全 文 ：BULLETIN OF BOTANICAL RESEARCH
第 16 卷　第 1 期 1996 年　1 月
Vol.16　No.1 Jan., 　1996
生物多样性的熵值结构①
欧乐其　何小双
THE ENTROPY STRUCTURE OF BIODIVERSITY
Orlóci lászló　He Xiao-shuang
〔摘　要〕] 　多样性一般定义为生物进化过程中物种表现型的丰富程度。从
广义上讲 ,生物进化与环境归类作用的结构复杂性对生物为多样性的解释是很有
益处的。我们认为区域水平与样地水平上的气候归类型与植被归类型是最基本的
划分因子 ,本文以中国东北部黑龙江省的山地植被为例 ,采用多等级熵值分配的方
法来证明所使用的方法。我们得出气候归类型与植被归类型所占有的熵值成分要
弱于物种表现型丰富度所占有的熵值成份。值得注意的是其相对的熵值贡献率对
每一个因子的最大熵值的变化规律 。分析结果表明气候归类型占有较低的熵值而
植被归类型占有较高的熵值。因此仅仅以在区域气候型为主要指标所划分的物种
生态位是不够的 ,而以小区域内的植被型为指标所产生的物种生态位具有较高利
用价值。
关键词　多样性;熵;气候归类型;植被归类型
Introduction
The ultimate challenge for contemporary science is to find w ay s to uphold global environ-
mental quality and to maintain biodiversity , The urgency to respond to this challenge is clear ,
considering that if no thing is done by the year 2050 as many as 2.2 million species could be lost
(Lean and Hinrichsen 1992)as a direct consequence of the deleterious industrial practices and
life-sty les of modern society .A corollary to this is the point that only about 1.4 million
species have so far been identified of a much higher number , and probably much fewer have ac-
tually been subjected to detailed study.
Quantification of biodiversity components under types of environmental sorting is integ ral
① 本文作者单位:Orlóci Lá szló is Visting Professor at the Open Research Laboratory , N.F.U., Harbin , China , and
Professor of Quanti tat ive Ecology at U.W.O.Present address for both authors:Department of Plan t S ciences , the University
of Western On tario , London , Canada N6A 5B7.He X.S.is a mem eber of faculty in the Botanical Inst itute , Northeast Forest
University and present ly a Graduate S tuden t at the Universi ty of Western .
1995年 8月收到本文。
of the evological approach.The know ledge gained from this has implications for prediction of
response to perturbation.Response could be potent in the sense that w hen a species becomes
ex tinct locally or g lobally , more is lost than just a population of o rganisms.The loss can trigger
chain reactions w ith effects w hich-as w e know from chaos theory , particularly f rom the w orks
of E.Lorenz w ith weather sy stems(Lorenz 1963)and R.May wi th ecosy stems(May 1976)
and because of the sensitive dependence of complex sy stems on even small perturbation-can be
catastrophic in their long-term consequences.
It is usually mentioned in discussions of the topic that entropy expresses disorder.In actual
fact the relationship is such that the entropy is low when disorder is low and high when disorder
is high in a system.In other w ords , increasing entropy is and ally of the march towards stabili-
ty in natural systems , in the direction of which the bio ta is pointed by evolut ion and environ-
mental so rting.By the same token , increasing entropy is also an ally of increasing unpre-
dictability .In this lies the paradox that exact predictability is not an inherent trait of biodiversi-
ty in highly evolved systems.All-in-all , as Masgalef (1958 , 1989)suggests , entropy -
based analysis captures the very essence of the bioenvironmental process.
Sources of biodiversity
Under the conditions of an ecological inquiry , it is logical to start contemplation of biodi-
versity with the sources that generate it.On the regional level.We identify three natural
sources:evolution , climate type sorting , and climax type sorting :
Evolution has produced fo rms of adaptation , the pheno types- such as for instance the life
-form of plants , their grow th form , or structural type(Raunkiaer 1934 , Box 1981)- that
enable indivduals to survive the unfavourable season.Evolution is a continuous process , but the
present-day conditions are responsible only for the phenotypes of the future.What w e see to-
day in phenotype richmess evolved in the past under conditions that now may not even exist.In
this sense , the sorting sources , climate type and climax type , wo rk upon a medium for w hose
evolution they are not responsible.In other w ords , phenotype richness is independent f rom
present sorting by climate type and climax type.
