(Upload on October 18 2021) [ 日本語 | English ]

Diversity (多様性)

Mount Usu / Sarobetsu post-mined peatland
From left: Crater basin in 1986 and 2006. Cottongrass / Daylily

[diversity index, species-area curve, biodiversity (生物多様性)]

Diversity: is evaluated by:
evenness, heterogeneity (homogeneity), richness, stability or complexity

Species richness

Number of species in a given area, such as community and plot
Relative abundance of species richness

Tropical → large number many → high species richness
Boreal → small number → low species richness

Diversity (Shmida & Wilson 1985)

How the number of species is determined in a given commmunity or plot

Evenness or equitability
Spatial distribution
→ diversity parameter includes one or more of these three factors

Functional diversity
the elements of biodiversity that influence how ecosystems function
Persistence (永続性)
species composition does not change through time in the absence of stress and disturbance
Resistance (耐性)
no change with "shoves" - stress/disturbance
Community Resistance, R (Frank & McNaughton 1991)

Rj = 1 - Σi=1mpij/2)

pij: relative dominance of species ith in community j
The species showing the temporal changes in dominance are significant (p < 0.05) are used for the calculation

Resilience (復元性, 弾力性)
the capacity of an ecosystem to recover from stress and disturbance = stability (s.s.)

Stability within bounds = no recognizable major changes in vegetation community over time resilience
Fig. Stability of ecosystem is a concept related to resilience. This figure shows the system is resistant to change over time. Ex. Boreal conifer forests self-replace within 50-80 years after wildfire; hence, the forests are highly resilient but not especially resistant to fire.
Ex. Tropical rainforests are resilient, stable gap-dynamics forests. The forests undergo gap dynamics spatio-temporally, but the characteristic species remain the same and so these forests exhibit long-term resilience and resistance to natural change.

⇒ most primary ecosystems are resistant and resilient to natural disturbances

loss of biodiversity may alter the ecosystem resilience
loss of resilience means increased uncertainty about future ecosystem condition
Hypothesis: biodiversity↑ = resilience↑?

Thresholds: the resilience capacity is overcome and the system moves to a new state

Ex. effects of tephra on wetland vegetation (Hotes et al. 2010)

Variability (変動性) ⇔ stability (安定性)
fluctuating scale expressed by the deviation of density, etc. → small = stable

Hypotheses on the changes of species diversity

S = f(+ habitat diversity, -(+) disturbance, + area, + age,

+ matrix heterogeneity, - isolation, - boundary discreteness)

S = number of species

      Manage for persistence                         Manage for change
Ecosystems are still recognizable                            Ecosystems have fundamentally
as being the same system (character)                           change to something differnt
          Resistance                    Resilience                     Transition
Fig. 1. Conceptual diagram of resistance, resilience and transition (Swanston et al. 2016; Nagel et al. 2017; Millar et al. 2007)
Lottery model (富くじモデル)
– no competition
Intermediate disturbance hypothesis, IDH (中規模撹乱仮説)

Fig. 1. The grazing optimization hypothesis (modified from IDH). Curve shows the change in production due to grazing based on data in Dyer (1975) and McNaughton (1979). (Hilbert et al. 1981)

Diversity-stability hypothesis

Hypothesis for the functional role of species diversity in ecosystem (Johnson, et al. 1996).
Diversity-stability hypothesis: ecological communities increase in energetic efficiency (productivity), and in ability to recover from disturbance, as the number of species in the system increases (MacArthur 1955, Elton 1958). In its original form (MacArthur 1995), this hypothesis did not include assertions of linearity in the effect of species richness on ecosystem function. The following hypotheses are alternatives to this hypothesis (Kareiva 1994, Tilman & Downing 1994, Collins 1995, Baskin 1995, Walker 1995).
Rivet hypothesis: likening species in an ecosystem to rivets holding an airplane together the removal of rivets beyond some threshold number may cause the airplane, or the ecosystem, to suddenly and catastrophically collapse (Ehrlich & Ehrlich 1981). Explicit in their presentation is the appreciation that a few extinctions may go unnoticed in terms of system performance because some species are redundant, generating a nonlinear relationship between species richness and ecosystem function.
Redundancy hypothesis: certain species have some ability to expand their 'jobs' in ecosystems to compensate for neighbor species that go extinct, sensu Ehrlich & Ehrlich (1981), (Walker 1992). Species may be segregated into functional groups; those species within the same functional group are predicted to be more expendable in terms of ecosystem function relative to one another than species without functional analogs.
Idiosyncratic hypothesis: the possibility of a null or indeterminate relationship between species composition and ecosystem function (Lawton 1994). This relationship is expected, for example, in communities featuring higher-order interactions.

