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Augmentation of Plant Genetic Diversity in Synecoculture: Theory and Practice in Temperate and Tropical Zones

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Part of the Sustainable Development and Biodiversity book series (SDEB,volume 22)


Natural vegetation forms a complex fractal structure of ecological niche distribution, in contrast to human-managed monoculture landscape. For the sustainable management of diverse plant genetic resources, including crop and wild species, the introduction of such ecologically optimum formation is important to compensate for the biodiversity loss and achieve higher ecological state that can provide sufficient ecosystem services for increasing human population. In this chapter, we first develop a conceptual and theoretical framework for the implementation and management of self-organized niche structures and develop an adaptive strategy of sustainable food production resulting from the statistical nature of ecosystem dynamics called power law. Second, we construct the integrative measures for the management of plant genetic resources for food and agriculture in ecological optimum that incorporate both phylogenic and phase diversities as important functional indicators of plant communities. This formalization leads to the extension of conventional concepts of biodiversity and ecosystem services toward human-assisted operational ecological diversity and utility and provides the definition and property of potentially realizable and utilizable plant genetic resources in the augmented ecosystems beyond natural preservation state. Finally, case studies from the synecoculture project in temperate and tropical zones are reported in reference to the developed framework, which draws out legislative requirements for future protection and propagation of plant genetic resources. The necessity of supportive information and communication technologies is also demonstrated. This article contains theoretical foundation and the results of the proof of concept experiments that are essential to establish a novel developmental and legislative framework for the sustainable use of plant genetic resources, overarching the protection of the natural environment and agricultural production mainstreaming biodiversity.


  • Plant genetic resources (PGR)
  • Ecological optimum
  • Power-law distribution
  • Synecoculture
  • Anthropogenic augmentation of ecosystems
  • Operational species diversity
  • Adaptive diversification
  • Ecological recapitulation principles
  • Open complex systems
  • Complexity measure
  • Information and communication technologies (ICT)
  • Traditional knowledge of indigenous peoples and local communities
  • Aichi biodiversity targets
  • United Nations sustainable development goals (SDGs)
  • The Nagoya Protocol on Access and Benefit-Sharing

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Fig. 1.1
Fig. 1.2
Fig. 1.3
Fig. 1.4
Fig. 1.5
Fig. 1.6
Fig. 1.7
Fig. 1.8
Fig. 1.9
Fig. 1.10
Fig. 1.11
Fig. 1.12
Fig. 1.13


  1. 1.

    A concrete example in one dimension is given in Appendix 3.

  2. 2.

    In measuring the surface on a fractal figure like Menger sponge, the usual Riemann integral is extremely difficult to handle. Originally, Riemann integral is defined based on infinite series, but a fractal figure is a function defined on the limit operation of infinite iteration of a map. Therefore, when attempting to perform Riemann integration of a fractal figure, it is necessary to calculate the “limit value of the function defined by the limit value,” which becomes analytically difficult. In reality, although the actual vegetation distribution has fractal feature, the lowest resolution is fixed to a finite value in actual data, but it is still complex to analytically calculate Riemann integral of long finite series on an iterative model. Besides a simple surface area, to calculate essential characteristics of a fractal figure such as fractal dimension, one needs to be based on the measure theory such as Hausdorff measure and related numerical implementation such as the box-counting method. Furthermore, to integrate on qualitative variables contained in ecosystem data, a method of counting qualitative variables must be set separately, which requires the formalization of measure integral.

  3. 3.

    Basic formalization of Lebesgue integral for vegetation data is detailed in Appendix 1.

  4. 4.

    Mathematical proof is given in Appendix 3, and a numerical example is simulated in Fig. 1.16 (right).

  5. 5.

    Artificially controlled stable monoculture can be described as a dynamical system, as derived in Appendix 1.

  6. 6.

    Other than stock trading, adaptive diversification is similar to recently prominent e-commerce strategy that is based on power-law distribution. Sales of Internet shopping sites such as is known to follow the power-law distribution, which is also called as “the long tail” (Anderson 2008).

  7. 7.

    Note that we can find other characteristics of geometrical mean that are compatible with the nature of biodiversity and ecosystems functioning. Another example of application in the context of food security concerning product diversity is developed in Appendix 2.

  8. 8.

    Virtual diversity does not exist in real ecosystems but only in human-prepared databases, which serves as the reservoir of resilience in the future adaptation of ecosystems. In stable ecosystems with saturated species diversity, virtual diversity does not make any significant contribution to ecosystem functions, just like redundant species in the redundancy hypothesis. It can, however, be a source of compensation for the loss of ecosystem functions under the rivet hypothesis in dynamical change and important accelerator of ecological transition. In either case, the virtual utility can contribute to enhancing ecosystem services for human purposes, such as the adaptation of the product portfolio to market value.

  9. 9.

    More theoretical details in Appendix 3.

  10. 10.

    A simple simulation on food security grounded on product diversity is shown in Appendix 2.

  11. 11.

    Three-dimensional generalization of the Cantor set is the Menger sponge discussed in Sect. 1.3 as an essential model of niche differentiation.


  • Akasaka M, Kadoya T, Ishihama F et al (2017) Smart protected area placement decelerates biodiversity loss: a representation-extinction feedback leads rare species to extinction. Conserv Lett 10(5):539–546

    CrossRef  Google Scholar 

  • Albert CH, de Bello F, Boulangeat I et al (2012) On the importance of intraspecific variability for the quantification of functional diversity. Oikos 121:116–126

    CrossRef  Google Scholar 

  • Anderson C (2008) The long tail. ISBN 9781401387259

    Google Scholar 

  • Arrhenius O (1921) Species and area. J Ecol 9:95–99

    CrossRef  Google Scholar 

  • Barnosky AD, Hadly EA, Bascompte J et al (2012) Approaching a state shift in Earth’s biosphere. Nature 486:52–58

    CAS  CrossRef  PubMed  Google Scholar 

  • Bastian M, Heymann S, Jacomy M (2009) Gephi: an open source software for exploring and manipulating networks. International AAAI Conference on Weblogs and Social Media.