Climate types sort the phenotypes into well def inde units that Merriam (1898)called ”life
-zones” and Clements (1928)referred to as ” fo rmations” .Life -zones or formations are
biobeoclimatic enti ties , having their unique climax types.Formations shift through latitudes
and elevations in response to the changing climate.Delcourt and Delcourt(1987)use fossil evi-
dence to describe histo ric formation shift s.Their records clearly show a close harmony of fo r-
mation dynamics and climate change.The lat ter w ere t riggered by the M ilankovitch (1947)
precession dynamics , fluctuating solar radiat ion , and peaks in atmospheric CO2 contents(Er-
ickson 1990).
Climax types are target states of vegetation succession w ithin a format ion.The two types
at t ract populations adapted to ” zonal” conditions (climatic climax)or ” azonal ” conditions
(edaphic climax).When nei ther is the case , the populat ions play a ” serial” role in succession.
1471 期　　　　　　　　　　　　　欧乐其等:生物多样性的熵值结构
Since the climax type is dependent on the climate type , succession is re-aimed at a new target
each time the climate changes substantially .We refer to Whit taker (1951)and M cIntosh
(1980)for review of the succession literature.
Diversity and its measurement
1.General concepts
The Sorting types are hierarchical in the sense that phenotypic diversity is basic , added to
w hich is diversity generated by climate type so rting and by climax type sorting.The diversity
measure must be able to handle this hierarchical nest ing by providing a wo rking definition fo r
the conditional diversity term.The family of measures known as Renyi s (1961)generalized
entropy and generalized informat ion are ideal fo r the purpose.Applications in a similar contex t
as ours and algo rithms are presented in Orlóci(1991).
2.Generalized entropy and information
M any of the ecological diversi ty indices are f rom this family .Fo r example , the familiar
Shannon index(Shannon 1948 , Margalef 1958)is a special case(order one)of generalized en-
tropy.The Shannon index is a point on the generalized entropy curve in the vicinity of an in-
f initesimal break where the value of the order variable (α)is exactly one.on one side of this
point lies the log Simpson index(α=2), and on the o ther , the log species richness index (α=
0).It is atα=0 that ent ropy at tains maximum value.One will do w ell to remember that struc-
tural discrimination at low order ent ropy is poor , being worst w henα=0.In a like manner , as-
sociation and interaction informat ion have different orders(Renyi 1961)and dif ferent pow ers of
resolution.The recognition of this should w arn of caution w hen one applies Kullback s I-di-
vergence information statistic(Kullback 1968).
We consider an s-valued frequency vector (f 1 f 2…fs)w ith sum n.Entropy of order αis
defined for this by the function
Hα= 11-αln ∑
s
j = 1
Pαj
where P j = f jn .While Hαis a measure of diversity , Reny i s generalized information Iα, de-
f ined by
Iα= 1α-1 ln ∑
s
j = 1
qj
α
P j
α-1
is a measure of interact ion , mutuali ty , or association depending on context.Iαis the amount of
information divergence between tw o dist ributions , P =(P1 P2…Ps)and P=(q1 q2…qs).Orlöci
(1991)discusses restrictions to be observed.
Special cases
The exact form of the diversity o r informat ion function will depend on the problem con-
148 植　　物　　研　　究　　　　　　　　　　　　　　　　16 卷
text.We def ine a hierarchical partition on entropy of order one.
1.Contribution by evolution
Biological evolution is a series of t ransformations of the genetic composi tion of populations ,
based principally on changing interactions wi th the environment and guided by natural selec-
tion.There are three main kinds of interrelated evolutionary changes:t ransfo rmation of species
in separate lines of (divergent)evolution , thereby enrichment of biota by new fo rms , and dis-
placement and replacement of forms(convergence).natural selection af fects the phenotype , so
the phenotype best adapted to the physical and biological conditions of i ts surrounding s have the
highest chance to survive and reproduce.Since phenotype richness is the product of evolution ,
the quantity
He = log2 n(bi ts)
is a measures of evolution s contribution to entropy in biodiversity.Symbol n desgnates the
number of pheno types(at the species level in our case).
