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Acknowledgements
Yoji
AizawaTsuyoshi Chawanya
Osaka University)

Hidetoshi
NishimoriDavid
Sherrington
Yoshiyuki Kabashima(Tokyo Institute of Technology)

Tomoyuki Obuchi(Tokyo Institute of Technology)     

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K. Tokita. Aioi-yama, Nagoya. May 27, 2016.
Canon EOS Kiss DN, EF50mmF1.8 STM, ISO800/F2.5/WB:auto, SS60", 25:45PM-26:35PM.
53 pictures' composite.

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Fumigation in a tropical rain forest
E.O.Wilson, 1995,
The Diversity ofLife

E.O.Wilson, 1995, The Diversity of Life
Size 

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insects

anthropods
, beetles!

crustaceans

J.B.S.Haldane• 1892-1964• Physiology, genetics, evolutionary biology,
mathematics, statistics, biostatistics• One of the inventors of the word“clone”.
• When a priest asked him what
is a message which God wants tosend to us, he answered…

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R. May, ¦¬®§«¥ £eec/dDk2001

Dynamics of complex biological networks

Drug discovery•Dynamics of large-scale multispecies system and theircontrol.
•Relationships between stability, behavior and patterns
and parameters.
•Modern challenge of nonlinear and nonequilibrium
physics

Environmental conservation
Sustainability

Chaos by a system with only 3 species !

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Rank-abundance relations
x(n): population x of n-th ranked
speciesMotivationlog(abundance)

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[1] S.P. Hubbell, The Unified Neutral Theory of Biodiversity and Biogeography (Princeton
Univ. Press)log(abundance)

Major Question on SADrank• Environmental
parameters(temperature, latitude,
elevation, pollution,
stress…)• Species interactions(competition,
predation, mutualism,
parasitism, aliens…)

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thit{0}$$\dot{\mathrm{x}}t.\sim$$\llcorner$$\ovalbox{\tt\small REJECT} \mathrm{f}\mathrm{f}\mathrm{f}\ovalbox{\tt\small REJECT} p_{1}\mathrm{g}$$\text{る_{}0}$

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

NA

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10000010000

S: 単語種数

Frequency

Heaps

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1

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A: 総単語数（〜面積）

Rank

図 2: 書物の単語数に現れる Zipf の法則．上から，シェークスピア戯曲集，ダーウィン／種の起原第６版，
ミルトン／失楽園，ウェルズ／タイムマシーン，キャ
ロル／不思議の国のアリス．直線は α = 1 のフィッティング．電子書籍データは，Project Gutenberg[9]
による．

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図 3: 書物における，１ページ目からの単語数 A と
単語種数 S の間に現れる「種数面積関係」．上の直線は α = 1，下は α = 0.9 のフィッティング．

 , 1499, pp.1-6, 2006

The Zipf’s law in gene expression

Furusawa &Kaneko, 2003(a) Human liver(b) Kidney(c) Human colorectal cancer(d) Mouse embryonic stem cells(e) C. elegans(f) Yeast (S. cerevisiae)

MacArthur と Wilson が創始した「島の生物地理学 [1]」の概念は，生態学以外のより一般的な群集に適用可能である．島の面積とそこに生息する種数についての種数面積関係と個体数分布を結ぶ解析を通じて，一般的な群集において普遍的に観察される分布（Zipf の法則 [2]) の起源を示唆する結果が得られた．
単語のベキ分布を説明する古典的な Simon モデル [3] を拡張して，大陸から島への確率的移入のダイナミクスを考えることにより，分布の指数が移入率や死亡率などにより表されることなども示す．

合がある．
係は，個体数分布 p(x)(個体数 x をもつ種の割合) と関係がある．総個体数 N
例するという仮定のもとで，個体数分布から種数面積関係 S(A) を導くこと
えば，Preston は実際の観測データによくあてはまる個体数分布として対数
1 島の生物地理学 (1) 式の種数面積関係を導い
案し，経験的に成り立つ「正準仮説」を用いて，ダーウィンがガラパゴス諸島を訪れて以来，島は進化生態学研究の最も重要なプラット，対数正規分布以外にさまざまな分布を検討して，対応する種数面積関係をフォームであり続けてきた．島は比較的閉じた生態系の有り様を教えてくれるので，生物多様性を駆動するメカニズムに興味をもつ生態学者は，古くから島に生息する種の数が何方，我々は，これまで検討されなかったベキ型の個体数分布
p(x) ∝ x−(1+α)
によって決められているかを調べてきた．MacArthur と Wilson は，その不朽の名著「島で成り立つことを見いだし，正準仮説を仮定することなく(1) 式の種数面積の生物地理学の理論 [1]」の中で，さまざまな島のいろいろな生物について，島にすむ特定グループの種数 S が，その島の面積 A にることを示した [6]．さらに，最近我々は有限の種数
S をパラメータとする，
よってzS =αCA1のベキ指数 z とベキ型個体数分布の指数との間の関係

