Abstract
Natural vegetation forms a complex fractal structure of ecological niche distribution, in contrast to humanmanaged 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 selforganized 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 humanassisted 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.
Keywords
 Plant genetic resources (PGR)
 Ecological optimum
 Powerlaw 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 BenefitSharing
This is a preview of subscription content, access via your institution.
Buying options
Notes
 1.
A concrete example in one dimension is given in Appendix 3.
 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 boxcounting 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.
Basic formalization of Lebesgue integral for vegetation data is detailed in Appendix 1.
 4.
 5.
Artificially controlled stable monoculture can be described as a dynamical system, as derived in Appendix 1.
 6.
Other than stock trading, adaptive diversification is similar to recently prominent ecommerce strategy that is based on powerlaw distribution. Sales of Internet shopping sites such as Amazon.com is known to follow the powerlaw distribution, which is also called as “the long tail” (Anderson 2008).
 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.
Virtual diversity does not exist in real ecosystems but only in humanprepared 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.
More theoretical details in Appendix 3.
 10.
A simple simulation on food security grounded on product diversity is shown in Appendix 2.
 11.
Threedimensional generalization of the Cantor set is the Menger sponge discussed in Sect. 1.3 as an essential model of niche differentiation.
References
Akasaka M, Kadoya T, Ishihama F et al (2017) Smart protected area placement decelerates biodiversity loss: a representationextinction feedback leads rare species to extinction. Conserv Lett 10(5):539–546
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
Anderson C (2008) The long tail. ISBN 9781401387259
Arrhenius O (1921) Species and area. J Ecol 9:95–99
Barnosky AD, Hadly EA, Bascompte J et al (2012) Approaching a state shift in Earth’s biosphere. Nature 486:52–58
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. https://www.aaai.org/ocs/index.php/ICWSM/09/paper/view/154/1009
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
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
Convention on Biological Diversity (CBD) (2000) The Cartagena protocol on biosafety to the convention on biological diversity. https://www.cbd.int/doc/legal/cartagenaprotocolen.pdf
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
Convention on Biological Diversity (CBD) (2010b) Aichi biodiversity targets. https://www.cbd.int/sp/targets/
Convention on Biological Diversity (CBD) (2017) The access and benefitsharing clearinghouse. https://absch.cbd.int/
Crutzen PJ (2002) Geology of mankind. Nature 415:23. https://doi.org/10.1038/415023a
Cushing JM, Costantino RF, Dennis B et al (2005) Chaos in ecology experimental nonlinear dynamics, vol 1. Theoretical ecology series. Academic Press. ISBN 9780121988760
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
DGM (Dedicated Grant Mechanism for Indigenous Peoples and Local Communities) (2017) Annual report. https://static1.squarespace.com/static/550abd2ce4b0c5557aa4f772/t/5a26f29b085229c33332f339/1512501925869/DGM_Report_Annual2017_LR_EN.pdf
FAO (Food and Agriculture Organization) (2001) Food balance sheets a handbook. http://www.fao.org/docrep/003/X9892E/X9892E00.htm#TopOfPage
FAO (Food and Agriculture Organization) (2011) Global food losses and food waste. http://www.fao.org/docrep/014/mb060e/mb060e00.pdf
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. http://www.fao.org/3/ai5248e.pdf
Farrior CE, Bohlman SA, Hubbell S et al (2016) Dominance of the suppressed: powerlaw size structure in tropical forests. Science 351:155–157
Flynn DFB, Mirotchnick N, Jain M et al (2011) Functional and phylogenetic diversity as predictors of biodiversityecosystemfunction relationships. Ecology 92:1573–1581
Funabashi M (2013) ITmediated development of sustainable agriculture systemstoward a datadriven citizen science. J Inf Technol Appl Educ 2(4):179–182
Funabashi M (2016a) Synecological farming: theoretical foundation on biodiversity responses of plant communities. Plant Biotechnol 33:213–234
Funabashi M (2016b) Synecoculture manual 2016 version (English version). Research and Education material of UniTwin UNESCO Complex Systems Digital Campus, elaboratory: Open Systems Exploration for Ecosystems Leveraging, No. 2
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
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. https://doi.org/10.5958/09761926.2017.00016.X
Funabashi M (2017b) Citizen science and topology of mind. Entropy 19(4). https://doi.org/10.3390/e19040181
Funabashi M (2017c) Open systems exploration: an example with ecosystems management. First Complex Systems Digital Campus World EConference, vol 2015, pp 223–243
Funabashi M, Hanappe P, Isozaki T et al (2017) Foundation of CSDC elaboratory: open systems exploration for ecosystems leveraging. First Complex Systems Digital Campus World EConference, vol 2015, pp 351–374
GLOBI (2017) https://www.globalbioticinteractions.org
