全 文 ：BULLETIN OF BOTANICAL RESEARCH
第 16 卷　第 1 期 1996 年　1 月
Vol.16　No.1 Jan., 　1996
中国种子植物区系定量化研究-Ⅳ.
应用模糊数学进行植物区系区划①
左家哺　傅德志　廖仁彩　　田伟政
QUANTITATIVE FLORISTICS-VI.ON THE
FLORISTIC REGIONALIZATION BY THE
METHODOF FOZZY MATHEMATICS
Zuo Jia-fu　Fu De-zhi　Liao Ren-cai　Tian Wei-zheng
〔摘　要〕　1.本文分析了植物区系区划研究的现状 ,定性分析有时将同一地
区如台湾归为不同的植物区;或者应用数值(或数学)方法进行区系区划则所用信
息量较少而简单 。2.由于任一区系是植物界在长期的自然地理(特别是古地理)
条件下尤其在植物物种(或居群)遗传变异对立统一综合作用下的产物 ,它是既相
互联系又相互区别 、既连续又间断的辨证统一的有机整体。所以 ,我们能应用模糊
数学方法并借助于计算机来进行植物区系区划 。3.本文确定了用于区系区划的
7项数量指标 ,即区系丰富性 、古老性 、种系发育性 、成分复杂性及特有性 、植被类
型以及生态地理等。4.通过以梵净山 、“百里杜鹃”林区 、宽阔水自然保护区 、茂兰
喀期特林区 、雷公山 、云南干热河谷 、苗儿山及大别山等 8个地点为例进行了区系
区划 。结果表明 ,模糊数学方法不仅能够完成植物区系区划 ,而且还能清楚地表明
各区系单元的等级 ,区划结果与传统研究一致 。
关键词　定量区系学;植物区系区划;模糊数学
HISTORY AND PRESEN T SITUATION
The f loristic regionalization , which has the history over 170 years , is an old problem.
Schouw (1823), a Denmark botanist , published the monog raph of “Grundzoge Einen Allger-
meinen Pf lanengeog raphie” which f irst studied the f lo ristic regionalization of the wo rld in 1823.
Hereafter , it has developed very fast.The most excellent botanists dealing w ith this problem
① 左家哺 ,廖仁彩,田伟政:湖南 ,衡阳 ,湖南林业专科学校(Hunan Forest ry Technical College , Hengyang 421005 Hu-
nan)。傅德志:北京 ,中国科学院植物研究所标本馆　(Herbarium , Insti tute of Botany Academia Sinica , Beijing 100093)。
1995年 8月收到本文。
are Engler and Diels(1936), Hayek(1926), Diels(1929), Rikli(1934), Good(1974)Szafer
(1951), Turil(1959), Schmithosen(1961), Mattick(1964), Neil(1969),Walter and Streka
(1970), Takhtajan (1978), Ehrendorfer (1971)and so on.They basically thought that the
w orld flora can be divided into 6 Kingdoms which corespond to the actual situation , but the
principal disputation among them is the southern border of the Holarctic Kindom , the northern
border of the Holantarctic Kingdom , the range and numbers of the Reg ion and Domain(Good ,
1974;Szafer , 1951;Takhtajan ,1978).
Frist , Handel-Mezzett i , an Aust ria taxonomist , studied the floristic regionalization of
China in 1931.Hereaf ter , Shengou Liu(1944),Huiling Li(1944), Xianxiao Hu(1948), and
Zhenyi Wu(C.Y.Wu)(1979)discussed the f loristic regionalization of some parts of the w hole
of China , especially Prof.Zhenyi Wu(1979)sy stematically studied this problem based on the
results on the research of the w hole bo tanists.But , the principal contention among them is the
belonging of Taiw an flora(Takhtajan , 1978;Wu , 1979;Good , 1974).
Above all the botanists have basically made the f loristic regionalization by using the meth-
ods of the qualitative analysis of the induction and balance according to the data of the f lo ristic
characteristics , origination , developement , resemblance and endemicity , and the floristic con-
sisting of vegetation even ecological geog raphical envi ronment in the area studied(Zuo ,1990a).
Therefore , the classical method(Schouw , 1823;Eng ler , 1936;Szafer ,1951;Good , 1974;Wu ,
1979)has some limitat ions to a certain extent , that is to say , it cann t transform some highly
abst raction and complex info rmation on the florist ic transform some highly abstraction and com-
plex information on the floristic geography into the simple law in precise terms(Zuo ,1990a).
