Species richness

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

Diversity (Shmida & Wilson 1985)

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

Functional diversity

Persistence (永続性)

Resistance (耐性)

Rj = 1 - Σi=1^m(Δpij/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)

Stability within bounds = no recognizable major changes in vegetation community over time
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.

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

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

Variability (変動性) ⇔ stability (安定性)

Hypotheses on the changes of species diversity

+ matrix heterogeneity, - isolation, - boundary discreteness)

S = number of species

Lottery model (富くじモデル)

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

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

                     Classic       Phytosociology  Theoretical
                     biogeography                  population
                                                   biology

  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

= 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

Types of α-diversities

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

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

tightly corresponded with species-area curve

Parker or Gini-Simpson = 1 - Σp_i²
Pielou = 1 - Σ{n_i(n_i - 1)}/N(N - 1)

MacArthur = -Σ(n_i/N)•log₂(n_i/N)

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

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

∴ N = R/H^– = R/U^– = U^–/H^– → U^– consists of two parts, H and O

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

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

Type 2: Evenness (均等度)

Ec = S/(logp₁ - logp_s)
Ec' = S/4√(Σ_i^s(logp₁ - logp_i)²/S)

S: species richness
p₁, p_s: 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(p_f, ρ_i) (= Czekanowski's index of proportional similarity)

D = Σ_i=1^s{ni·(n_i - 1)}/{N·(N - 1)}, or Σ_i=1^sp_i → 1 - D

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

Simpson's reciprocal index, 1/D = 1/Σ_i=1^sp_i

→ 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, D_f = 1/(e_ip_i²) (→ Simpson)

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

Relative phenological diversity, D'_f = (D_f – D_fmin)/(D_fmax – D_fmin)

if phenological gropus are classified into four categories, then
D'_f = (D_f – 1)/3, (D_fmax = 4, D_fmin = 1)

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

(Jost 2006, Tuomisto 2010)

True diversity, ^qD_γ (真の多様性)

⇒ Hill numbers (effective number of species)

x (diversity in terms of x)__Σ_i=1^sp_i⁰ (diversity in terms of p_i)

exp(x)________________ exp(-Σ_i=1^sp_ilnp_i)

1/x___________________1/Σ_i=1^sp_i²

1/(1 - x)_______________1/Σ_i=1^sp_i²

[(1 - (q - 1)x]^{1/(1 - q)}______(Σ_i=1^sp_i^q)^{1/(1 - q)}

exp(x)________________(Σ_i=1^sp_i^q)^{1/(1 - q)}

Differentiating assemblages based on five compounds of diversity

(Guisande et al. 2017)

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

calculated by DER library in R

Dark diversity

observed community < species pool < regional richness

↑dispersal/habitat filtering/biotic interactions↑

Qualitative data

1. β_w = S/α_mean - 1 (Whittaker 1960)
2. β_c = [g(H) + l(H)]/2 (Cody 1975)
3. β_R = S²/(2γ + S) - 1 (Routledge 1977)
   β_I = log(T) - (1/T)Σ_ie_iloge_i - (1/T)Σ_jα_jlogα_j
   β_E = exp(β_I) - 1
4. β_T = [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
α_mean: the average number of species found in samples along H

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

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

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

PA = 1/2·(c/a + c/b) or c(a + b)/2ab

CC = c/√(ab)

CD = c/a (a > b)

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

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

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

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

PS = 2·Σimin(xi, yi)/Σi(xi + yi)

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

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)

= [Σ_i=1(p_i + q_i)log(p_i + q_i) – Σ_i=1p_ilogp_i - Σ_i=1q_ilogq_i]/(2·log2)

Dissimilarity ⇔ Similarity (非類似度と類似度)

= 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)

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

(Villéger et al. 2014)

= 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 b ≥ c

Generalized dissimilarity modeling, GDM (一般化非類似度モデル)

(Ferrier et al. 2007)

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

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)

β = 7 - (6 + 3 + 5)/3 ≈ 2.33

β = 7/{(6 + 3 + 5)/3} = 1.5

(Darlington 1957, Preston 1960)

Species-area curve (種数-面積曲線)

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

Zipf's law (ジップの法則)

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

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

y ∝ x^{-(1 + α)}, logx + αlogy = C
f(y) ∝ y^{-(1 + 1/(1 + α))}

y ∝ r^x (r < 1), x + logy = C
f(y) ∝ 1/y

Species-dominance curves (種優占度曲線)

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

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