Guimarães PR Jr, Pires MM, Jordano P et al (2017) Indirect effects drive coevolution in mutualistic networks. Nature 550:511–514
Hashiguchi Y (2005) Islands need “food selfsufficiency ability”. J Island Stud 2005(5):33–53
Houlton BZ, Morford SL, Dahlgren RA (2018) Convergent evidence for widespread rock nitrogen sources in Earth’s surface environment. Science 360:58–62
Jaenicke H, Ganry J, HoeschleZeledon I et al (eds) (2009) International symposium on underutilized plants for food security, nutrition, income and sustainable development. Arusha, Tanzania. ISBN 9789066057012
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
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
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
MeCab (2017) http://taku910.github.io/mecab/
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 GeoInformation 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. http://www.iirs.gov.in/iirs/sites/default/files/StudentThesis/chandan_final.pdf
NRC (National Research Council) (1993) Managing global genetic resources: agricultural crop issues and policies. The National Academies Press, Washington, DC. https://doi.org/10.17226/2116
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
Pecl GT, Araújo MB, Bell JD et al (2017) Biodiversity redistribution under climate change: impacts on ecosystems and human wellbeing. Science 355. https://doi.org/10.1126/science.aai9214
Pereira HM et al (2010) Scenarios for global biodiversity in the 21st century. Science 330:1496. https://doi.org/10.1126/science.1196624
Petherick A (2012) A note of caution. Nat Clim Change 2:144–145
Prusinkiewicz P, Lindenmayer A (2012) The algorithmic beauty of plants. Springer, ISBN 9781461384762
Putman RJ, Wratten SD (1984) Principles of ecology. University of California Press, California
R Core Team (2015) R: A language and environment for statistical computing. R Foundation for Statistical Computing, Vienna, Austria. http://www.Rproject.org/
Reich PB, Tilman D, Isbell F et al (2012) Impacts of biodiversity loss escalate through time as redundancy fades. Science 336:589–592
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
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
Rippke U, RamirezVillegas J. Jarvis A et al (2016) Timescales of transformational climate change adaptation in subSaharan African agriculture. Nat Clim Change 6:605–609
Rohde RA, Muller RA (2005) Cycles in fossil diversity. Nature 434:208–210
Scanlon TM, Caylor KK, Levin SA et al (2007) Positive feedbacks promote powerlaw clustering of Kalahari vegetation. Nature 449:209–212
Seuront L (2010) Fractals and multifractals in ecology and aquatic science. CRC Press. ISBN 9781138116399
Steffen W, Richardson K, Rockström J et al (2016) Planetary boundaries: guiding human development on a changing planet. Science 347:1259855
Takayasu H, Sato A, Takayasu M (1997) Stable infinite variance fluctuations in randomly amplified Langevin systems. Phys Rev Lett 79:966–969
TINA (2017) https://iscpif.fr/chavalarias/projects/tinasoft/
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, elaboratory: Open Systems Exploration for Ecosystems Leveraging, No. 5
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, elaboratory: Open Systems Exploration for Ecosystems Leveraging, No. 7
Turner GM (2008) A comparison of the limits to growth with 30 years of reality. Glob Environ Chang 18(3):397–411
UA (African Union) (2015) Lignes directrices pratiques de l’Union Africaine pour la mise en oeuvre coordonnée du Protocole de Nagoya en Afrique. http://www.absnitiative.info/fileadmin/media/Knowledge_Center/Pulications/African_Union_Guidelines/UA_Lignes_Directrices_Pratiques_Sur_APA__20150215.pdf
UN (United Nations) (2015) Sustainable development goals. https://sustainabledevelopment.un.org/sdgs
UN (United Nations) (2017) UN member states. https://www.un.org/depts/dhl/unms/whatisms.shtml
UNEP (United Nations Environment Programme) (2017) http://web.unep.org/regionalseas/whatwedo/conservationbiodiversityareasbeyondnationaljurisdictionbbnj
Whittaker RH (1960) Vegetation of the Siskiyou mountains, Oregon and California. Ecol Monogr 30:280–338
Wu H, Sun Y, Shi W et al (2013) Examining the satellitedetected urban land use spatial patterns using multidimensional fractal dimension indices. Remote Sens 5:5152–5172. https://doi.org/10.3390/rs5105152
Yong RN, Mulligan CN, Fukue M (2006) Geoenvironmental sustainability. CRC Press, United States
ZuppingerDingley D, Schmid B, Petermann JS et al (2014) Selection for niche differentiation in plant communities increases biodiversity effects. Nature 515:108–111
Acknowledgements
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: KaiYuan 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).
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Appendices
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 powerlaw 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
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
Then, the Lebesgue integral of the probability satisfying \( p\left( x \right) \ge \alpha \) is given by
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
The occurrence probability that is not less than \( \alpha \) when measured with \( P \) is given as