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0.8というベキ関数で表されることを指摘していS=102
る．ただし C, z は正定数である．
0.6
103
z
この関係は，島に限らず，ある生態系にお
γy
104
0.4
105
いて，調査面積と観測される種数の間にも成106り立つことが知られており，「種数面積関係」 0.2とよばれている．大陸など総種数が多い大き
0
0
0.5
1
1.5
2
な生態系においては，指数 z は比較的小さな䃇
値になり，島などのように比較的種数の少ない生態系では大きな値をとることが多いとさ 図 1: 種数面積関係のベキ指数 z と個体数分布の

γy

z =

dy
e −1
=
dµ
(1 + γ)e − 1y ≡ ln(S + 1),µ ≡ ln A,

(1)

γ≡

1−α
α

(2)

)[7]．種数 S が大きい極限では，ベキ指数 α の関係．

れる．

z=

!
α (for α < 1)

1

(3)

1 "(for α ≥ 1) , 1499, pp.1-6, 2006
Irie and Tokita, Int. J. Biomath. 5, 1260014 (9 pages) , 2012

Motivation

RAR

SAD

Species abundance distributions
P(x): how many species having
population x?

Molecular biological evidence
Left-skewed lognormal distribution of gene expression
Blake, et al, Nature 422 (2003)
4 days3 days4 days2 daysBlack: 1 day
elapse of theexperiment

Saccharomyces cerevisiae

Economical evidence
Car-species abundance distributions
Gaston, Blackburn & Lawton (1993)
Heathrow Airport

log(population of car species)YorkSAD (species abundance distributions)• Isao Motomurageometric series• Fisher(1943): logseries• Preston(1948): lognormal• Simon(1955): power law (Zipf’s law)
• MacArthur(1957): “Broken stick”=exponential• May(1975): SAD -> SAR(species-area relationships)• Hubbell(1979-): “Neutral model”=“zeo-sum multinomial”• Diserud & Engen (2000): Dynamic Model• Tokita(2004), Yoshino, Galla & Tokita(2008), Obuchi,
Kabashima & Tokita (2016): Random community modelin tropical rain forests

Preston plot

SAD of 687 invertebrates over 1970 Koreanriver sites.
Park, et al, 2007
Food webLeft-skewed
lognormal
Nee, Harvey & May, 1991

Korea Peninsula

http://www.wamis.go.kr/

N

128
km

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•

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Random Community Model
Modern theory: Global analysis

N⇥ (thermodynamic limit)cf.) productivity, maturity orrate(mutualistic/competition)

Nonequilibrium phase transition
May’stransition point(many metastable fixed points)RDTime delay

Correlation function

Colorednoise

Characteristic function

Generating functional (path integral)
A field theoretical technique for quantum field theory and othermany complex systems like spin glasses, neural networks.

Trajectories of N-species deterministic dynamics

Initial states

averaged over random parameters(interactions, resource consumption,
resource variability)
Attractor

One-species effective dynamics
with time delay and colored noise

Applicable to nonequilibrium systems with no Hamiltonianalgebraic eq.

Self-consistent equations for orderparameters

Opper & Diederich, 1992; Galla 2006; Yoshino, Galla & KT 2008

Perfect
coexistence
only for u → ∞Rank-abundance relations

productivity

lower Simpson index
u largeru largeru largerhigher Simpson index

zero-sum gamerandom pay-off

Random Community ModelwithRandom Sparse Interactions

Obuchi, Kabashima and Tokita, 2016

All-to-all interactions and sparse interactions
All-to-all interactions

Sparse interactions

Degree c = N − 1 ∼O(N ); N = 6

Degreec = 3 ≪ N ; N = 22

Statistical mechanics of a sparsely-interacted system has progressed
recently (Kabashima and Takahashi 2001, Mezard and Montanari 2009).
Obuchi (Titech)

Degree of Interactions

Aug. 29, 2015

6 / 20Results: Phase diagram
The area of perfect coexistence is emerged for finite value of uc incontrasted to uc = ∞ for RRD w/all-to-all Jij .

uc vs. ∆uc vs. c

Degree c = 3

No mutualism bias ∆ = 0

Dashed curves: 2nd order perturbation expansion up to u−2 .
Symbols: Non-perturvative solution by the cavity method.
Both parameters are the unstability factor except for the singularpoint ∆ = −1(1), i.e., every Jij = −1(1), respectively, i.e.,completely neutral case.
Obuchi (Titech)

Degree of Interactions

Aug. 29, 201514 / 20

Results: SAD: Direct numerical simulation
SAD for RRD on Square Lattice

SAD for RRD on Random Graph

P!S"
4

P!S"
3.5
3.0

3

2.5
2.0

2

1.5
1.0

1

0.5
0.6

0.8

1.0

1.2

1.4

Degree c = 4, u = 10, ∆ = 0,
N = 128 × 128 = 16384.
Jij = ±1,

Obuchi (Titech)

1.6

S

0.6

0.8

1.0

1.2

1.4

S

Degree c = 3, u = 7, ∆ = 0,
N = 16000,
P (Jij ) = N (0, 0.1)(Gaussian).

Degree of Interactions

Aug. 29, 201517 / 20

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