Time theory and Productivity hypothesis

Table. Diversity/ecosystem function studies (Johnson et al. 1996)
EcosystemDisturbance typeDiversity measureProductivity measureStability measureDiversity/ productivity relationshipDiversity/ stability relationship
California annual grassland1Annual variationSaSCa (aboveground)SC(NPP)-1plants (-)plants (0)
New York old fields1Annual variation; N-P-K fertilizerS/log2 N;S and H'NPPa (aboveground, belowground)ΔNPPplants (-)
herbivores (-)
carnivores (-) (0)
plants (+)
herbivores (-)
carnivores (-)
Serengeti2Δprecipitation; grazingH'NPP (aboveground)ΔSCNAaplants (+)
Yellowstone grasslands3drought; grazingH'SC (aboveground)Δrelative abundancesNAplants (+)
British grasslands4, 5fertilization; mowing; ΔprecipitationSSC (aboveground)Δvegetation; composition4 ΔSC4, 5plants (-)plants (-)4, (-)4, 5
Minnesota grasslands6, 7drought6, ΔS7SSC (aboveground)DSC; recovery rateb, NA7plants (-)6, (+)7plants (+)6, NA7
Costa Rican tropical forest8annual variationSsoil fertilitycΔsoil fertilityNAplants (+)
Ecotron9SSNPPNAplants (+)NA
a. S, species richness; SC, standing crop; NPP, net primary production; N, total community biomass; Δ, change in parameter, H', Shannon index; NA, not assessed; +, positive association; -, negative association; 0, no relationship.
b. Resistance was measured as ln(Δb/BΔt) and resilience was measured as ln(b/B), where b is biomass at time t after disturbance and B is biomass before disturbance.
c. Components of soil fertility included: available nitrogen (N), phosphorous (P) and potassium (K) content; and pH and cation exchange capacity.

Theory of spatial heterogeneity vs competition hypothesis / predation hypothesis


Diverstiy index (多様性指数)

Table. Types of community diversity index (Shmida & Wilson 1985)
  • Type: Definition (Index), Scale
  • α-diversity: the species richness of samples representing communities (H', C), 102-104 m2
  • β-diversity: The amount of biotic change among units. The differentiation of units on the α-diversity scale (Similarity index)
  • γ-diversity: The diversity of landscapes (Same with α-diversity), 106-108 m2
+ Genetic diversity, that refers to variation in heritable characteristics of a species. This diversity has three levels:

genetic variation in a single individual
genetic differences among individuals in a population
genetic differences among populations

Genetic diversity is related to minimum viable population size.
Table 1. Characteristics of three approaches to the study of diversity (Shmida & Wilson 1985)
                     Classic       Phytosociology  Theoretical
                     biogeography                  population

  Scale of           Regional      Among-          Within-
    observation                    communities     communities
  Type of diversity  γ            β              α
  Primary proposed   Historical    Environmental   Species
    determinants                                   interactive
  Role of noise      Irrelevant    Important       Unobserved

α-diversity (α多様性)

= within-habitat diversity or habitat diversity
The species, population response to habitat variables within this hypervolume, as expressed in a population measure, describes its habitat. = The species richness of samples representing communities

Size: 102-104 m² e.g., forest (gap, mound, tip-up, litter etc.)
± disturbance → habitat management, + size, ± age (since volcanic eruption, flood etc.)
+ matrix heterogeneity (spatial heterogeneity)
- isolation
- discreteness of ecotones

Rarefaction: A statistical interpolation method of rarefying or thinning a reference sample by drawing random subsets of individuals (or samples) to standardize the comparison of biological diversity on the basis of a common number of individuals or samples

Types of α-diversities

Type 0: Using number of species only

species richness, S = number of species
species density, D = number of species per unit area Species density vs nitrogen
Relationship between species density and nitrogen in soil until nitrogen is not excess.