2.Contribution by climate type sorting
The amount of ent ropy from this particular source is
Hc =-∑Sc
j = 1
f j.
n
log2
f j .
n
bi ts
Symbol fj.signif ies the number of phenotypes found under climate type j.For Hc to be maxi-
mal , the climate types should have equal representations by phenotypes.The evenness of this
representation is
0 Hclog 2 Sc 1
We no te that climate type sorting acts upon phenotypes w hich are products f rom the past fo r
w hich the present climate is not responsible.Therefo re a conditional term (Hc/e need not be
considered.
3.Contribution by climax type sorting
This is the outcome of succession dynamics.The entropy generated by i t is
Hs =-∑Ss
k = 1
log2
f .k
n bits
Hs is maximal w hen the climax types have equal representations by phenotypes.The entropy
generated by climax type sorting within the climate types is
Hs/ c =-∑Sc
j = 1∑
S
s
k = 1
log2
f ik
f i.bits
This is maximal w hen climax types have equal representation by phenotypes w ithin the climate
types.The joint ent ropy in the 2-way sorting , assuming a single multistate cri terion in each
so rt ing type , is
Hc , s =-∑Sc
j = 1∑
Ss
j = 1
f jk
n
log2
f jk
n
=Hc +Hs/c =Hc +Hc -I(c , s)bits
In this
1491 期　　　　　　　　　　　　　欧乐其等:生物多样性的熵值结构
I(c;s)=∑Sc
j = 1∑
Ss
j = 1
f jk log2
f jkn
f j.f .kbi ts
which is informat ion of o rder one.The evenness of phenotype representation in the climax
types given the climate types is measurable by the f raction.
0 Hs/ c
log2
S
s
1
The Heilongjiang vegetation landscape
The geog raphic region under consideration covers a good po rt ion of the mountainous ter-
rain and uplands , about 450 , 000 square kilometers in total , in northeastern China.The no rth-
ern and eastern boundaries of the region coincide w ith the rivers Heilong jiang and Wusuli.To
the west , it is contiguous w ith the Greater Xingan Range and to the south with Jilin province.
The latitude-longi tude reading , 43°25′—53°33′N and 121°11′—135°05′E give rough geo-
g raphic delineation.
The climate of the region is determined by the East Asian monsoon with alternating domi-
nance of tw o main ai rmass , the cold , high pressure Siberian(Winter monsoon), and the w arn
low pressure Pacific(Summer monsoon).As a result , the summers are w arm humid , and the
w inters are cold dry .The mean annual precipitation in the region is about 500 mm of w hich
about 70%is concentrated in the summer.The north has accumulated heat(average day tem-
peratures exceeding 10℃)of about 1 ,300°to 1 ,600°, and the southern 1 ,600°to 2 ,900°.
The orographic characterist ics of the land surface and the climate create pattern in the veg-
etation.This is highlighted by species reponse to heat and moisture w ith def inite regulari ty.
Four climate-def ined vegetation fo rmations have been described from the region by Li(1993):
1.The Subalpine forest formation is restricted to elevations mainly above 1 ,150 m.The
mean annual preciptation is substantially higher (550mm , humidity index 0.80-1.20), and
the mean annual temperature substantially lower (-3°to -8℃)than the regional averages.
The leading plant species include Pinus pumila , Betula exil is , Empertrum nigrum var.
japonicum , Arcyous alpina , Betula ermanii , Artemisia lagocephala , Sabina davurica ,
Vaccinium uliginosum , Vaccinicum v it is-idaea , Dryopteris fragrans.
2.The Boreal formation , an extension of the northern humid cold-temperate coniferous
forests of Siberia into the region , forms a narrow belt w ithin roughly 49°40′—53°33′N in the
elevation belt of 450-1 ,150 m.The mean annual temperature varies f rom -2°to -6℃.The
annual mean precipitation is about 420 mm (humidity index 0.7-1.10).The Spring and fall
seasons are arid and w indy , the w inter is dry and ex t remely cold.
