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GEOLOGY, ECOLOGY, AND LANDSCAPES INWASCON https://doi.org/10.1080/24749508.2021.1923270 RESEARCH ARTICLE Extending the concept of entropy-negentropy for the assessment of ecological dominance and diversity at alpha, beta and gamma levels a b c b Vinod Kumar , Ashwani Kumar Thukral , Anket Sharma and Renu Bhardwaj a b Department of Botany, Government Degree College, Jammu, India; Department of Botanical and Environmental Sciences, Guru Nanak Dev University, Amritsar, India; State Key Laboratory of Subtropical Silviculture, Zhejiang A & F University, Hangzhou, China ABSTRACT ARTICLE HISTORY Received 3 September 2020 Several measures of ecological diversity have been defined at alpha, beta, and gamma levels Accepted 24 April 2021 and less attention has been paid to characterise their ecological dominance. In this paper, we extend the concept of negative entropy (negentropy) for the measurement of ecological KEYWORDS dominance and diversity at the three hierarchical levels of community characterization. Entropy; negentropy; Negentropy is a measure of energy, and gives a convex curve for binary negentropy function, dominance; Shannon’s whereas Shannon’s entropy gives a typical concave curve. Similarly, we have defined indices for diversity; Simpson’s Simpson’s and Brillouin’s dominance functions at alpha, beta and gamma levels. The results of dominance; Brillouin’s information; binary entropy diversity indices followed a trend for different sites as: Harike > Beas > Goindwal Sahib, while plot; beta diversity; trend obtained for dominance is Goindwal Sahib > Beas > Harike. The pooled results of both community; phytosociology indicated that Harike showed maximum ecological information by the application of Shannon’s diversity and Simpson’s inverse diversity, while results of Simpson’s diversity remain same for all sites. Introduction Dominance The vital objective of ecology is to delineate the Common indices used for the measurement of dom- mechanisms which are essential for the sustainabil- inance are Simpson’s index, Berger-Parker index and ity of ecosystems (Tilman et al., 2014). Any damage McIntosh index. to biodiversity destroys the functionality of ecosys- tems, and in severe circumstances leads to the mass Entropy as a measure of diversity extinction of species and the loss of whole ecosys- tems (Dunne & Williams, 2009). Biodiversity in its Shannon’s entropy (H’) is one of the most preferred widest sense is defined as the variety of life forms measures of diversity, and is given by the equation, at different levels, ranging from the organismic X X n n i i levels to the species (DeLong, 1996; Pandita et al., H ¼ p ln p ¼ i ¼ 1K ln i i N N 2019). By applying diversity indices, we can com- i¼1 pare diverse spatial sites and temporal periods etc. where p is the probability of occurrence of (Daly et al., 2018). These measures are thus impor- a species, and K is the number of species in tant tools for ecological monitoring and conserva- a community. tion, and to deal with any biodiversity calamity Simpson’s index of diversity (C’) and Simpson’s (Morris et al., 2014). Ecological dominance and inverse index of diversity (D’) are represented by the diversity are extensively used community features equation as: at alpha, beta and gamma levels (Thukral et al., 2019), and these diversity measures are widely 0 2 C ¼ 1 p applied in the ecology from the last few decades i¼1 (Semeniuk &, Cresswell 2013). Some of the mea- sures used for the determination of community D ¼ diversity are Shannon’s entropy (Shannon- P i¼1 i Weiner’s index) (Shannon, 1948), Simpson’s index (Simpson, 1949), Brillouin’s index (Brillouin, 1953, Simpson’s index of dominance (C) is also known as 1962), Renyi’s entropy (Renyi, 1961), Berger- Simpson’s index with replacement and is used if the Parker’s index (Berger & Parker, 1970) etc. sample is quite large. CONTACT Vinod Kumar vinodverma507@gmail.com Department of Botany, Government Degree College, Jammu, Ramban 182144, India © 2021 The Author(s). Published by Informa UK Limited, trading as Taylor & Francis Group on behalf of the International Water, Air & Soil Conservation Society(INWASCON). This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 2 V. KUMAR ET AL. K � � Table 1. Some beta diversity indices from PAST software C ¼ p ¼ (Hammer et al., 2001). i¼1 K Whittaker (1960) 1 Harrison et al. (1992) α where, p is probability of occurrence of a species in i N 1 gðSÞþlðSÞ Cody (1975) the community, K is the number of species and n is gðSÞþlðSÞ th Wilson-Shmida (1984) the number of individuals of the i species, and