Mean Information
The mean information of all events of \( S \) is
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 \( \left( {S, \varvec{E}, P^{\prime}} \right) \):

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
The mean yield of vegetation with a yield above a certain value \( \beta \), such as \( Y \ge \beta \), is given as
If \( Y\left( S \right) \) is a skewed distribution with respect to \( S \), such as powerlaw 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
Example: For tagged ecosystem data, the occurrence probability of various combinations of tags can be calculated. In order to handle cooccurrence 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 \( A = \left\{ {s \in SO_{1} \left( s \right) \ge \beta } \right\} \) is given by

Calculation of Expected Value of an Objective Function by Nonuniform Probability Density Function
The expected value of the objective function \( O\left( s \right) \) with respect to the nonuniform probability density function \( P\left( s \right) \) on \( s \in S \) is given as
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 twodimensional map, as a model of vegetation succession:
where \( \varvec{T} \) is a nonsingular map. If we want to include other numerical data \( \varvec{R}^{\varvec{m}} \) such as environmental parameters, this extends to
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 realvalue (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 \( \varvec{T} \).
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 \( \delta \):
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 Selfsufficiency Measure with a Geometric Mean
The conventional definition of food selfsufficiency rate (SSR) is based on the arithmetic mean, whether it be caloriebased or productionbased (FAO 2001). As an application of the concept of selective information developed in Sect. 1.4, we simulate a novel food selfsufficiency rate related to the geometric mean.
The problem of arithmetic means is that it does not correctly reflect the notion of selfsufficiency 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 selfsufficient “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 selfsufficiency regarding the diversity of food products:
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 \( SSR_{k} \),
It transforms to the mean selective information \( I_{\text{s}} \) if we measure the selfsufficiency risk as a mean information cost for the search, such as
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 selfsufficiency 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 \( I^{\prime}_{\text{s}} \):
where
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 powerlaw vegetation. The algorithm is defined as follows:

1.
Define the parameters \( a \) and \( b \) of Pareto distribution (see Sect. 1.2).

2.
Sample \( k^{{max} } \) values from the Pareto distribution and define them as \( p_{k } \left( {k = 1, \ldots , k^{{max} } } \right) \).

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.
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.
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 selfsufficient crops approaches to \( k^{{max} } \), since all food products need to achieve selfsufficiency 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 selfsufficiency risk zero around \( \# \left\{ {{\text{SSR}}_{k} \ge 100\% } \right\} = 44\;{\text{to}}\;45 \). This critical value is invariant with the parameters of powerlaw vegetation \( a \) and \( b \), due to the logarithmic property of selective information and regularization process of the algorithm.
This simulation suggests that the selfsufficient 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 powerlaw 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 powerlaw 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 selfsufficiency 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 multifaceted 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 onetimeonly events such as natural disasters, in which we need to select a failsafe 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 \( k \):
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 onedimensional Cantor set^{Footnote 11} \( f_{c} \) as follows, which divides the surface \( C \) represented as a line segment into two isometric parts:
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
To avoid parameter dependency, we normalize the niche surface ratio of a species \( k \) by dividing \( X_{k} \) with \( C \) as
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).
Such situation is particularly familiar with rapidly expanding urban land use planning, where complex intersections with multifractal 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 smallscale local production of various food products in the burgeoning frontiers of urban development. To raise awareness and increase susceptibility to synecoculture, practices in smallscale 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 infield 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 huntinggathering. In either case, coping with smallscale diversified practices are strongly influenced by powerlaw 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.
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,
Then, the condition that the right side does not diverge to infinity is given by
By definition, \( a > 0 \), then for all \( a > 0 \), \( 1 /H\left( X \right) \) converges to a finite value as follows:
which gives
Rights and permissions
Copyright information
© 2019 Springer Nature Switzerland AG
About this chapter
Cite this chapter
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. https://doi.org/10.1007/9783319964546_1
Download citation
DOI: https://doi.org/10.1007/9783319964546_1
Published:
Publisher Name: Springer, Cham
Print ISBN: 9783319964539
Online ISBN: 9783319964546
eBook Packages: Biomedical and Life SciencesBiomedical and Life Sciences (R0)