Nonparametric asymptotic estimators: Estimators of total species richness (including Chao1, Chao2, abundancebased coverage estimator (ACE), incidence-based coverage estimator (ICE), and the jackknife) that do not assume a particular form of the species abundance distribution (such as a log-series or log-normal distribution). Instead, these methods use information on the frequency of rare species in a sample to estimate the number of undetected species in an assemblage
Type 1: Using number of species and total number of individuals
Fisher's α: S = αloge(qn/α)

n: number of individuals per unit area
q: area, or number of plots

Gleason (1922), D = S/logN, or rarely S/log10A (A: unit area, or plot size)

tightly corresponded with species-area curve

Margalef (1957), D = (S - 1)/lnN
Menhinick (1964) D = logS/logN, or S/√N
Simpson = 1/Σ(ni/N)

Parker or Gini-Simpson = 1 - Σpi²
Pielou = 1 - Σ{ni(ni - 1)}/N(N - 1)

MacArthur = -Σ(ni/N)•log2(ni/N)

Type 2: Considering domonance with type-1 diversity

Whittaker, Shannon, Simpson, MacArthur, MacIntosh, Hurlbert, Bulla

N = R/U·(1 + O/H)

N: species richness, R: total resources, H–, averaged distance between the interrupted lines, C: number of neighboring species, R: range of resources, U: range of resources utilized by each species, O: overlaps of resource utilization

R = Σi=1NHi = N·(1/N)Σi=1NHi = NH

N = R/H = R/U = U/HU consists of two parts, H and O

Σi=1NUi = Σi=1NHi + 2Σi=1N–1Oi, i+1 → divided by N

U = H + 2O: the values are different between the two edges
O- is slightly lower than O

Ds = (DR/Dvλ = 1/Σpi,

l: Rayleigh quotient on matrix α, α-: averaged competition coefficient, C: number of neighboring species

Type 2: Evenness (均等度)
Whittaker's evenness (Whittaker 1952)

Ec = S/(logp1 - logps)
Ec' = S/4√(Σis(logp1 - logpi)²/S)

S: species richness
p1, ps: relative dominance of species i
(base = 10)

Bulla's evenness index, E (Bulla 1994, Feisinger et al. 1981)

E = (O - minO)/(1 - minO),
O = Σmin(pf, ρi) (= Czekanowski's index of proportional similarity)

Simpson (1949): D, 1 - D, 1/D

D = Σi=1s{ni·(ni - 1)}/{N·(N - 1)}, or Σi=1spi → 1 - D

randamly selected one individual → Pi = ni/N, the probability that species i is selected
randamly selected one individual again → iPi = (ni - 1)/(N - 1), the probability that species i is selected
→ selected two individuals continuously → Pi × iPi, probability that both of them are species
∴ 1 - D, the probability that the two species are different

Simpson's reciprocal index, 1/D = 1/Σi=1spi

→ non-sensitive to changes in species richness (becoming remarkable when species richness < 10 ↔ sentitive to the abundance of dominant species
→ follwing the law of log-normal distribution (and probably broken-stick model)

Phenological diversity, Df = 1/(eipi2) (→ Simpson)

where pi is the relative frequency of each phenological group (in a given plot)

Relative phenological diversity, D'f = (DfDfmin)/(DfmaxDfmin)

if phenological gropus are classified into four categories, then
D'f = (Df – 1)/3, (Dfmax = 4, Dfmin = 1)


Evenness, J' = H'/H'max = H'/logS

(Jost 2006, Tuomisto 2010)

True diversity, qDγ (真の多様性)

q (integer): called order of the diversity, qDγi=1spiq)1/(1 - q), q ≥ 0

⇒ Hill numbers (effective number of species)

q = 0: 0Dγ = Σi=1spi0 = s
q = 1: 1Dγ = limq→11Dγ = exp(-Σi=1s(pi·lnpi)) = exp(H')
q = 2: 2Dγ = 1/i=1spi2)
Table 1. Conversion of common indices to true diversities (Jost et al. 2006)
Species richness xΣi=1spi0

x (diversity in terms of x)__Σi=1spi0 (diversity in terms of pi)

Shannon entropy x ≡ - Σi=1spilnpi

exp(x)________________ exp(-Σi=1spilnpi)

Simposon concentration xΣi=1spi2


Gini-Simposon index x ≡ 1 - Σi=1spi2

1/(1 - x)_______________1/Σi=1spi2

HCDT (or Tsallis) entropy x(1 - Σi=1spiq)/(q - 1)

[(1 - (q - 1)x]1/(1 - q)______i=1spiq)1/(1 - q)

Renyi entropy x ≡ -lni=1spiq)/(q - 1)

exp(x)________________i=1spiq)1/(1 - q)