Permafrost in the soils is w ide-spread.The leading species include Larix gmelini , Betu-
la ermanii , Betual platyphyla , Pinus pumila , Pinus sylvestris var .mongolica , Vac-
cinicum vit is-idaea , Ledum palustre var.angustum , Cornus alba , Pyrola incarnata ,
Maianthemum bi folium , Pleuroziopsis rutheniea , Hylocomium splendens , Ptil ium orita -
150 植　　物　　研　　究　　　　　　　　　　　　　　　　16 卷
oastrense.
3.The M ixed Conifer Hardw ood formation humid fo rest w ithin the geographic limits
roughly delineated by 43°23′—49°40′N below 450 m.Here the mean annual temperature is -
2°to 4℃ and the mean annual precipitation equals the average for the region.Typeical species
include Pinus Koraiensis , Quecus mongolica , Tilia amurensis , Fran xinus mandshurica ,
Acer mono , Populus ussuriensis , Phylodendron amurensis , Juglans mandshurica , Carpinus
cordatra , Ulmus propoingua , Vitis amurensis , Panax ginseng , Aralia elata , Sm ilacina
tri folia , Viola verecunda , Climacium dendroides.
4.The Forest Steppe formation is cont iguous with the previous formation under a semi-
areid climate.The mean annual precipitation is about 350 mm , the humidity index 0.4—0.8 ,
and the mean annual temperature 0°to 5℃.The leading species include Quercus mongolica ,
Populus davidiana , Betula platyphyl la , Ulmus japonica , Fili fol ium sibiricum , Cleisto-
genes suqarrosa , Aneurolepidium chinesis , Carex duriuscula , Lespedeza hedysaroides , St ipa
grandis , St ipa baicalensis , Artem isia gmelini i.
Diversity partition
We use 646 species (160 trees and shrubs , 485 herbs)from Li s list.The species are
scored acco rding to climate type and climax type.The climate type score identifies optimal fo r-
mation(Subalpine , Bo real , Mixed Conifer Hardw ood , Forest Steppe)and the climax type
score describes successional status whether climatic climax , edaphic climax , or serial.The
scores are summarized in a 4x3 contingency table with rows for climate types and columns fo r
the climax types.An element in the table is the count of species common to the respective cli-
mate type and climax type category .
Table 1　presents some results.It is st ructured by the dotted lines into blocks.The first
block contains a single number , the entropy level in species richness.In the second block , the
total ent ropy att ributed to climate type sorting and climax type so rt ing is parti tioned hierarchi-
cally .We use conditional ent ropy Hs/c to establish addit ivity.In these terms , the total ent ropy
in the 646 species collection is 11.71 bits of w hich 79%is att ributed to phenotype richness and
21% to climate type and climax type sorting .The results are g rouped as observed and expect-
ed.“Observed” refers to di rect analysis of the data and “ expected” to the hypothetical state of
so rt ing w ith all types equally favoured.The “expected” entropy , log24 o r log23 , is thus the
maximum value at tainable under the sorting schemes.The observed value divide by the expect-
ed is “evenness” , an expression of observed entropy in comparable terms.
Table 1.Entropy partitions of total diversity according to definit ions in the tex t.The data
is a list of 646 species f rom Li(1993)scored fo r climate type and climax type.The expected
value corresponds to a state of sorting w here all states are equally favoured.Legend to symbols:
He-entropy ow ing to evolut ion;Hc-Entropy generated by climate type sorting ;Hslc-en-
tropy generated by climax type sorting , given the climate types;Hc , s -joint ent ropy (Hc+
Hs/c);Gt.- total ent ropy of biodiversity (He+Hc , s).All entropy values are in bits.Evenness
1511 期　　　　　　　　　　　　　欧乐其等:生物多样性的熵值结构
is unit less.
Table 1.　Entropy partitons of total diversity according to definitions in the text.The data is
a l ist of 646 species from Li(1993)scored for climate type and climax type.The
expected value corresponds to a state of sorting where all st ates are equally
favoured.Legend to symbols:He-entropy owing to evolution;Hc-entropy gen-
erated by climate type sorting;Hslc-entropy generated by climax type sorting ,
given the climate types;Hc , s -joint entropy(Hc+Hslc);Gt.-total entropy of
biodiversity(He+Hc , s).All entropy values are in bits.Evenness is unitless.