  • Cadotte MW, Carscadden K, Mirotchnick N (2011) Beyond species: functional diversity and the maintenance of ecological processes and services. J Appl Ecol 48:1079–1087

    CrossRef  Google Scholar 

  • Carmona CP, Guerrero I, Morales MB et al (2017) Assessing vulnerability of functional diversity to species loss: a case study in Mediterranean agricultural systems. Funct Ecol 31:427–435

    CrossRef  Google Scholar 

  • Convention on Biological Diversity (CBD) (2000) The Cartagena protocol on biosafety to the convention on biological diversity.

  • Convention on Biological Diversity (CBD) (2010a) The Nagoya protocol on access to genetic resources and the fair and equitable sharing of benefits arising from their utilization (ABS) to the convention on biological diversity

    Google Scholar 

  • Convention on Biological Diversity (CBD) (2010b) Aichi biodiversity targets.

  • Convention on Biological Diversity (CBD) (2017) The access and benefit-sharing clearing-house.

  • Crutzen PJ (2002) Geology of mankind. Nature 415:23.

    CAS  CrossRef  PubMed  Google Scholar 

  • Cushing JM, Costantino RF, Dennis B et al (2005) Chaos in ecology experimental nonlinear dynamics, vol 1. Theoretical ecology series. Academic Press. ISBN 978-0-12-198876-0

    Google Scholar 

  • de Bello F, Lavorel S, Albert CH et al (2011) Quantifying the relevance of intraspecific trait variability for functional diversity. Methods Ecol Evol 2:163–174

    CrossRef  Google Scholar 

  • DGM (Dedicated Grant Mechanism for Indigenous Peoples and Local Communities) (2017) Annual report.

  • FAO (Food and Agriculture Organization) (2001) Food balance sheets a handbook.

  • FAO (Food and Agriculture Organization) (2011) Global food losses and food waste.

  • FAO (Food and Agriculture Organization) (2016) FAO guideline: voluntary guidelines for mainstreaming biodiversity into policies, programmes and national and regional plans of action on nutrition.

  • Farrior CE, Bohlman SA, Hubbell S et al (2016) Dominance of the suppressed: power-law size structure in tropical forests. Science 351:155–157

    CAS  CrossRef  PubMed  Google Scholar 

  • Flynn DFB, Mirotchnick N, Jain M et al (2011) Functional and phylogenetic diversity as predictors of biodiversity-ecosystem-function relationships. Ecology 92:1573–1581

    CrossRef  PubMed  Google Scholar 

  • Funabashi M (2013) IT-mediated development of sustainable agriculture systems-toward a data-driven citizen science. J Inf Technol Appl Educ 2(4):179–182

    Google Scholar 

  • Funabashi M (2016a) Synecological farming: theoretical foundation on biodiversity responses of plant communities. Plant Biotechnol 33:213–234

    CrossRef  Google Scholar 

  • Funabashi M (2016b) Synecoculture manual 2016 version (English version). Research and Education material of UniTwin UNESCO Complex Systems Digital Campus, e-laboratory: Open Systems Exploration for Ecosystems Leveraging, No. 2

    Google Scholar 

  • Funabashi M (2016c) Chapter 4.1. In: Tokoro M, Takahashi K (eds) Water cycle and life: creating water environment in 21st century. [Mizu daijunkan to kurashi: 21 seiki no mizu kankyo wo tsukuru (in Japanese)]. Maruzen Planet, Japan, pp 95–112

    Google Scholar 

  • Funabashi M (2017a) Synecological farming for mainstreaming biodiversity in smallholding farms and foods: implication for agriculture in India. Indian J Plant Genet Resour 30(2):99–114.

    CrossRef  Google Scholar 

  • Funabashi M (2017b) Citizen science and topology of mind. Entropy 19(4).

    CrossRef  Google Scholar 

  • Funabashi M (2017c) Open systems exploration: an example with ecosystems management. First Complex Systems Digital Campus World E-Conference, vol 2015, pp 223–243

    Google Scholar 

  • Funabashi M, Hanappe P, Isozaki T et al (2017) Foundation of CS-DC e-laboratory: open systems exploration for ecosystems leveraging. First Complex Systems Digital Campus World E-Conference, vol 2015, pp 351–374

    Google Scholar 

  • GLOBI (2017)

  • Guimarães PR Jr, Pires MM, Jordano P et al (2017) Indirect effects drive coevolution in mutualistic networks. Nature 550:511–514

    CrossRef  PubMed  Google Scholar 

  • Hashiguchi Y (2005) Islands need “food self-sufficiency ability”. J Island Stud 2005(5):33–53

    CrossRef  Google Scholar 

  • Houlton BZ, Morford SL, Dahlgren RA (2018) Convergent evidence for widespread rock nitrogen sources in Earth’s surface environment. Science 360:58–62

    CAS  CrossRef  Google Scholar 

  • Jaenicke H, Ganry J, Hoeschle-Zeledon I et al (eds) (2009) International symposium on underutilized plants for food security, nutrition, income and sustainable development. Arusha, Tanzania. ISBN 978-90-66057-01-2

    Google Scholar 

  • Larkin DL, Bruland GL, Zedler JB (2016) Heterogeneity theory and ecological restauration. In Palmer MA, Zedler JB, Falk DA (eds) Foundations of restoration ecology. Island Press. ISBN 9781610916974

    Google Scholar 

  • Laurance WF (2009) Beyond island biogeography theory. In: Losos JB, Ricklefs RE (eds) The theory of island biogeography revisited. Princeton University Press, United States, pp 214–236

    CrossRef  Google Scholar 

  • Laurance W, Mesquita R, Luizão R et al (2004) The biological dynamics of forest fragments project: 25 years of research in the Brazilian Amazon. Tropinet 15(2/3):1–3

    Google Scholar 

  • MeCab (2017)

  • Nayak C (2008) Comparing various fractal models for analyzing vegetation cover types at different resolutions with the change in altitude and season. Master Thesis, Faculty of Geo-Information Science and Earth Observation of the University of Twente (ITC), Enschede, the Netherlands, and Indian Institute of Remote Sensing (IIRS), National Remote Sensing Agency (NRSA), Department of Space, Dehradun, India.

