Moran or Wright–Fisher processes are probably the most well known models to study the evolution of a population under environmental various effects.Our object of study will be the Simpson index which measures the lev...Moran or Wright–Fisher processes are probably the most well known models to study the evolution of a population under environmental various effects.Our object of study will be the Simpson index which measures the level of diversity of the population,one of the key parameters for ecologists who study for example,forest dynamics.Following ecological motivations,we will consider,here,the case,where there are various species with fitness and immigration parameters being random processes(and thus time evolving).The Simpson index is difficult to evaluate when the population is large,except in the neutral(no selection)case,because it has no closed formula.Our approach relies on the large population limit in the“weak”selection case,and thus to give a procedure which enables us to approximate,with controlled rate,the expectation of the Simpson index at fixed time.We will also study the long time behavior(invariant measure and convergence speed towards equilibrium)of the Wright–Fisher process in a simplified setting,allowing us to get a full picture for the approximation of the expectation of the Simpson index.展开更多
The Gini-Simpson quadratic index is a classic measure of diversity, widely used by ecologists. As shown recently, however, this index is not suitable for the measurement of beta diversity when the number of species is...The Gini-Simpson quadratic index is a classic measure of diversity, widely used by ecologists. As shown recently, however, this index is not suitable for the measurement of beta diversity when the number of species is very large. The objective of this paper is to introduce the Rich- Gini-Simpson quadratic index which preserves all the qualities of the classic Gini-Simpson index but behaves very well even when the number of species is very large. The additive partitioning of species diversity using the Rich-Gini- Simpson quadratic index and an application from island biogeography are analyzed.展开更多
The weighted Gini-Simpson quadratic index is the simplest measure of biodiversity which takes into account the relative abundance of species and some weights assigned to the species. These weights could be assigned ba...The weighted Gini-Simpson quadratic index is the simplest measure of biodiversity which takes into account the relative abundance of species and some weights assigned to the species. These weights could be assigned based on factors such as the phylogenetic distance between species, or their relative conservation values, or even the species richness or vulnerability of the habitats where these species live. In the vast majority of cases where the biodiversity is measured the species are supposed to be independent, which means that the relative proportion of a pair of species is the product of the relative proportions of the component species making up the respective pair. In the first section of the paper, the main versions of the weighted Gini-Simpson index of biodiversity for the pairs and triads of independent species are presented. In the second section of the paper, the weighted Gini-Simpson quadratic index is calculated for the general case when the species are interdependent. In this instance, the weights reflect the conservation values of the species and the distribution pattern variability of the subsets of species in the respective habitat induced by the inter-dependence between species. The third section contains a numerical example.展开更多
The distribution of biodiversity at multiple sites of a region has been traditionally investigated through the additive partitioning of the regional biodiversity, called γ-diversity, into the average within-site biod...The distribution of biodiversity at multiple sites of a region has been traditionally investigated through the additive partitioning of the regional biodiversity, called γ-diversity, into the average within-site biodiversity or α-diversity, and the biodiversity among sites, or β-diversity. The standard additive partitioning of diversity requires the use of a measure of diversity which is a concave function of the relative abundance of species, like the Shannon entropy or the Gini- Simpson index, for instance. When a phylogenetic distance between species is also taken into account, Rao’s quadratic index has been used as a measure of dissimilarity. Rao’s index, however, is not a concave function of the distribution of relative abundance of either individual species or pairs of species and, consequently, only some nonstandard additive partitionings of diversity have been given using this index. The objective of this paper is to show that the weighted quadratic index of biodiversity, a generalization of the weighted Gini-Simpson index to the pairs of species, is a concave function of the joint distribution of the relative abundance of pairs of species and, therefore, may be used in the standard additive partitioning of diversity instead of Rao’s index. The replication property of this new measure is also discussed.展开更多
The number and composition of species in a community can be quantified withα-diversity indices,including species richness(R),Simpson’s index(D),and the Shannon-Wiener index(H΄).In forest communities,there are large ...The number and composition of species in a community can be quantified withα-diversity indices,including species richness(R),Simpson’s index(D),and the Shannon-Wiener index(H΄).In forest communities,there are large variations in tree size among species and individu-als of the same species,which result in differences in eco-logical processes and ecosystem functions.However,tree size inequality(TSI)has been largely neglected in studies using the available diversity indices.The TSI in the diameter at breast height(DBH)data for each of 99920 m×20 m forest census quadrats was quantified using the Gini index(GI),a measure of the inequality of size distribution.The generalized performance equation was used to describe the rotated and right-shifted Lorenz curve of the cumulative proportion of DBH and the cumulative proportion of number of trees per quadrat.We also examined the relationships ofα-diversity indices with the GI using correlation tests.The generalized performance equation effectively described the rotated and right-shifted Lorenz curve of DBH distributions,with most root-mean-square errors(990 out of 999 quadrats)being<0.0030.There were significant positive correlations between each of threeα-diversity indices(i.e.,R,D,and H’)and the GI.Nevertheless,the total abundance of trees in each quadrat did not significantly influence the GI.This means that the TSI increased with increasing spe-cies diversity.Thus,two new indices are proposed that can balanceα-diversity against the extent of TSI in the com-munity:(1−GI)×D,and(1−GI)×H’.These new indices were significantly correlated with the original D and H΄,and did not increase the extent of variation within each group of indices.This study presents a useful tool for quantifying both species diversity and the variation in tree sizes in forest communities,especially in the face of cumulative species loss under global climate change.展开更多
文摘Moran or Wright–Fisher processes are probably the most well known models to study the evolution of a population under environmental various effects.Our object of study will be the Simpson index which measures the level of diversity of the population,one of the key parameters for ecologists who study for example,forest dynamics.Following ecological motivations,we will consider,here,the case,where there are various species with fitness and immigration parameters being random processes(and thus time evolving).The Simpson index is difficult to evaluate when the population is large,except in the neutral(no selection)case,because it has no closed formula.Our approach relies on the large population limit in the“weak”selection case,and thus to give a procedure which enables us to approximate,with controlled rate,the expectation of the Simpson index at fixed time.We will also study the long time behavior(invariant measure and convergence speed towards equilibrium)of the Wright–Fisher process in a simplified setting,allowing us to get a full picture for the approximation of the expectation of the Simpson index.