Observ ed Expected G lobal
entropy entropy evenness
He 9.34
Hc 1.19 2.00 0.60
H slc 1.28 1.58 0.81
Hc , s 2.47 3.58 0.69
G t 11.71
%He 79
%Hc , s 21
Inferences
One aspect to be observed is the low joint entropy contribut ion(2.47 bi ts)of climate type
and climax type so rting to the total(11.71 bits).In this respect , species richness as an en-
tropy source far overrides in importance the other sources in natural vegetation.Relative impo r-
tance should how ever be considered in the context of the maximum possible contribution(3.58
bits)which the sorting sources can provide jointly.It is important to realize that climate type
so rt ing and climax type sorting af fect community structure and org anization , and because of
this , the level of evenness achieved by the dif ferent so rt ing types is consequential.In the exam-
ple , climate type sorting is much less even(0.60)than climax type sorting (0.81).This sug-
gests insuff icient richness of species in the f lora fo r a more complete utilization of the available
climatic niches , but suf ficient richness to provide high utilization of the climax type niches.
The low evenness in climate type so rting points up an impo rtant imbalance w hich may con-
tribute to region-wide instability and an undesired vegetation response under adverse ef fects ,
such as climate warming.①
Acknowledgment　We thank M.Anand for comments , and the Nat ional Research Council of
Canada and Northeast Forest ry Universi ty for suppo rt.
152 植　　物　　研　　究　　　　　　　　　　　　　　　　16 卷
① 1The thermal rise that w e estimated for the region follow ing the technique described in Orló ci(1994)is about 7°to 9℃
in the annual everage tem peratu re , corresponding to the M anabe 2x CO2 scenario or 2.5℃ rise in the global average temperature
in about 6 more decades.
ABSTRACT
The no tion of biodiversi ty conno tes in common usage phenotype richness produced by the
process of evolution.We believe it is useful for specific purposes to think of biodiversity in
broader terms , having in point of fact the implication of st ructural complexities for w hich evo-
lution and environmental sorting of the phenotypes are joint ly responsible.We consider these on
the regional and site levels w here climate types and climax types are the primary sorting agents.
In o rder to quantify the components contributed by evolution and by the so rting types , we sug-
gest hierarchical ent ropy partit ions.We use data f rom the upland and montane vegetations of
Heilong jiang province in northeast China to illustrate the method.The results are int riguing.
For example.the entropy components at tributed to climate type sorting and climax type sorting
are dw arfed by the level of entropy in pheno type richness.While this is expected , the relative
contributions in relation to their respective maximum values are surprising.The entropy com-
ponent contributed by climate type sorting is low ;the component contributed by climax type
so rt ing is high.The lat ter indicates a highly even utilization of the climax type niches , while
the lat ter points to insuff icient richness to allow a more even utilization of the climate niches in
the region.
Key words　Biodiversi ty;Entropy partition;Climate type;Climax type
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《植 物 研 究》 编 辑 委 员 会
名誉主编　周 以 良 (Chou Yiliang)
主　　编　聂 绍 荃 (Nie Shaoquan)
副 主 编　祖 元 刚 (Zu Yuangang)
编　　委
(以姓氏笔划为序)
　　王忠笑 (Wang Zhongxiao)
艾达尔 (Kent T.Adair)
李书馨 (Li Shuxin)
李锡文 (Li Xiwen)
陈心启 (Chen Xinqi)
邱　虹 (Qiu Hong)
杨国亭 (Yang Guoting)
赵奇僧 (Zhao Qiseng)
董世林 (Dong Shilin)
火树华 (Huo Shuhua)
付立国 (Fu Liguo)
李普斯克姆 (Barney L.Lipscomb)＊
刘鸣远 (Liu Mingyuan)
陈守良 (Chen Shouliang)
杨昌友 (Yang Changyou)
林有润 (Lin Youren)
黄普华 (Huang Puhua)
鲍弗德 (David E.Boufford)＊
　　＊美 (U.S.A)
154 植　　物　　研　　究　　　　　　　　　　　　　　　　16 卷