Application of fundamental equations to species−area theory
 Xubin Pan^{1}Email author
DOI: 10.1186/s1289801600975
© The Author(s) 2016
Received: 31 May 2016
Accepted: 27 August 2016
Published: 7 October 2016
Abstract
Background
Species−area relationship (SAR), endemicsarea relationship (EAR) and overlaparea relationship (OAR) are three important concepts in biodiversity study. The application of fundamental equations linking the SAR, EAR and OAR, can enrich the axiomatic framework of the species−area theory and deepen our understanding of the mechanisms of community assembly.
Results
Two fundamental equations are derived and extended to power law model and random replacement model of species−area distribution. Several important parameters, including the overlap index and extinction rate, are defined and expressed to enrich the species−area theory. For power law model, both EAR and OAR have three parameters, with one more parameter of the total area than SAR does. The EAR equation is a monotonically increasing function for parameter c and z, and a monotonically decreasing function for parameter A. The extinction rate, with two parameters, is a monotonically increasing function for parameter z, and a monotonically decreasing function for parameter A. The overlap index is a monotonically increasing function for parameter A, and a monotonically decreasing function for parameter z, independent of parameter c.
Conclusions
The general formats of SAR, EAR, OAR, overlap index, overlap rate, sampling rate and extinction rate, are derived and extended to power law model and random replacement model as the axiomatic framework of species−area theory. In addition, if the total area is underestimated, the extinction rate will be overestimated.
Keywords
Endemicsarea relationship Overlaparea relationship Power law Random replacement Real total area Sampling rate Overlap rate Extinction rate Overlap indexBackground
Species−area relationship (SAR) is a core concept in biodiversity and species distribution [1], and endemicsarea relationship (EAR) is a useful tool in biodiversity conservation and habitat preservation [2–5]. Besides SAR and EAR, overlaparea relationship (OAR), which refers to the number of overlap species in two areas, is also a relevant and important concept [6–8]. To link SAR and EAR and develop a complete species−area theory, two fundamental equations are established to describe species distribution and interrelation between two compensatory areas [7]. Now the species−area theory has been reconstructed by the set theory, integrating SAR, EAR, OAR, alpha diversity, beta diversity, and gamma diversity [8]. Although OAR curves for two areas of the same size are described and zeta diversity as the average number of species shared by multiassemblages is proposed, the expanding concept that compares two or more areas of different sizes has not been fully discussed yet [8, 9]. Furthermore, to investigate the spatial characteristics of species richness, it is necessary to integrate the two fundamental equations into the species−area model with distribution information or assumption. Then more parameters can be defined and expressed with empirical data, which can enrich the axiomatic framework of the species−area theory and deepen our understanding of the mechanisms or processes of community assembly.
In addition, debate still exists over the estimation of extinction rate based on the SAR, which is higher than observed extinction rate [10–13]. One explanation for the overestimation is that some species are “committed to extinction” instead of going extinct due to habitat clearing [14–16]. However, another reason has been ignored in this debate [8]. According to the power law model, the SAR is a twoparameter equation, whereas the EAR is a threeparameter equation. It does not consider total area in SAR, while total area and its corresponding total species number are crucial factors to determine species disappearing and extinction rate. However, the impact of total area on the extinction rate is still unknown without the specific species−area model and sensitivity analysis.
In this paper, power law and random replacement functions, both of which are widely used species−area models, were selected for the application of two fundamental equations [17–19]. Then several important parameters were defined and expressed to enrich the species−area theory. For power law model, sensitivity analysis of parameters was conducted for EAR, extinction rate and overlap index, and the extinction rate based on different total areas were assessed for overestimate comparison.
Methods
These fundamental Eqs. (2 and 3) for species−area theory were applied to power law model and random replacement model of SAR. To enrich the species−area theory, several parameters were proposed, including overlap index, overlap rate, sampling rate and extinction rate, which were defined by equations in the general format, power law model and random replacement model.
For power law model, sensitivity analysis of parameters was conducted for EAR, extinction rate and overlap index, and the extinction rate based on different total areas was assessed for overestimate comparison. The data can be downloaded from the Supplementary of Data.
Results
Application of two fundamental equations to power law model
Application of two fundamental equations to random replacement model
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 _{ Aa } = S _{ A } = E _{ A } S _{ Aa } + 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 − 2^{ z }  \(h = 2  \frac{1}{{1  \frac{{ \sum \nolimits_{i = 1}^{{S_{A} }} \left(\frac{1}{2}\right)^{{N_{i} }} }}{{S_{A} }}}}\) 
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 ^{’} = 2^{1−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  \(\lambda = 1  c(1  \frac{a}{A})^{z}\)  \(\lambda = \frac{{ \sum \nolimits_{i = 1}^{{S_{A} }} (\frac{a}{A})^{{N_{i} }} }}{{S_{A} }}\) 
Sensitivity analysis for power law model
Discussion
Extinction rate estimate and overestimate comparison
Area of habitat loss  c = 25 z = 0.25 A = 9  c= 25 z= 0.25 A = 49  c= 25 z= 0.25 A = 100  