  • NRC (National Research Council) (1993) Managing global genetic resources: agricultural crop issues and policies. The National Academies Press, Washington, DC.

  • Paroda RS, Tyagi RK, Mathur PN et al (eds) (2017) Proceedings of the ‘1st international agrobiodiversity congress: science, technology and partnership’, New Delhi, India, November 6–9, 2016. Indian Society of Plant Genetic Resources, New Delhi and Bioversity International, Rome, 152 pp

    Google Scholar 

  • Pecl GT, Araújo MB, Bell JD et al (2017) Biodiversity redistribution under climate change: impacts on ecosystems and human well-being. Science 355.

    CrossRef  PubMed  Google Scholar 

  • Pereira HM et al (2010) Scenarios for global biodiversity in the 21st century. Science 330:1496.

    CAS  CrossRef  PubMed  Google Scholar 

  • Petherick A (2012) A note of caution. Nat Clim Change 2:144–145

    CrossRef  Google Scholar 

  • Prusinkiewicz P, Lindenmayer A (2012) The algorithmic beauty of plants. Springer, ISBN 9781461384762

    Google Scholar 

  • Putman RJ, Wratten SD (1984) Principles of ecology. University of California Press, California

    Google Scholar 

  • R Core Team (2015) R: A language and environment for statistical computing. R Foundation for Statistical Computing, Vienna, Austria.

  • Reich PB, Tilman D, Isbell F et al (2012) Impacts of biodiversity loss escalate through time as redundancy fades. Science 336:589–592

    CAS  CrossRef  PubMed  Google Scholar 

  • Reuter MA, Hudson C, Hagelüken C et al (2013) Metal recycling: opportunities, limits, infrastructure. A Report of the Working Group on the Global Metal Flows to the International Resource Panel. UNEP

    Google Scholar 

  • Richards CM, Falk DA, Montalvo AM (2016) Population and ecological genetics in restoration ecology. In Palmer MA, Zedler JB, Falk DA (eds) Foundations of restoration ecology. Island Press, ISBN 9781610916974

    Google Scholar 

  • Rippke U, Ramirez-Villegas J. Jarvis A et al (2016) Timescales of transformational climate change adaptation in sub-Saharan African agriculture. Nat Clim Change 6:605–609

    CrossRef  Google Scholar 

  • Rohde RA, Muller RA (2005) Cycles in fossil diversity. Nature 434:208–210

    CAS  CrossRef  PubMed  Google Scholar 

  • Scanlon TM, Caylor KK, Levin SA et al (2007) Positive feedbacks promote power-law clustering of Kalahari vegetation. Nature 449:209–212

    CAS  CrossRef  PubMed  Google Scholar 

  • Seuront L (2010) Fractals and multifractals in ecology and aquatic science. CRC Press. ISBN 9781138116399

    Google Scholar 

  • Steffen W, Richardson K, Rockström J et al (2016) Planetary boundaries: guiding human development on a changing planet. Science 347:1259855

    CrossRef  Google Scholar 

  • Takayasu H, Sato A, Takayasu M (1997) Stable infinite variance fluctuations in randomly amplified Langevin systems. Phys Rev Lett 79:966–969

    CAS  CrossRef  Google Scholar 

  • TINA (2017)

  • Tindano A, Funabashi M (eds) (2016) Proceedings of the 1st African forum on synecoculture (English version). Research and Education material of UniTwin UNESCO Complex Systems Digital Campus, e-laboratory: Open Systems Exploration for Ecosystems Leveraging, No. 5

    Google Scholar 

  • Tindano A, Funabashi M (eds) (2017) Proceedings of the 2nd African forum on synecoculture (English version). Research and Education material of UniTwin UNESCO Complex Systems Digital Campus, e-laboratory: Open Systems Exploration for Ecosystems Leveraging, No. 7

    Google Scholar 

  • Turner GM (2008) A comparison of the limits to growth with 30 years of reality. Glob Environ Chang 18(3):397–411

    CrossRef  Google Scholar 

  • UA (African Union) (2015) Lignes directrices pratiques de l’Union Africaine pour la mise en oeuvre coordonnée du Protocole de Nagoya en Afrique.

  • UN (United Nations) (2015) Sustainable development goals.

  • UN (United Nations) (2017) UN member states.

  • UNEP (United Nations Environment Programme) (2017)

  • Whittaker RH (1960) Vegetation of the Siskiyou mountains, Oregon and California. Ecol Monogr 30:280–338

    CrossRef  Google Scholar 

  • Wu H, Sun Y, Shi W et al (2013) Examining the satellite-detected urban land use spatial patterns using multidimensional fractal dimension indices. Remote Sens 5:5152–5172.

    CrossRef  Google Scholar 

  • Yong RN, Mulligan CN, Fukue M (2006) Geoenvironmental sustainability. CRC Press, United States

    CrossRef  Google Scholar 

  • Zuppinger-Dingley D, Schmid B, Petermann JS et al (2014) Selection for niche differentiation in plant communities increases biodiversity effects. Nature 515:108–111

    CAS  CrossRef  PubMed  Google Scholar 

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Kousaku Ohta and Tatsuya Kawaoka contributed as a research assistant at Sony CSL. Experiments of synecoculture were conducted in collaboration with in Japan: Takashi Otsuka, Sakura Shizen Jyuku; in Taiwan: Kai-Yuan Lin, Asian SustaInable Agriculture Research and production Center (ASIARC); and in Burkina Faso: André Tindano, Association de Recherche et de Formation du Développement Rural Autogéré (AFIDRA) and Centre Africain de Recherche et de Formation en Synécoculture (CARFS).