文摘The Gini-Simpson quadratic index is a classic measure of diversity, widely used by ecologists. As shown recently, however, this index is not suitable for the measurement of beta diversity when the number of species is very large. The objective of this paper is to introduce the Rich- Gini-Simpson quadratic index which preserves all the qualities of the classic Gini-Simpson index but behaves very well even when the number of species is very large. The additive partitioning of species diversity using the Rich-Gini- Simpson quadratic index and an application from island biogeography are analyzed.
文摘The weighted Gini-Simpson quadratic index is the simplest measure of biodiversity which takes into account the relative abundance of species and some weights assigned to the species. These weights could be assigned based on factors such as the phylogenetic distance between species, or their relative conservation values, or even the species richness or vulnerability of the habitats where these species live. In the vast majority of cases where the biodiversity is measured the species are supposed to be independent, which means that the relative proportion of a pair of species is the product of the relative proportions of the component species making up the respective pair. In the first section of the paper, the main versions of the weighted Gini-Simpson index of biodiversity for the pairs and triads of independent species are presented. In the second section of the paper, the weighted Gini-Simpson quadratic index is calculated for the general case when the species are interdependent. In this instance, the weights reflect the conservation values of the species and the distribution pattern variability of the subsets of species in the respective habitat induced by the inter-dependence between species. The third section contains a numerical example.
文摘The distribution of biodiversity at multiple sites of a region has been traditionally investigated through the additive partitioning of the regional biodiversity, called γ-diversity, into the average within-site biodiversity or α-diversity, and the biodiversity among sites, or β-diversity. The standard additive partitioning of diversity requires the use of a measure of diversity which is a concave function of the relative abundance of species, like the Shannon entropy or the Gini- Simpson index, for instance. When a phylogenetic distance between species is also taken into account, Rao’s quadratic index has been used as a measure of dissimilarity. Rao’s index, however, is not a concave function of the distribution of relative abundance of either individual species or pairs of species and, consequently, only some nonstandard additive partitionings of diversity have been given using this index. The objective of this paper is to show that the weighted quadratic index of biodiversity, a generalization of the weighted Gini-Simpson index to the pairs of species, is a concave function of the joint distribution of the relative abundance of pairs of species and, therefore, may be used in the standard additive partitioning of diversity instead of Rao’s index. The replication property of this new measure is also discussed.
基金supported by the National Natural Science Foundation of China(32101260).
文摘The number and composition of species in a community can be quantified withα-diversity indices,including species richness(R),Simpson’s index(D),and the Shannon-Wiener index(H΄).In forest communities,there are large variations in tree size among species and individu-als of the same species,which result in differences in eco-logical processes and ecosystem functions.However,tree size inequality(TSI)has been largely neglected in studies using the available diversity indices.The TSI in the diameter at breast height(DBH)data for each of 99920 m×20 m forest census quadrats was quantified using the Gini index(GI),a measure of the inequality of size distribution.The generalized performance equation was used to describe the rotated and right-shifted Lorenz curve of the cumulative proportion of DBH and the cumulative proportion of number of trees per quadrat.We also examined the relationships ofα-diversity indices with the GI using correlation tests.The generalized performance equation effectively described the rotated and right-shifted Lorenz curve of DBH distributions,with most root-mean-square errors(990 out of 999 quadrats)being<0.0030.There were significant positive correlations between each of threeα-diversity indices(i.e.,R,D,and H’)and the GI.Nevertheless,the total abundance of trees in each quadrat did not significantly influence the GI.This means that the TSI increased with increasing spe-cies diversity.Thus,two new indices are proposed that can balanceα-diversity against the extent of TSI in the com-munity:(1−GI)×D,and(1−GI)×H’.These new indices were significantly correlated with the original D and H΄,and did not increase the extent of variation within each group of indices.This study presents a useful tool for quantifying both species diversity and the variation in tree sizes in forest communities,especially in the face of cumulative species loss under global climate change.
