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Table 1 General format, power law and random placement models for SAR, EAR and OAR

From: Application of fundamental equations to species−area theory

Parameters General format Power law model Random placement model
SAR
EAR
S a  + E A-a  = S A  = E A
S A-a  + E a  = S A  = E A
S a  = ca z
E a  = cA z − c(A − a)z
\(S_{a} = S_{a}^{1} = S_{A} - \sum \limits_{i = 1}^{{S_{A} }} \left(1 - \frac{a}{A}\right)^{{N_{i} }}\)
\(E_{a} = S_{a}^{N} = \sum \limits_{i = 1}^{{S_{A} }} (\frac{a}{A})^{{N_{i} }}\)
OAR \(O_{a, A - a} = S_{a} - E_{a}\)
\(O_{a, A - a} = S_{a} + S_{A - a} - S_{A}\)
\(O_{a, A - a} = S_{A} - E_{a} - E_{A - a}\)
\(O_{a, A - a} = O_{A - a, a}\)
\(O_{a, A - a} = ca^{z} + c(A - a)^{z} - cA^{z}\) \(O_{a, A - a} = S_{A} - \sum \limits_{i = 1}^{{S_{A} }} \left[ {\left( {1 - \frac{a}{A}} \right)^{{N_{i} }} + \left( {\frac{a}{A}} \right)^{{N_{i} }} } \right]\)
h \(h = \frac{{O_{a, A - a} }}{{S_{a} }} = \frac{{S_{a} - E_{a} }}{{S_{a} }} = 1 - \frac{{E_{a} }}{{S_{a} }}\) \(h = 1 - \frac{{A^{z} - (A - a)^{z} }}{{a^{z} }}\) \(h = 1 - \frac{{ \sum \nolimits_{i = 1}^{{S_{A} }} \left(\frac{a}{A}\right)^{{N_{i} }} }}{{S_{A} - \sum \nolimits_{i = 1}^{{S_{A} }} \left(1 - \frac{a}{A}\right)^{{N_{i} }} }}\)
h, a = A/2 \(h = \frac{{2S_{A/2} - S_{A} }}{{S_{A/2} }} =\) 2\(\frac{{S_{A} }}{{S_{A/2} }}\) h = 2 − 2z \(h = 2 - \frac{1}{{1 - \frac{{ \sum \nolimits_{i = 1}^{{S_{A} }} \left(\frac{1}{2}\right)^{{N_{i} }} }}{{S_{A} }}}}\)
h \(h^{'} = \frac{{O_{a, A - a} }}{{S_{A} }} = \frac{{S_{a} + S_{A - a} }}{{S_{A} }} - 1\) \(h^{'} = \frac{{ca^{z} + c(A - a)^{z} }}{{cA^{z} }} - 1\) \(h^{'} = \frac{{2S_{A} - \sum \nolimits_{i = 1}^{{S_{A} }} \left[ {\left( {1 - \frac{a}{A}} \right)^{{N_{i} }} + \left( {\frac{a}{A}} \right)^{{N_{i} }} } \right] }}{{S_{A} }} - 1\)
h′, a = A/2 \(h^{'} = \frac{{O_{a, A - a} }}{{S_{A} }} = \frac{{2S_{A/2} }}{{S_{A} }} - 1\) h  = 21−z − 1  
\(\eta\) \(\eta = \frac{{S_{a} }}{{S_{A} }} = 1 - \frac{{E_{A - a} }}{{S_{A} }}\) \(\eta = c(\frac{a}{A})^{z}\) \(\eta = 1 - \frac{{ \sum \nolimits_{i = 1}^{{S_{A} }} (1 - \frac{a}{A})^{{N_{i} }} }}{{S_{A} }}\)
λ \(\lambda = \frac{{E_{a} }}{{S_{A} }} = 1 - \frac{{S_{A - a} }}{{S_{A} }}\)
λ = (1 − h)η
\(\lambda = 1 - c(1 - \frac{a}{A})^{z}\) \(\lambda = \frac{{ \sum \nolimits_{i = 1}^{{S_{A} }} (\frac{a}{A})^{{N_{i} }} }}{{S_{A} }}\)
  1. SAR species−area relationship; EAR endemic-area relationship; OAR overlap-area relationship; S a is the number of species in area a; E a is the number of species only in the area a, but not in the area A-a; O a, Aa is the number of species both in the area a and Aa; c (a = 1, c = S 1 ) and z (0 ≤ z≤1) are constants; A is the total area, S A is total number of the species in the area A, and E A is the total number of specific species in area A; h is overlap index; h’ is the ratio of overlapping species number over the total species number, overlap rate; \(\eta\) is ratio of S a over S A , sampling rate; \(\lambda\). is extinction rate; N i is the number of individuals of the specific species i; S 1 a and S N a are species−area curve and endemics-area curve across all species in A [3, 6, 7, 10, 12, 16, 17]