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Correspondence to Masatoshi Funabashi .

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Appendix 1: Construction of Lebesgue Integral in an Ecosystem Data Set

This section summarizes the basics of Lebesgue integral, which gives the mathematical basis of analysis in ecosystem data with power-law configuration. The integration of the probability measure can be implemented by a search algorithm, and an IT platform such as content management system (CMS) is required for the analysis of massive data (Funabashi 2017b).

  • Lebesgue Integral as Probability Integral Over Real Number and Integration of the Probability Measure on a Set

In order to give the calculation of the probability by Lebesgue integral on the real parameter space, consider Lebesgue measurable space \( \left( {X, \varvec{B}, \mu } \right) \).

The normalization condition by the Riemann integral of the probability density function \( p\left( x \right) \) on \( X = \varvec{R}{:}\left( { - \infty ,\varvec{ }\infty } \right) \), \( \varvec{x} \in X \) is given by

$$ \mathop \int \limits_{ - \infty }^{\infty } p\left( x \right){\text{d}}x = 1. $$

The Riemann integral is defined by the limit value of infinite series. Therefore, fractal figure, etc., which involves the limit operation in the definition of the function \( p\left( x \right) \) itself, will encounter double limit operations, and the analysis becomes extremely difficult. Since vegetation pattern has the property of fractal figure, the calculation of analytical solution is difficult when extrapolating measurement value to fractal figure model.

In such a case, it can be calculated using the Lebesgue integral. Let \( \mu \) be the Lebesgue measure on \( \varvec{R} \), then the normalization condition of probability by Lebesgue integral is

$$ \mathop \int \limits_{X}^{{}} p\left( x \right){\text{d}}\mu = 1. $$

Then, the Lebesgue integral of the probability satisfying \( p\left( x \right) \ge \alpha \) is given by

$$ \mathop \int \limits_{{\left. X \right|_{{\varvec{p}\left( \varvec{x} \right) \ge\varvec{\alpha}}} }}^{{}} p\left( x \right){\text{d}}\mu . $$

This can be calculated for the distribution of \( p\left( x \right) \) with a complex configuration, such as fractal distribution on \( \varvec{R} \), by the Lebesgue convergence theorem. Intuitively, since the phase structure of the completely additive class \( \varvec{B} \) is defined, the convergence of the infinite sequence can be treated in a topologically simple manner, by putting the limit operation that defines the fractal function outside of the integration. The probability density function \( p\left( x \right) \) on \( X = \varvec{R}^{\varvec{n}} {:}\left( { - \infty ,\varvec{ }\infty } \right)^{\varvec{n}} \) is also given in the same way.

In order to calculate the probability on a symbolic set \( S \), not on \( \varvec{R}^{\varvec{n}} \), it is necessary to determine the completely additive class \( \varvec{E} \) on \( S \) and the probability measure \( P \). In the measurable space \( \left( {S, \varvec{E}, P} \right) \), let \( 1_{S} \) be the definition function of \( S \), then the normalization condition of the probability measure is

$$ \mathop \int \limits_{S}^{{}} 1_{S} {\text{d}}P = 1 . $$

The occurrence probability that is not less than \( \alpha \) when measured with \( P \) is given as

$$ \mathop \int \limits_{{\left. S \right|_{P\left( S \right) \ge \alpha } }}^{{}} 1_{S} {\text{d}}P. $$
  • Mean Information

The mean information of all events of \( S \) is

$$ \mathop \int \limits_{S}^{{}} - { \log }\left( P \right){\text{d}}P . $$

When \( P \) is binarized with a certain threshold value with respect to the occurrence probability \( P\left( s \right) \) of the element \( s \in S \), the mean information is redefined as follows, on a newly binarized probability measure \( P^{\prime} \) on the measurable space \( \left( {S, \varvec{E}, P^{\prime}} \right) \):

$$ \mathop \int \limits_{S}^{{}} - \log \left( {P^{\prime}} \right){\text{d}}P^{\prime}. $$
  • Mean Yield Per Surface

The mean yield per surface on \( S \) can be defined by constituting a yield function \( Y{:}S \to \varvec{R} \), such as

$$ \mathop \int \limits_{S}^{{}} Y{\text{d}}P. $$

The mean yield of vegetation with a yield above a certain value \( \beta \), such as \( Y \ge \beta \), is given as

$$ \mathop \int \limits_{{\left. S \right|_{Y\left( S \right) \ge \beta } }}^{{}} Y{\text{d}}P. $$

If \( Y\left( S \right) \) is a skewed distribution with respect to \( S \), such as power-law distribution, attention should be paid to interpretation since the fluctuation of mean yield could be enormous in actual data.

  • Occurrence Probability of Objective Function

To obtain the occurrence probability of vegetation having an objective function \( O{:}S \to \varvec{R} \) greater than or equal to the constant value \( \beta \), the measure space \( \left( {S, \varvec{E}, O} \right) \) of the objective function should be constructed on the same completely additive class \( \varvec{E} \) as \( \left( {S, \varvec{E}, P} \right) \), and given by

$$ \mathop \int \limits_{{\left. S \right|_{O\left( S \right) \ge \beta } }}^{{}} 1_{S} {\text{d}}P. $$

Example: For tagged ecosystem data, the occurrence probability of various combinations of tags can be calculated. In order to handle co-occurrence of tags, it is necessary to construct a completely additive class \( \varvec{E} \) of resolution that satisfies \( s_{i} \cap s_{j} = \emptyset \left( {s_{i} , s_{j} \in S} \right) \).

