0e-7 were used for the fitting.

Figures Figure 1
Figures

Figure 4 Productivity
Figure 4 and   are the productivity measured by the sale price,   and   are product biomass per unit price for each crop (  =   •   and   =   •   ).