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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

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
  1. McNaughton SJ. 1993. Biodiversity and function of grazing ecosystems. In: Biodioersity and ecosyskm function (Ecological studies 99) (Schulze E-D and Mooney HA, eds), pp. 361-383, Springer-Verlag
  2. McNaughton SJ. 1977. Diversity and stabillty of ecological communities: a comment on the role of empiricism in ecology, Am. Nat. 111: 515-525
  3. Frank DA and McNaughton SJ. 1991. Stability increases with diversity in plant communities: empirical evidence from the 1988 Yellowstone drought. Oikos 62: 360-362
  4. Silvertown J. et al. 1994. Rainfall, biomass variation, and community composition in the Park Grass Experiment. Ecology 75: 2430-2437
  5. Dodd M et al. 1994. Community stability: a 60-year record of trends and outbreaks in the occurrence of species in the Park Grass Experiment. J Ecol 83: 277-285
  6. Tilman D and Downing JA. 1994. Biodiversity and stability in grasslands. Nature 367: 363-365
  7. Tilman D, Wedin D and Knops J. 1996. Productivity and sustainability influenced by biodiversity in grassland ecosystems. Nature 379: 718-720
  8. Ewel JJ, Mazzarino MJ and Berish CW. 1991. Tropical soil fertility changes under monocultures and successional communities of different structure. Ecol Appl 1: 289-302
  9. Naeem S et al. 1994. Declining biodiversity can alter the performance of ecosystems. Nature 368: 734-737

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)

Types of community diversity index (Shmida & Wilson 1985)
α-diversitythe species richness of samples representing communitiesH', C102-104 m2
β-diversityThe amount of biotic change among units.
The differentiation of units on the α-diversity scale
Similarity index
γ-diversityThe diversity of landscapesSame with α-diversity106-108 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 biogeographyPhytosociologyTheoretical population biology
Scale of observationRegionalAmong-communitiesWithin-communities
Type of diversityγβα
Primary proposed determinantsHistoricalEnvironmentalSpecies interactive
Role of noiseIrrelevantImportantUnobserved

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
Table. Types of α-diversities
Type 0: Using number of species only

Number of species (species richness)

Type 1: Using number of species and total number of individuals

Gleason, Margalef, Menhinick

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 0
S, species richness = 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
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 = 1 - Σpi²
Pielou = 1 - Σ{ni(ni - 1)}/N(N - 1)

MacArthur = -Σ(ni/N)•log2(ni/N)

Type 2: Evenness
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

β-diversity (β多様性)

= 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)

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
→ simolarity = 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

[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)

αavg + β = γ

β = 7 - (6 + 3 + 5)/3 ≈ 2.33

(Baselga 2010)

αavg × β = γβ = γ/αavg

β = 7/{(6 + 3 + 5)/3} = 1.5

Species relationships (種数関係)

Species-area relationship (種数-面積関係)

Species-area curve
N = CSα → logN = logC + αlogS

N: species richness
S: area

→ intercept = C', and slope = α

(Darlington 1957, Preston 1960)


[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

1897 Paleto: income of individuals
1913 Auerbach: population of cities
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