  • Conditional Probability

If the two objective functions \( O_{1} \) and \( O_{2} \) are constructed as the measure \( O_{1} {:}S \to \varvec{R} \) and \( O_{2} {:}S \to \varvec{R} \) on the completely additive class \( \varvec{E} \) of \( S \), the integral of the target measure \( O_{2} \) under the condition \( A = \left\{ {s \in S|O_{1} \left( s \right) \ge \beta } \right\} \) is given by

$$ \mathop \int \limits_{A}^{{}} 1_{S} {\text{d}}O_{2} . $$
  • Calculation of Expected Value of an Objective Function by Non-uniform Probability Density Function

The expected value of the objective function \( O\left( s \right) \) with respect to the non-uniform probability density function \( P\left( s \right) \) on \( s \in S \) is given as

$$ \mathop \int \limits_{S}^{{}} O\left( s \right){\text{d}}P = \mathop \int \limits_{S}^{{}} O\left( s \right)f\left( s \right){\text{d}}\mu \left( s \right) . $$

where \( f{:}S \to \varvec{R} \) is the probability density function of \( P \) for \( \mu \), which is called the Radon–Nikodym derivative. The set measure \( \mu \) is, for example, \( \mu \left( s \right) = \# \left\{ s \right\} \) in case of a symbolic set.

  • On Vegetation Succession: Perron–Frobenius Operator and Climax Community

The vegetation transition can be represented as a symbolic dynamical system that is the temporal change of the direct product space \( \varvec{R}^{\varvec{m}} \times {\mathbf{Str}}^{\varvec{n}} \) of numerical (\( \varvec{R} \)) and symbolic (\( {\mathbf{Str}} \)) data of soil and vegetation variables. Then, as an example, the climax community can be described using the Perron–Frobenius operator of the symbolic dynamical system. For simplicity, we consider a symbolic dynamical system \( \varvec{T} \) of \( n \) kinds of plant species on a two-dimensional map, as a model of vegetation succession:

$$ \varvec{T}{:}\varvec{R}^{2} \times {\mathbf{Str}}^{\varvec{n}} \to \varvec{R}^{2} \times {\mathbf{Str}}^{\varvec{n}} \varvec{ }, $$

where \( \varvec{T} \) is a non-singular map. If we want to include other numerical data \( \varvec{R}^{\varvec{m}} \) such as environmental parameters, this extends to

$$ \varvec{T}{:}\varvec{R}^{2} \times \varvec{R}^{\varvec{m}} \times {\mathbf{Str}}^{\varvec{n}} \to \varvec{R}^{2} \times \varvec{R}^{\varvec{m}} \times {\mathbf{Str}}^{\varvec{n}} . $$

Let \( \varvec{V} \) be a completely additive class on \( X{:}\varvec{R}^{2} \). \( \varvec{V} \) corresponds to a list of every niche structure of vegetation with real-value (infinite) resolution. Let \( g\left( X \right) \) be the density function on \( X \) (Radon–Nikodym derivative). Practically, in case of a map which displays only the presence or absence of vegetation in practice, \( g\left( X \right) \) is a binary function.

Here, we call the \( {\text{PF}} \) that is defined as follows for the Lebesgue measure \( m \) on \( X \) as the Perron–Frobenius operator of vegetation succession \( \varvec{T} \).

$$ \mathop \int \limits_{V}^{{}} {\text{PF}} \cdot g\left( x \right){\text{d}}m = \mathop \int \limits_{{T^{ - 1} \left( V \right)}}^{{}} {\text{PF}} \cdot g\left( x \right){\text{d}}m. $$

The stationary density function satisfying \( {\text{PF}} \cdot g = g \) gives the area of each species that potential natural vegetation (climax community) comprises.

Practically, in consideration of intrinsic fluctuations, if the following relation holds with respect to the function norm of \( {\text{PF}} \cdot g = g \) and the upper bound \( \delta \) of fluctuation, the vegetation can be judged as climax community for a time scale \( t \) and inherent fluctuation \( \delta \):

$$ \left| {{\text{PF}}^{t} \cdot g - g} \right| \le \delta . $$

This relationship can also represent a stabilized monoculture method by human control, as detailed in Sect. 1.4.

If Perron–Frobenius operator \( {\text{PF}} \) for vegetation succession \( \varvec{T} \) can be completely determined, the potential natural vegetation and the response to the vegetation strategy of synecological farming can be uniformly described. In practice, since the real ecosystem is too complex to model as a dynamical system, the stationary density function that determines the Perron–Frobenius operator can only be numerically approximated by the convergence of the function norm under inevitable fluctuation. Therefore, it can be described as a dynamical system of conditional probability, and it becomes necessary to connect to probabilistic analysis such as information geometry (Funabashi 2017b). By interpreting vegetation succession \( \varvec{T} \) and corresponding Perron–Frobenius operator \( {\text{PF}} \) as a probability map, finding deterministic structure as much as possible from these can be considered as the primary direction of model refinement.

Appendix 2: Food Self-sufficiency Measure with a Geometric Mean

The conventional definition of food self-sufficiency rate (SSR) is based on the arithmetic mean, whether it be calorie-based or production-based (FAO 2001). As an application of the concept of selective information developed in Sect. 1.4, we simulate a novel food self-sufficiency rate related to the geometric mean.

The problem of arithmetic means is that it does not correctly reflect the notion of self-sufficiency with respect to the diversity of food products: Suppose there exist food items that cannot be produced in a social community but crucial for the survival. Then, the community could not survive when the importation is prohibited, even if the total SSR over the whole food products is superior to 100%. This is typically the case with food production in limited geographical scale such as in small island (Hashiguchi 2005).

