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

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]