For the above-ment ioned reasons , Frankenberg(1978a ,1978b), a German bo tanist , put
forward the method of the integ rative planar diagram of the spect ra of the f loristic elements i.e.
distributional areas based on the uni ts of the areal square net w hich w as used to study the floris-
tic regionalization and phytogeographical dif ferentiation of Xizang(Tibet)by Du Zheng(1985),
Pingsheng Hsu and Bomao M iao(1988)took the dist ributional pat terns of the Chinese Vibur-
num as an example to study the floristic regionalization of China and divided the areog raphy of
the Chinese Viburnum into the eighty -eight quadrate areas , each taken as an Operational
Taxonomic Unit(OTU)and at tribtes of OTU if species existed or not , by the unw eighted pair
-g roup method using arthmatic averages (UPGMA).Sijun Hao , Ping shen Hsu and Bomao
M iao (1989)deal w ith a numerical study of the floristic affinities of seed plants of the Zhoushan
Archipelago by means of both cluster and principal component analy sis (PCA).The autho r
took the distributional patterns of Guizhou (Kweichow)Michelia as example to study the
floristic regionalization of the southern areas of Yangtze Rive(Zuo , 1988a);took the distribu-
tional patterns of the Chinese Keteleeria is as an example to study the floristic regionalization of
the southern Qing lin-Bashan mountainous range and divieded the areog raphy of the Chinese
Keteleeria into six Regions , and also discussed the floristic belonging of Taiw an and Hainan Is-
land(Zuo ,1989a).Though the above-mentioned learners have studied the florist ic regional-
ization by using mo re strict mathematical method and have go t some significant conclutions , the
1231 期　　　左家哺等:中国种子植物区系定量化研究-VI.应用模糊数学进行植物区系区划
common problem is that they use the information of the florist ic regionalization less and simple.
So , it is necessary to find out better method to study the floristic regionalizat ion.
BASIC IDEAS
Acoording to tw o aspects of the int rinsic features of plant species and the plant phytogenet-
ic principles judging f rom the theo ry of the plant evolution and origination , the f lora is the sum
of the w hole phy totaxonomic uni ts(such as species , genera or families and so fo rth)in some
(geological)period in a phystog raphical area(or administ rative region);it is the time and space
outcomes of the evolution and breeding , the developement and unceasingly dilation of the plan-
tae in the integ rative action of the long term natural geog rphical conditions(especially the paleo-
geographical conditions), especially the unity of opposites on the genetics and variation of the
plant species(or population);i t has gone throught the innumerable evolutionary processes f rom
the plant taxa being devoid , lit tle , simple and low -grade to possession , many , complex and
high-grade;at last , it has formed the organic ent ity(or system)of the dialectical unity w hich
are not only mutual cantace but also mutual differentiat ion and no t only consecution but also in-
consecutio in level(geograpical and vertical dist ribution of every phy totaxonomi units , f lo ristic
consti tution and combination and its historical development processes;it is a statical system also
a dynamical one(Zuo ,1991).Therefo re , which choria they belong to can all be a specious be-
cause among the choria , such as Kingdom or Reg ion or Domat ion and so forth , has not an um-
bridgeable gap especially w hen both the inlaid f lo ristic geographical and phsiog raphical condi-
tions appear.From what has been said above , we can use the fuzzy mathematical method to
study any flo ras to synthesize , induce , summarize and analyse automatically the impo rtant in-
formation of the f lora as many as possible with the help of computer and divide floristic area into
some choria (Go ,1985;Chen ,1984).
QUAN TITAT IVE DEGREES
It is the key to success or failure to determine quantitative deg ree of the f loristic regional-
ization by using the fuzzy mathematical method.Acco rding to the classical regionalization on
the flora (Szafer , 1951;Good , 1974;Takhta , 1978;Wu , 1979;Wu &Wang , 1983;Wu &
Zhang ,1979;Chen (ed.), 1980)and the quantitat ive description and treatment of the funda-
mental charcateristics on the f lora by the autho r (Zuo ,1993), the follow ing seven quantitative
deg ress are considered as the stat istic parameter of the fuzzy mathematical regionalizat ion on the
flora.
　　1.Degree of Floristic Abundance
The floristic abundance is a basic problem of i ts regionalization , We have considered the
numbers of the families , genera and species per square kilometre in some flo ra o r other as i ts de-
g rees of the f loristic regionalization.
　　2.Degree of Floristic ancient
The f loristic acientness is no t only a very impo rtant problem fo r it s regionalization but also
124 植　　物　　研　　究　　　　　　　　　　　　　　16 卷
a very knot ty one because its fossil data are lacking .
