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

Masatoshi Funabashi

doi:10.1007/978-3-319-96454-6_1

Genetic Diversity in Horticultural Plants pp 3-46 | Cite as

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

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Masatoshi Funabashi

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First Online: 18 October 2019

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

Abstract

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.

Keywords

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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Notes

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: 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).

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}} $:

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

where

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

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.

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 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.

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 set¹¹ $ 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).

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
**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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Masatoshi Funabashi
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1.Sony Computer Science Laboratories, Inc.TokyoJapan

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

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