Differentiating assemblages based on five compounds of diversity

(Guisande et al. 2017)

heterogeneity (including species richness)
phylogenetic/taxonomic diversity, PD

the sum of the lengths of all phylogenetic branches that are members of the corresponding minimum spanning path

functional diversity

calculated by DER library in R

Dark diversity
= the set of species that are absent from a given site but present in the surrounding region and potentially able to inhabit particular ecological conditions

observed community < species pool < regional richness

↑dispersal/habitat filtering/biotic interactions↑

β-diversity (β多様性)

Similarity = between-habitat diversity
The species, population response within its niche hypervolume describes its niche.
The amount of biotic change among units. The differentiation of units on the α-diversity scale is β-diversity
Interspecific association and similarity between communities
Qualitative data
= binary data, or presence/absence data
Measuring β-diversity with presence-absence data (Wilson & Shmida 1984)
  1. βw = S/αmean - 1 (Whittaker 1960)
  2. βc = [g(H) + l(H)]/2 (Cody 1975)
  3. βR = S²/(2γ + S) - 1 (Routledge 1977)
    βI = log(T) - (1/T)Σieilogei - (1/T)Σjαjlogαj
    βE = exp(βI) - 1
  4. βT = [g(H) + l(H)]/2 (β-turnover, Wilson & Shmida 1984)

H: given range of a habitat gradient
g(H): the number of new species encountered or gained along H
l(H): the number of species that drop out or are lost along H
αmean: the average number of species found in samples along H

Coefficient of community (群集類似係数), C (Jaccard 1912)

= association coefficient (Agrell 1945), faunistic relation factor (Webb 1950)
C = a/(b + c + a)

Community coefficient or Quotient of similarity (類似係数), QS (Sφrensen 1948)

QS = 2a/(b + c + 2a)
C and QS range from 0 to 1 → 0 = completely diffrent between the two groups, 1 = completely same

Simpson's coefficient, SC (Simpson 1943

SC = c/b (ab)
To reduce the effects of a and d when the sample sizes are greatly different between the two groups

Percentage of affinity (相関率), PA (Masamune 1931, 正宗 1934)

PA = 1/2·(c/a + c/b) or c(a + b)/2ab

Coefficient of closeness (親和係数), CC (Otuka 1936)

CC = c/√(ab)

Coefficient of difference (差異係数), CD (Savage 1960)

CD = c/a (a > b)

Kulezynsky similarity measure (カルチンスキー類似度), SM (Kulezynsky 1927)

SM = c/(a + b - 2c)

Debatable points (Southwood 1966)
  1. the values become extremely low when the differences bewteen the two groups are large and are difficult to compare between the groups by similarity
  2. rare species are overestimated
Quantitative data
Simple correlation coefficient or Peason's φ (単純相関係数法) (Motomura 1935), or Product-moment correlation coefficient (積率相関係数), Cp (Goodall 1973)

φ= Σk(x1k - X1/n)·(x2k - X2/n)/[Σk(x1k - X1/n)2Σk(x2k - X2/n)2]1/2
correlation coefficient: assuming a normal distribution → rarely occuring in the distribution of species in a community → non-sophisticated and unacceptable

Kendall's τ coefficient = rank correlation
Similarity ratio, SR (Janssen 1975)

SR = Σixi·yi/(Σixi2 + Σiyi2 - Σixi·yi)

Percentage similarity, PS (Sorensen 1948)

PS = 2·Σimin(xi, yi)/Σi(xi + yi)

Percentage difference, PD (Odum 1950) or Bray-Curtis dissimilarity index

PD = Σi|Nai + Nbi|/(Na + Nb)

Na and Nb: total number of species in A and B, respectively
Nai and Nbi: number of individuals in ith species on A and B, respectively

1 = completely different species composition between the two communities
→ similarity = 1 - PD dominant species contribute more to the similarity → effective to investigate the effects of dominat species

Percentage similarity (種間相関示数), PS (Whittaker 1952)

PS = 1 - 0.5·Σi|pai - pbi| = Σimin(pai, pbi) = Σmin(nx, ny)/(Nx + Ny)

pai, pbi: dominance in communities A and B, respectively

→ application to succession (Tsuyuzaki 1991)

Index of nichee overlap (S) (ニッチ重複指数)
Coefficient of community, S1 = Σi=1min(pi, qi)
Morisita's index, S2 = 2·Σi=1piqi/(Σpi2 + Σqi2)
Horn's index, S3