To properly adopt the notion of self-sufficient “ability” with respect to the survival of a community in isolation, the following geometric mean \( I_{\text{g}} \) can provide a simple definition of risk that threatens self-sufficiency regarding the diversity of food products:

$$ I_{\text{g}} {:} = \sqrt[{k^{{max} } }]{{\mathop \prod \limits_{k = 1}^{{k^{{max} } }} l\left( {SSR_{k} } \right)}} , $$

where \( SSR_{k} \) is the SSR defined with the percentage of \( k \)-th food item \( \left( {k = 1, \ldots , k^{{max} } } \right) \), and to cut above 100% of each \( SSR_{k} \),

$$ l\left( x \right){:} = \left\{ {\begin{array}{*{20}l} 1 \hfill & {{\text{if}}\;{\text{x}} \ge 100{\text{\% }}} \hfill \\ {0.01 \cdot x} \hfill & {\text{else}} \hfill \\ \end{array} } \right.. $$

It transforms to the mean selective information \( I_{\text{s}} \) if we measure the self-sufficiency risk as a mean information cost for the search, such as

$$\begin{aligned} I_{\text{s}} {:} & = {{ - \log \left( {I_{\text{g}}^{{k^{{max} } }} } \right)} \mathord{\left/ {\vphantom {{ - \log \left( {I_{\text{g}}^{{k^{{max} } }} } \right)} {k^{{max} } }}} \right. \kern-0pt} {k^{{max} } }} = {{ - \log \left( {\mathop \prod \limits_{k = 1}^{{k^{{max} } }} l\left( {{\text{SSR}}_{k} } \right)} \right)} \mathord{\left/ {\vphantom {{ - \log \left( {\mathop \prod \nolimits_{k = 1}^{{k^{{max} } }} l\left( {{\text{SSR}}_{k} } \right)} \right)} {k^{{max} } }}} \right. \kern-0pt} {k^{{max} } }} \\ &= - \mathop \sum \limits_{k = 1}^{{k^{{max} } }} \log l\left( {{\text{SSR}}_{k} } \right)/k^{{max} } .\end{aligned} $$

The mean selective information \( I_{\text{s}} \) represents the mean information cost to search all food products under given SSR. It diverges to infinity when there is a single food item with \( {\text{SSR}}_{k} = 0\% \), while coincides with 0 when all food items’ \( {\text{SSR}}_{k} = 100\% \). It represents the situation that there exists sufficient food in the world, but the distribution is not equitable for the self-sufficiency of the global population (FAO 2011).

Actual trade of food products between and within social communities may substitute some items with others. We define a natural extension of the mean selective information \( I_{\text{s}} \) to the domain including \( {\text{SSR}}_{k} > 100\% \), as the extended mean selective information \( I^{\prime}_{\text{s}} \):

$$ I^{\prime}_{\text{s}} = - \mathop \sum \limits_{k = 1}^{{k^{{max} } }} \log l^{\prime}\left( {{\text{SSR}}_{k} } \right)/k^{{max} } , $$


$$ l^{\prime}\left( x \right){:} = 0.01 x . $$

This formulation is equivalent to define the exchange rate of a product with others according to the ratio of selective information representing the search cost. For the value \( {\text{SSR}}_{k} > 100\% \), the selective information turns into a negative value and can be interpreted as a search gain that cancels out the search cost of other products.

We simulated the \( I_{\text{s}} \) and \( I^{\prime}_{s} \) for the different levels of production from power-law vegetation. The algorithm is defined as follows:

  1. 1.

    Define the parameters \( a \) and \( b \) of Pareto distribution (see Sect. 1.2).

  2. 2.

    Sample \( k^{{max} } \) values from the Pareto distribution and define them as \( p_{k } \left( {k = 1, \ldots , k^{{max} } } \right) \).

  3. 3.

    Create a new series \( p^{\prime}_{{k_{1} k_{2} }} \) with respect to each value of \( p_{k} \) as a regularization factor, such as \( p^{\prime}_{{k_{1} k_{2} }} {:} = p_{{k_{1} }} /p_{{k_{2} }} \) \( \left( {k_{1} , k_{2} = 1, \ldots , k^{{max} } } \right) \).

  4. 4.

    Calculate \( I_{\text{s}} \) and \( I^{\prime}_{\text{s}} \) for each \( k_{2} \) with respect to \( k_{1} = 1, \ldots , k^{{max} } \), with the use of \( {\text{SSR}}_{{k_{1} k_{2} }}^{{}} = 100 \cdot p^{\prime}_{{k_{1} k_{2} }} \). For each \( k_{2} \), the number of \( {\text{SSR}}_{{k_{1} k_{2} }} \) that exceeds 100% should be attributed as \( \# \left\{ {{\text{SSR}}_{{k_{1} k_{2} }} | {\text{SSR}}_{{k_{1} k_{2} }} \ge 100\% , k_{1} = 1, \ldots , k^{{max} } } \right\} \), which can be simplified to a single parameter \( k \) such as \( \# \left\{ {{\text{SSR}}_{k} \ge 100\% } \right\} \) \( \left( {k = 1, \ldots , k^{{max} } } \right) \).

  5. 5.

    Repeat from 2 to 4 for \( N_{\text{s}} \) times and take the mean values and standard deviations of \( I_{\text{s}} \) and \( I^{\prime}_{\text{s}} \) for each \( k_{2} \) with respect to \( N_{\text{s}} \) samplings.

The results are shown in Fig. 1.14. As a representative example, \( k^{{max} } = 120 \) was chosen to represent the commonly utilized crop species diversity in world agriculture (Yong et al. 2006). \( N_{\text{s}} = 193 \) corresponds to the number of member states of the United Nations (UN 2017). Naturally from the definition, the value of \( I_{\text{s}} \) converges to zero as the number of self-sufficient crops approaches to \( k^{{max} } \), since all food products need to achieve self-sufficiency for the survival. On the other hand, \( I^{\prime}_{\text{s}} \) admits the compensation between crops by the extended definition of search cost, which achieves self-sufficiency risk zero around \( \# \left\{ {{\text{SSR}}_{k} \ge 100\% } \right\} = 44\;{\text{to}}\;45 \). This critical value is invariant with the parameters of power-law vegetation \( a \) and \( b \), due to the logarithmic property of selective information and regularization process of the algorithm.