λ (%)  Overestimate^{a}  λ (%)  Overestimate^{a}  λ_{100} (%)  
0.52  1.48  10.34  0.27  1.04  0.13 
1.00  2.90  10.56  0.51  1.05  0.25 
9.00  100.00  41.91  4.95  1.12  2.33 
25.00  16.34  1.36  6.94 
Based on EAR, however, small total area for habitat preservation does lead to potential high species extinction rate. Thus large total area should be adopted for the Natural Protected Areas (NPAs). UNESCOMAB World Network of Biosphere Reserves, suggests to apply a zonation system to NPAs, which consists of a core zone, a buffer zone and a transition zone. Normally, both the buffer zone and transition zone do not have any different or concerned species that are not in the core zone, thus the total number of species will not increase when the protected area is expanded from core zone to include the buffer zone and transition zone. But both the buffer zone and transition zone can relieve the impact of anthropic activities on the core zone, and this result can be derived from the species−area theory.
Since EAR and OAR involve species in two complementary areas, one more parameter, the total area, has been added in their expressions compared with SAR. If the concepts of EAR and OAR are expanded to arbitrary two areas (they can be treated as complementary in the point of mathematics), then the h′ will be transferred to the Jaccard index, and Sørensen index can also be expressed by \(\frac{{2O_{a, A  a} }}{{S_{A} + 2O_{a, A  a} }}\) [23–25]. If the concepts of EAR and OAR are expanded to more areas, then zeta diversity and new beta diversity can handle this circumstance [8, 9].
Conclusions
Fundamental equations for species−area theory are applied to power law model and random replacement model of SAR. To enrich the species−area theory, several parameters are proposed, including overlap index, overlap rate, sampling rate and extinction rate, which are defined by equations in the general format, power law model and random replacement model. For power law model, both EAR and OAR have three parameters, with one more parameter of the total area than SAR does. If the total area is underestimated, the extinction rate will be overestimated. The EAR equation is a monotonically increasing function for parameter c and z, and a monotonically decreasing function for parameter A. Extinction rate, which has two parameters, is a monotonically increasing function for parameter z, and a monotonically decreasing function for parameter A. The overlap index is a monotonically increasing function for parameter A, and a monotonically decreasing function for parameter z, independent of parameter c.
Abbreviations
 SAR:

species−area relationship
 EAR:

endemicsarea relationship
 OAR:

overlaparea relationship
Declarations
Acknowledgements
Thanks go to Mr. Fengqiao Liu for language polishing, and the editor and two anonymous reviewers for their insightful comments on the manuscript.
Competing interests
The author declares that he has no competing interests.
Availability of data and materials
The dataset supporting the conclusions of this article is included within its Additional file 1.
Funding
This work is supported by Beijing Nova Programme (Z1511000003150107). The author also wants to thank the financial support for his study from Chinese Government Award for Outstanding SelfFinanced Students Abroad, Texas A&M University Kingsville, Tsinghua University and Chinese Academy of Inspection and Quarantine.
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The Creative Commons Public Domain Dedication waiver (http://creativecommons.org/publicdomain/zero/1.0/) applies to the data made available in this article, unless otherwise stated.
Authors’ Affiliations
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