Gymnosperms in seed a plant are generally recognized the acient g roups , they have 13
familles now the follow ing:Cycadaceae , Ginkgoaceae , Araucariaceae , Pinaceae , Taxodiaceae ,
Cupressaceae , Podocarpaceae , Nageiaceae (Fu , 1992), Cephalotaxaceae , Taxaceae ,
Ephedraceae , Gnetaceae and Welwi tschiaceae.
Which family is the ancient g roups of angiosperm have many controversies over differing
opinions(Eng ler , 1964;Hutchinson , 1973).So ,we have considered 39 families of angiosperm
that are put forward by Smith(1967)as ancient g roup the following:Magnoliaceae , Illiciaceae ,
Schisandraceae ,Annonaceae ,Myristicaceae ,Aristolochiaceae ,Chloranthaceae , Calycanthaceae ,
Lauraceae , Hernandiaceae , Piperaceae , Saururaceae , Lardizabalaceae , Sargentodoxaceae ,
Menispernaceae , Ranunculaceae , Circaesteraceae , Berberidaceae , Papaveraceae , Fumariaceae ,
Nymphaceae , Ceratophy llaceae , Trochodendraceae , Tetracentraceae ,Eupteleaceae ,Cericidip-
hyllaceae , Eucommiaceae , Winteraceae , Degeneriaceae , Eupomatiaceae , Himantandraceae ,
Canellaceae ,Austrobaileyaceae , Ambo rellaceae , Trimeniaceae , Lacto ridacezae , Como rtegaceaeand
Gyrocarpaceae.
From what has been said above , the author takes the proportions that the existing of the
above-mentioned numbers of families and their numbers of genera and species included in some
flora or other occupy the total of this flora respectively as the degree of the floristic.
　　3.Degree of Floristic Growth of Speciation
Because the big or small ratio betw een the numbers of families and genea or genera and
species in some flora or o ther reflects the long o r sho rt histo rical process of the g row th of the
genera or speciation regarding family or genus as floristic unit to a certain ex tent (Zhang ,
1988).It is somewhat impo rtant significant to reflect the floristic grow th of speciation in i ts re-
gionalization.The deg ree of the f lo ristic grow th of speciation can be defined by the numbers of
the genera in unit family and numbers of species in unit genus in some f lo ra o r other i.e.the
following:
S i =qn-1
qn
(1)
　　4.Degree of Complex of Floristic Elements
The types of the flo ristic elements and the propo rt ion consisting of them are a principal
factor to w eigh up the floristic regionalization.It can be def ined by the propo rtions of the num-
bers of each area-type in some flora or o ther occupying the total in this f lora.
　　5.Degree of Endemicis of Floristic Elements
Generlly speaking the existing of the endemic elements and its numbers in some f lora o r
o ther are closely related w ith the evolutionary histo ry of paleogeog raphy , the climatic change
and ecological geographical facto rs in this flora(Stace , 1980;Zhang , 1988).So , the endemicity
1251 期　　　左家哺等:中国种子植物区系定量化研究-VI.应用模糊数学进行植物区系区划
of the florist ic elements , which and the floristic ancientness are same , is not only a very impo r-
tant problem for i ts regionalizat ion but also a very knot ty one .The author has considered the
numbers of the endemic genera of China in some flora or other occupying the to tal genera in this
flora as the degree of floristic endemici ty.
　　6.Degree of Vegetation-type
Because the vegetation which grows in an area is the integ retive ref lect ion of the histoical
and modern physiog raphy and the plant evolution and dif ferentiation (Hou , 1988;Walter ,
1979;Wu(ed.), 1980), the dominant vegetationtypes w hich grow s in an area fo rm an impo r-
tant basis of the reference for our study of the floristic regionalization.According to the Vegeta-
tion systems of classification that is set up by the Prof.Zhenyi Wu(Wu(ed.), 1980), if there is
a certain vegetation-type w e take it as 1 , o therwise as 0.
　　7.Degree of Ecological Geography
As the facto rs of the ecological geography reflect the development and the dist ribution of
the flora , so it has some significances to reflect the ecolog ical geography in the florist ic regional-
ization.Of those the states of the w ater and heat ref lect most(Zuo , 1989b ,1989c).The paper
takes the average annual rainfall(ri)and temperature (ti)in an area of the the factor of the eco-
logical geography , and their degrees calculate the following respectively :
RA i = r i∑k
i=1r i
(2)
TA i = ti∑k
i =1 t i
(3)
Notes:i=1 ,2 , …… ,K flora;RA i and show quantitative deg ree of average annual rainfall and
temperature in an area of the flo ra respectively .