Fig. 1.14
figure 14

Simulation of the mean selective information \( I_{\text{s}} \) and the extended mean selective information \( I^{\prime}_{\text{s}} \) with respect to the number of self-sufficient food products \( \# \left\{ {{\text{SSR}}_{k} \ge 100\% } \right\} \) \( \left( {k = 1, \ldots , k^{{max} } } \right) \). Mean ± standard deviation of \( I_{\text{s}} \) and \( I^{\prime}_{\text{s}} \) is shown with the red and blue plot, respectively, and those of \( I^{\prime}_{\text{s}} \) with the green plot. The value of \( \# \left\{ {{\text{SSR}}_{k} \ge 100\% } \right\} \) where \( I^{\prime}_{\text{s}} \) crosses zero (intersection between blue and green lines) is shown to be invariant with respect to different values of parameters \( a \) and \( b \) that define Pareto distribution

This simulation suggests that the self-sufficient ability defined with respect to the risk index \( I^{\prime}_{\text{s}} \) can be achieved by securing SSR of about 37% of the necessary product diversity within a closed territory if we take on the strategies of adaptive diversification with power-law productivity. By exchanging \( k^{{max} } \) and \( N_{\text{s}} \) in the algorithm, we can simulate the case of international trades between \( N_{\text{s}} \) countries (following power-law productivity) with variance on \( k^{{max} } \) crop diversity, which derives qualitatively the same behavior of \( I^{\prime}_{\text{s}} \): 37% of the member states need to achieve self-sufficiency for each crop, in order to attain global sufficiency under the exchange rate defined in \( I^{\prime}_{\text{s}} \).

Parameters: The values of \( a \) and \( b \) are shown in the figure legend. \( k^{{max} } = 120 \), \( N_{\text{s}} = 193 \).

Appendix 3: Ecological Scarcity and Land Utilization

How can we most efficiently allocate the land use distribution (e.g., between city, farmland, and protected area) for the protection of biodiversity? For that purpose, let us think about the ecological value of a species. The value of a species in ecosystems has multi-faceted criteria, over multiple ecosystem functions and services, on various spatial–temporal scales. There is an only limited amount of known value compared to the unknown part that cannot be measured advance. Take for example ecological resilience, there could be an infinite unknown possibility of ecological disturbance in the future in which each species may play constructive roles for the recovery and development of new ecosystems.

An indeterminable amount of our ignorance primarily limits the argument on resilience. If we are to formulate the total amount of functions and these values of a species with respect to the future resilience of ecosystems, an only expression such as “all species are equally invaluable at the largest limit of spatial–temporal scale” would be allowed to eliminate any specific bias. We call this premise as the “value equivalence principle of species,” which mathematical formulations commonly adopt for multiple diversity indexes in ecological study. This standpoint is also vital in the management process of one-time-only events such as natural disasters, in which we need to select a fail-safe strategy without prior knowledge of future change (Funabashi 2017c).

Now, suppose that all species are equally invaluable for a whole ecosystem. The simplest ideal scenario of the conservation is to allocate each species an equal surface, or uniform distribution of habitat surface over species, which maximizes Shannon’s diversity index measured on the proportion of habitats. The actual distribution of species would differ under multiple social and ecological factors, which can be quantified with the following measure of ecological scarcity \( R_{k} \) of a species \( k \):

$$ R_{k} {:} = \frac{C}{{X_{k} }}, $$

where \( X_{k} \) is the habitat surface of the species \( k \), and \( C \) is the total surface taking arbitrary constant value. Here, ecological scarcity \( R_{k} \) represents how much a species deserves conservation value with respect to the smallness of actual habitat under the value equivalence principle of species.

Let us consider the fractal dynamics of niche differentiation. For simplicity, we consider the recurrence relation of one-dimensional Cantor setFootnote 11 \( f_{c} \) as follows, which divides the surface \( C \) represented as a line segment into two isometric parts:

$$ C_{n} {:} = f_{c} \left( {C_{n - 1} } \right){:} = \frac{{C_{n - 1} }}{3} \cup \left( {\frac{2}{3} + \frac{{C_{n - 1} }}{3}} \right) $$

for \( n \ge 1 \), and \( C_{0} = \left[ {0, C} \right] \).

This means that by deleting the open middle third \( \left( {{\text{C/}}3, {\text{C/}}3} \right) \) from the interval [0, C], this transformation leaves two line segments: [0, C/3] and \( \left[ {2C /3, C} \right] \).

After \( n \) iteration of \( f_{c} \), the habitat \( X_{k} \) will divide into \( 2^{n} \) separated niches, and the ecological scarcity becomes

$$ R_{k} = \left( {\frac{3}{2}} \right)^{n} . $$

To avoid parameter dependency, we normalize the niche surface ratio of a species \( k \) by dividing \( X_{k} \) with \( C \) as

$$ \frac{{X_{k} }}{C} = \left( {\frac{2}{3}} \right)^{n} . $$

The relationship between the number of different niches, its surface ratio, and ecological scarcity of a species is depicted in Fig. 1.15. Starting from a monoculture situation, as niche differentiation progresses with \( n \) and increases the complexity of fractal configuration (Fig. 1.15 left), ecological scarcity increases in an exponential manner (Fig. 1.15 right).