= [Σi=1(pi + qi)log(pi + qi) – Σi=1pilogpi - Σi=1qilogqi]/(2·log2)

Euclidean distance, S4 = 1 – [Σi=1(piqi)2/2]1/2

Dissimilarity ⇔ Similarity (非類似度と類似度)

C = a/(b + c + a) ⇒
Def. Jaccard dissimirality, βdiss ≡ 1 - C

= 1 - a/(b + c + a) = {(b + c + a) - a)/(b + c + a)
= (b + c)/(b + c + a)
= (number of unshared species)/(total number of species)
→ dependence of species richness (b and c) and overlap (a)

Def. βturnover ≡ 2 × min(b, c)/(a + 2 × min(b, c)) (Baselga 2012)

= (2 × c)/(a + 2 × c) when bc (∵ min(b, c) = c)

(Villéger et al. 2014)

Def. pturnβturnover/βdiss

= 2 × min(b, c)/(a + 2 × min(b, c)) × (b + c + a)/(b + c)
= (2 × c)/(a + 2 × c) × (b + c + a)/(b + c) when bc

Generalized dissimilarity modeling, GDM (一般化非類似度モデル)
A statistical technique for modelling spatial variation in biodiversity between pairs of geographical locations or spatial patterns of turnover in community composition (β-diversity) (Library gdm in R package)

(Ferrier et al. 2007)

Fig. 5. Applications of generalized dissimilarity modelling
Table 1. Approaches, utility, input data types, examples and associated software packages (Thomassen et al. 2010)

Canonical trend surface analysis: Modelling of biological variation across landscape
Principal coordinates of neighbour matrices: Modelling of biological variation across landscape; purely spatial modelling step of which the results are used in subsequent (regression) analyses
Tree regression, random forest: Modelling of biological variation across landscape; relating environmental heterogeneity to biotic differences
GDM: Modelling of biological variation across landscape; relating environmental heterogeneity to biotic differences

[landscape ecology (景観生態学)]

γ-diversity (landscape diversity, γ多様性)

The variables of habitats and niches may be combined to define as axes an (m + m')-dimensional ecotope hyperspace. The part of this hyperspace to which a given species is adapted is its ecotope hypervolume. When a population measure is superimposed on this hypervolume, the ecotope of the species is described.

Relationships between α, β and γ diversities

(α, β, γ多様性間の関係)

    Community 1: A B C D E F
    Community 2:       D E F
    Community 3: A B     E F G

α1 = 6, α2 = 3, α3 = 5
γ = 7

(Veech et al. 2002, Baselga 2010)

Additive partitioning: αavg + β = γ

β = 7 - (6 + 3 + 5)/3 ≈ 2.33

Proportional partitioning: αavg × β = γβ = γ/αavg

β = 7/{(6 + 3 + 5)/3} = 1.5

Species relationships (種数関係)

[ island biogeography ]

Species-area relationship (種数-面積関係)

(Darlington 1957, Preston 1960)

Species-area curve (種数-面積曲線)
1921 Arrhenius O: linear log-log relationship between area and richness
N = CSα → logN = logC + αlogS

N: species richness
S: area
⇒ intercept = C', and slope = α


[optimal plot size]

Species-dominance relationship (種数-優占度関係)

Zipf's law (ジップの法則)

the law on the relationships between rank and size

Ex.1: 1897 Paleto: income of individuals
Ex.2: 1913 Auerbach: population of cities
Ex.3: 1922 Willis: number of species in genera, freqencies of word usage

Rank-size relation
x: rank (in decreasing order of size y)
y: size = Y(x)
Size distriubtion function

the ratio of size that ranges between y and y + dy
= f(y)dy
the number that ranges between y and y + dy
= n(y)dy
f(y) = n(y)/N

N: total number of species

f(y)dy = -d/dy·Y-1(y)dy

yx-(1 + α), logx + αlogy = C
f(y) ∝ y-(1 + 1/(1 + α))


Zipf's law: α = 0 in general

yrx (r < 1), x + logy = C
f(y) ∝ 1/y

Species-dominance curves (種優占度曲線)
x = rank and y = relative dominance on Zipf's law

Geometric series (等比級数則)
Logarithmic series (対数級数則)
Log-normal series (対数正規則)

1978 Sepkoski

S = aN - αN2, E = bN - βN2

a, α, b, β: constants

dN/dt = r(1 - N/K)N → diversification curve

K: constant, r = a - b