METHOD AND STEP
　　1.Set Fuzzy Matrix
Given all the floras as the universe of discouse U , and all the parameters of the quantitative
target as the universe of discourse V.If Xibin the condition of vb on the universe of discouse U ,
so it forms the fuzzy matrix A of he floristic regionalizat ion i.e.the fo rmula (4):
A =〔X ib〕 kxn (4)
Notes:n show es the numbers of the w hole parameters of eight quanti tative targets.
which showes a fuzzy relat ionship from the universe of discourse of discouse U to V.
　　2.Set Similarity Relationship Matrix of Fuzzy
This paper uses the method of maxium value i.e.fo rmula(5)to set the similarity relation-
126 植　　物　　研　　究　　　　　　　　　　　　　　16 卷
ship matrix R of fuzzy i.e.formula(6).
r ij =
1:When i = j
∑n
b=1
min (X ib , X jb)/∑n
b=1
max(X ib , X jb);When i ≠ j s(5)
R =〔r ij〕 kxn (6)
　　3.Set Hartrix of Fuzzy Equivalent Relation
Generally speaking , the matrix of fuzzy similarity relationship only satisfies the ref lexivity
and symmetry but doesn t satisfy the t ransmission (Go , 1985;Chen , 1984;Zuo , 1989d;Li &
Wu ,1986;Lo et al., 1987).It can st ructure a matrix of fuzzy equivalent relation.i.e.R＊ to
use the method of transi tive closure i.e.t(R)below :
r ij = Vm
i=1(r ij ∧ r ij) (7)
When R2
h=R2h+1 , then t (R)=r2h and it satisfies transmission.
So , R2h=R＊ and it is a matrix of fuzzy equivalent relation for the f loristic
4 、Regionalizational Treatment
Through
r(λ)i j = 1;when 　r′i j ≥λ;
0;when 　　r′ij <λ (8)
as elements comvert R＊ ,we get a λ-cut matrix R(λis the confidence level)which show s the
general equivalent relat ion , what s more , 0≤λ≤r ij≤1 , also more simple and clear.
When w e take the value ofλeach time , some floras in the universe of discouse U can be
divided into one type or several types untill the minium value ofλ.Whenλ-values reduce to 0
from 1 , the division is the process w ill become a motional dendrog ram , According to the dif fer-
ence of theλ-increment i.e.■λand practicable condition of the phenomenon lines , that is to
say , we can determine the confidence level of the floristic regionalization , and it can be divided
into every cho ria.
EXAMPLE
This paper uses eight locali ties i.e.eight f lo ras(Fig.1),which are Fanjing shan Mountain
(Zhuo , 1990), “Hundred Li Rhododendron” Forest Area(Liu , 1985), Kuankuoshui Nature
Reserve(Zhuo , 1985), Maolan Krast Frast Area (Zhou , 1987), Leigongshan M ountain (Li ,
1985)and Dabieshan Mountain(Tao , 1983), as example to study the floristic reginaloization.
We calculate the following 18 parmeters reg ard as o riginal data per locality :1.Numbers of
families per square kilometre;2.Numbers of genera per square kilometre ;3.Numbers of
species per square kilometre;4.Proportion of Cosmopolitan;5.Proportion of tropoic genera;
1271 期　　　左家哺等:中国种子植物区系定量化研究-VI.应用模糊数学进行植物区系区划
6.Propo rtion of temperate genera;7.Proportion of Tethysian and Pan-mediterranea genera;
8 , Proportion of Endemic genera in China;9.Proportion of Ancient families;10.Proportion
of Ancient families including genera;11.Proportion of Ancient families including species;12.
Ration betw een numbers of families and genera;13.Ration between numbers of genera and
species;14.Existing Everg reen broad-leaved forest take as 1 , atherwise as 0;15.Existing
Everg reen and deciduous broad-leaved mixed fo rest take as 1 , otherw ise as 0;16 , Existing
Savannah take as 1 , o therwise as 0;17.Proportion of average annual rainfall occupies the to-
tal;18.Propo rt ion of average annual temperature occupies the total.Secondary , according to
the above-mentioned method , we can put the data into the computer and get the results of R
and R＊ as follow s:
F ig.1.The localities (or flo ras)of 8 areas studied (1)=Fanjingshan Mountain , 2=”Hundred Li Rhododen-
dron” Forest A rea , 3=Kuanduoshui Natrue Reserv e, 4=Mao lan Krast Forest Area , 5=Leig ong shan Mountain ,
6=Yuanmou Dry-ho t River valley , 7=Miaoershan Mountain , 8=Dabieshan Mountain)
R =
1.000 0.6103 0.7735 0.5021 0.7222 0.2826 0.5710 0.5555
1.0000 0.6122 0.4138 0.7066 0.2975 0.5897 0.5095
1.0000 0.5681 0.7162 0.3193 0.5243 0.6738
1.0000 0.6106 0.4134 0.6489 0.6785
1.0000 0.3558 0.6721 0.7439
1.0000 0.3310 0.4077
1.0000 0.5990
1.0000
es (9)
128 植　　物　　研　　究　　　　　　　　　　　　　　16 卷
R ＊ =
1.0000 0.7066 0.7735 0.6875 0.7222 0.4134 0.6721 0.7222
1.0000 0.7066 0.6785 0.7066 0.4134 0.6721 0.7066
1.0000 0.6785 0.7222 0.4134 0.6721 0.7222
1.0000 0.6785 0.4134 0.6721 0.6785
1.000 0.4134 0.6721 0.7439
1.0000 0.4134 0.4134
1.0000 0.5621
1.0000
d - (10)
When w e take the different value ofλf rom big to small one in R＊-matrix each time , i t w ill
become a motional dendrog ram as Figure 2.