Fig. 1.15
figure 15

Relationship between niche surface ratio \( X_{k} /C \), ecological scarcity \( R_{k} \), and niche number \( 2^{n} \) of a species \( k \) after \( n \)–time niche differentiation with recurrence relation \( f_{c} \). Left Niche partitions on the scale of surface ratio versus ecological scarcity \( R_{k} \). The gray rectangles that correspond to the direct product sets \( \frac{{C_{n} }}{C} \times \left[ {0, R_{k} } \right] \) are superimposed for different \( n = \left\{ {0, 1, 2, \ldots , 9} \right\} \). The intervals \( C_{n} /C \) correspond to the niche partitions after \( n \)-th iteration measured on the scale of niche surface ratio on X-axis. The number of differentiated niches is aligned at right Y-axis in correspondence with the value of \( R_{k} \) in left Y-axis. Right Number of niche partitions \( 2^{n} \) versus niche surface ratio \( X_{k} /C \) and ecological scarcity \( R_{k} \). As the niche number increases with \( n \), niche surface ratio decreases, and ecological scarcity increases in an exponential manner. Monoculture situation (yellow background) is dominated by a single species with low ecological scarcity, while highly differentiated mixed polyculture situations (cyan background) consist of niches with high ecological scarcity, where the role of synecoculture on the conservation efforts becomes increasingly important

Such situation is particularly familiar with rapidly expanding urban land use planning, where complex intersections with multi-fractal features between city, farmland, and natural ecosystems are often at the forefront of the development (e.g., Wu et al. 2013). In finely fragmented lands with various heterogenic environments, conventional monoculture farming methods are inefficient to perform, while synecoculture can provide combined solutions by making use of the particularity of each niche condition with various vegetation portfolios, and combining with other modes of food production according to the adjacent land use that provides different accessibility for the distribution of the products. This strategy can harmonize the conservation of rare species with high ecological scarcity and small-scale local production of various food products in the burgeoning frontiers of urban development. To raise awareness and increase susceptibility to synecoculture, practices in small-scale fragmented land such as family gardens, abandoned farmland, and city greenbelt are important, which will strengthen the future option of environmental conservation in smart green cities. Figure 1.16 left shows a schematic diagram of the patterns of the combination of synecoculture with other adjacent land use. Not only the in-field strategy of adaptive diversification in synecoculture (Sect. 1.4, Fig. 1.4), we can also produce local food from surrounding practices and environments, such as conventional agriculture, urban farming, and hunting-gathering. In either case, coping with small-scale diversified practices are strongly influenced by power-law fluctuation, which needs to count on the stability of harmonic mean for the management (Fig. 1.16 right, see also Sect. 1.4). To realize human augmentation of ecosystems at the boundaries between city and natural environments, we need to deeply understand these properties of the power law prevalent in urban and vegetation dynamics and presuppose the benefits to both sides in the formation of public opinions and policymaking.

Fig. 1.16
figure 16

Left Synecoculture as an interface between urban and natural environments. Conceptual diagram of growing margins at the intersections between city, farmland, and natural environment is depicted as self-organized fractal landscape (red triangles), where synecoculture can play an important role for both conservation of ecologically scarce species and local food production. According to the adjacent land use, synecoculture can be combined with 1. conventional agriculture (yellow circle); 2. urban farming (violet circle); and 3. hunting–gathering activities (blue circle) in the mixed topography. Right Convergence of harmonic mean and divergence of arithmetic mean in Pareto distribution. Ten thousand independent samplings were performed from a Pareto distribution with \( a = 0.1 \) (i.e., no finite arithmetic mean, see Sect. 1.2), \( b = 0.01 \). The dynamics of the arithmetic mean of previous samplings (horizontal axis) show huge discontinuous fluctuation several times triggered by rare big events, while that of the harmonic mean (vertical axis) is confined in a small stable range and converges through time. Color gradient represents the time step of the sampling

Mathematical proof for the convergence of the harmonic mean of Pareto distribution is given as follows:

For a harmonic mean \( H\left( X \right) \) of \( f\left( x \right){:} = \frac{{ab^{a} }}{{x^{a + 1} }} \), \( \left( {x \in X{:} = \left( {b,\infty } \right]} \right) \) in Sect. 1.2,

$$ \begin{aligned} \frac{1}{H\left( X \right)} & {:} = \mathop \int \limits_{ - \infty }^{\infty } \frac{1}{x}f\left( x \right)dx \\ & = \mathop \int \limits_{b}^{\infty } \frac{{ab^{a} }}{{x^{a + 2} }}dx \\ & = \frac{{ab^{a} }}{a + 1}\left[ { - x^{{ - \left( {a + 1} \right)}} } \right]_{b}^{\infty } \left( {a \ne - 1} \right). \\ \end{aligned} $$

Then, the condition that the right side does not diverge to infinity is given by

$$ \begin{array}{*{20}c} { - \left( {a + 1} \right)} & < & 0 \\ \end{array} , $$
$$ \begin{array}{*{20}c} a & > & { - 1} \\ \end{array} $$

By definition, \( a > 0 \), then for all \( a > 0 \), \( 1 /H\left( X \right) \) converges to a finite value as follows:

$$ \begin{aligned} \frac{1}{H\left( X \right)} & = \frac{{ab^{a} }}{a + 1}\left\{ {0 - \left( { - b^{{ - \left( {a + 1} \right)}} } \right)} \right\} \\ & = \frac{a}{{\left( {a + 1} \right)b}} , \\ \end{aligned} $$

which gives

$$ \begin{array}{*{20}c} {H\left( X \right)} & = & {\frac{{b\left( {a + 1} \right)}}{a}} \\ \end{array}. $$

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Funabashi, M. (2019). Augmentation of Plant Genetic Diversity in Synecoculture: Theory and Practice in Temperate and Tropical Zones. In: Nandwani, D. (eds) Genetic Diversity in Horticultural Plants. Sustainable Development and Biodiversity, vol 22. Springer, Cham.

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