F ig.2.The clur ter mo tional dendrog ram of the floristic regionalization by the fuzzy mathematics(No.localities
and Figure 1 are same)
RESULTS AND DISCUSSION
From above-mentioned motional dendrog ram (Figure 2), whenλ=0.4134 , eight locali-
ties can be divided into two types;one is Yuanmou Dry -hot River valley ;the other is Fan-
jingshan Mountain , “Hundred Li Rhododendron” Forest Area , Kuankuoshui Nature Reserve ,
naolan K rast Forest Area , Leigong shan M ountain , M iaoershan Mountain and Dabieshan
M ountain.It is just identical with the ranges and bounds of both Sino-Himalayan and Sino-
Japan Fo rest Subkingdoms (Wu , 1979;Wu &Wang , 1983).Because the increment of λ-
value between M aolan Drast Forest Area and Miaoershan Moutain is the smallest w hich equals
0.0064 w ithin the types too:one is M aolan Krast Forest Area and Miaoershan M outain;The
o ther is Fanjing shan M ountain , “ Hundred Li Rhododendron” Fo rest Area , Kuanduoshui Na-
1291 期　　　左家哺等:中国种子植物区系定量化研究-VI.应用模糊数学进行植物区系区划
ture Reserve , Leigongshan Mountain and Dabieshan Mountain.The former is just situated in
the range of Yunnan-Guizhou-Guangxi Region(Wu , 1979;Wu &Wang , 1983), the later
is just situated in the range of Central China Reg ion(Wu , 1979;Wu &Wang , 1983).
From what has been said above , the method of the fuzzy mathematies not only studies the
floristic regionalization but also shows clearly the grades every chorias.We think if the f lora has
a g reat number and it is continuous , then the method of the fuzzy mathematis will divide the
Kingdom , Subkingdom , Region and Domain more object ively .
ACKNOWLEDGMENTS
We are g reateful to thank Mis.Suhua Wu in Appendix M iddle School of Hengyang Teach-
er s College for resiving some English in this paper.
ABSTRACT
1.In the paper the histories and current situations of the floristic regionalization have been
reviewed.The general mothod is perceptual analysis , that the same flo ra such as Taiw an is be-
longed to the different Kingdom sometimes(Good , 1974;Takhtajan , 1978;Wu ,1979;Wu &
Wang , 1983)that the mathematical(or numerical)methods apply the inforamational capacities
w hich are very lit tle and simple.2.Because any flora is an o rganic ent ity of the dialectical uni-
ty w hich is the mutual contact and difference , mutual consecution and inconsecution , we can
use method of fuzzy mathenatics w ith the help of the computer to divide the floras.3.This pa-
per determines 7 quantitative degrees , which are the florist ic abundance , acientness and grow th
of speciat ion , the complex and endemicity of flo ristic elements , the vegetat ion-type and eco-
logical geography , for the f lo ristic regionalization by the method of fuzzy mathematics to study
the floristic regionalization 4.Through the study of Fanjingshan Mountain , “Hundred Li
Rhododendron” Fo rest Area , Kuankuoshui Nature Reserve , Maolan Krast Forest Area ,
Leigongshan M ountain , Yuanmou Dry-hot River Valley M iaoershan M ountain and Dabieshan
M ountain and so forth 8 floras fo r example , the results of the f lo ristic regionalization by the
method fo fuzzy mathematics show that this method not only studies the f loristic regionalization
but also show clearly the grades every chorias , that it and classical study are the same.
Key words　Quantitaltive floristics;Flo ristic regionalization;Fuzzy mathematics
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