N is 2α gðSÞþlðSÞ Mourelle and Ezcurra (1997) 2αðN 1Þ the total number of individuals of all the species. The Harrison et al. (1992) max maximum value of Simpson’s index for a single species N 1 αmax Williams (1996) 1 community is 1, and the minimum value for an evenly distributed species is (1/K) . If the sample size is small, Simpson’s index of dominance without replacement is 0 have S ¼ E . Thus, in terms of energy, we can max max used, rewrite the entropy equation as, nðn 1Þ i i i¼1 0 0 C ¼ E þ S ¼ E max NðN 1Þ Berger-Parker index of dominance (D ) is the BP In ecological systems, dominance and diversity proportion of number of individuals of the dominant characterize the numerical structure of different spe- species (n ) to the total number of individuals of all D cies. Shannon’s index of diversity (H’) is a measure of the species (N), entropy. Since a community with even distribution of species has the maximum entropy (H’ ), we can n max D ¼ BP extend the concept of negentropy (E’) to the study of ecological dominance: For a single species community, the value of Berger- Parker dominance index is 1 (maximum), and for an 0 0 0 E ¼ H H max evenly distributed community, the value is 1/K, where K is the number of species. McIntosh’s U is Diversity and dominance are two inseparable a dominance index, aspects of biological community studies, and are gen- vffiffiffiffiffiffiffiffiffiffiffiffi erally studied together. However, diversity indices are uX t 2 more common in literature (Kumar et al., 2017; U ¼ p McIntosh 1 Parkash & Thukral, 2010; Sarangal et al., 2012; Thukral, 2017). Further, community characterization where pi represents the proportional number of indi- at β and γ levels is generally restricted to diversity only, viduals of different species present in a community. some of the indices used are given in (Table 1). The present work, therefore, focuses on defining new eco- logical dominance indices at alpha level, and extend- Relation between entropy and negentopy ing the concept of ecological dominance at β and γ The concept of negative entropy was given by levels. Schrödinger (1944), and later elaborated by Brillouin (1953), who introduced the term negentropy. Among all the probability distributions, normal distribution Methods has the maximum entropy, negentropy is always non- Dominance and diversity negative (Quarati et al., 2016). The relation between entropy (S) and negentropy (J) of a system is given by Drawn from the relation between entropy and negen- the equation, tropy, we can write the relationships between ecologi- cal dominance and diversity for some known diversity J ¼ S S max and dominance indices as given in (Table 2). where, S is the maximum entropy possible. This max also implies that sum of entropy and negentropy of Table 2. Negentropy measures of four common indices used a system is equal to the theoretical maximum in ecological diversity and dominance. entropy, i.e., Entropy as Negentropy as Maximum Measure Diversity Dominance entropy J þ S ¼ S max K K Shannon’s P P ln K 0 0 H ¼ p ln p E ¼ ln K p ln p i i i i i¼1 i¼1 If entropy measures order in a system, negentropy K K Simpson’s P P 1 0 2 2 measures the disorder. Negentropy may also be trea- C ¼ 1 p C ¼ p i i i¼1 i¼1 ted as the energy not utilised (E’) by the system and 0 1 1 Simpson’s P P K D ¼ D ¼ K K K 2 2 p p i¼1 i i¼1 i inverse available for work, i.e., J ¼ E . Because, maximum N! ln N! N! ln N! Brillouin’s H ¼ ln E ¼ ln n1 !n2 !...nK ! N n1 !n2 !...nK ! N order and maximum disorder are equal, we GEOLOGY, ECOLOGY, AND LANDSCAPES 3 Levels of diversity γðH Þ β ðH Þ ¼ mult: αðH Þ Community diversity can be studied at three levels, alpha (α), beta (β) and gamma (γ). Alpha diversity is where, α(H’), β(H’) and γ(H’) are the exponential α β γ defined as the diversity of a single sample or commu- functions, e , e and e of the three diversities respec- nity. For a landscape consisting of M communities, tively. Similarly, the additive model for Simpson’s β Shannon’s α diversity, αðH Þ, is the average commu- diversity, β(C’), Simpson’s inverse β diversity, β(D’), nity diversity: Brillouin’s β diversity, β(H) of a landscape consisting of M communities, may be defined, respectively, as, 0 0 αðH Þ ¼ H 0 0 0 β ðC Þ ¼ γðC Þ αðC Þ add: j¼1 0 0 0 Similarly, Simpson’s α diversity of a landscape α β ðDÞ ¼ γðDÞ αðDÞ add: (C’), Simpson’s inverse alpha diversity α(D’), and Brillouin’s α diversity α(H) consisting of β ðHÞ ¼ γðHÞ αðHÞ add: M communities may be defined, respectively, as, We have similarly defined the dominance equations 0 0 at α, β and γ levels. In this paper we have used only the αðC Þ ¼ C additive models of entropy and negentropy for diver- j¼1 sity and dominance measurements. We have used our models in different communities under four hypothe- 0 0 tical situations: αðDÞ ¼ D j¼1 (1) Each community having one different species with equal number of individuals of each species. αðHÞ ¼ H j¼1 (2) Only one species having equal number of indi- viduals in all the communities. Gamma diversity is the diversity of the landscape (3) Five different species, each having different consisting of two or more communities considering numbers of individuals. that the landscape is a single community. The relation- (4) Communities having different number of spe- ship between α, β and γ diversities may be described cies, each species having different number of using additive or multiplicative models (Whittaker, individuals. 1972). The additive model gives the value of γ diversity as, Results γ ¼ αþ β Derivation of negentropic indices of dominance where, β diversity is the difference of diversities at two levels (γ-α). The entropies at α and γ levels cannot In this paper, we extend the concept of negentropy to be negative, but their difference, mentioned in this the study of community dominance and diversity at manuscript as (β ) diversity, can be positive, nega- add alpha, beta and gamma levels. We have used Euler’s tive or zero. Using the additive model, Shannon’s β constant (e) as the base for logarithm to give entropy diversity, β(H’) of a landscape consisting of or negentropy in nats for Shannon’s and Brillouin’s M communities, may also be written as, indices. Alternatively, the logarithms could be used with base 2 (units as bits or Shannon’s), or base 10 0 0 0 β ðH Þ ¼ γðH Þ αðH Þ add: (units as decits). Using Shannon’s entropy, we define negentropic index of dominance (E’), as follows: where, α(H’), β(H’) and γ(H’) are Shannon’s addi- tive α, β, and γ diversities, respectively. The multi- 0 0 0 E ¼ H H max plicative model is based on effective number of H’ species in a community (β ) i.e., e . For α, β and where, E’ is the negentropy, H’ is the entropy or mult. γ Shannon’s entropies, the multiplicative beta diver- diversity, and H’ is the theoretical maximum max sity is. entropy of the community if all the species are evenly distributed. If a community consists of K species, max- γ α e ¼ e :e imum Shannon’s diversity is given as H ¼ ln K , max therefore, where e is ratio of two diversities and is given by γ α e =e . Shannon’s multiplicative model is also writ- 0 0 E ¼ ln K H ten as: 4 V. KUMAR ET AL. We have given the negentropies derived as mea- Binary entropy was calculated for different values of sures of dominance in (Table 2). For a landscape p varying from 0 to 1, and a plot between entropy (H’) consisting of M communities, α dominance may be and probability of the first class (p) is called binary defined as the average community dominance: entropy function plot (Figure 1). We can extend the binary entropy function to binary negentropy function 0 0 using the relation, αðEÞ ¼ E j¼1 E ¼ ln 2 ðp ln pþ q ln qÞ: th where E’ is negentropy of the j community. A plot between the negentropy (E’) and the prob- Negentropic β dominance, β(E’) of a landscape con- ability (p) may be called binary negentropy function. sisting of M communities, may be defined as, Binary entropy plot between probability and 0 0 0 β ðEÞ ¼ γðEÞ αðEÞ add: Shannon’s entropy gives a typical concave curve, whereas binary negentropy plot gives a convex curve where γ(E’) is negentropic dominance index of the (Figure 1). Binary entropy plot may also be called landscape. Like Shannon’s entropy, the negentropies, binary information plot in a wider sense of usage. α and γ, are non-negative, but the negentropy 0 (Figure 1) depicts that in the initial phase when β ðEÞ can be positive, negative or zero, depending add: Shannon’s entropy increases then negentropy or dom- on the magnitudes of α and γ negentropies. inance decreases, but at the end Shannon’s entropy Negentropies for Simpson’s dominance (C), declines, while negentropy or dominance increases. In Simpson’s inverse dominance (D) and Brillouin’s information theory, entropy is regarded as a measure dominance (E) may be defined accordingly at β and of uncertainty. To put it instinctively, suppose pi = 0, γ levels: the event is assured never happen, and thus there is no β ðCÞ ¼ γðCÞ αðCÞ ambiguity at all, leading to entropy of zero. If pi = 1, add: the result is again definite; consequently, the entropy is β ðDÞ ¼ γðDÞ αðDÞ zero as well. In case of pi = ½, the results indicate that add: ambiguity is maximum, if one were to place a fair bet β ðEÞ ¼ γðEÞ αðEÞ: on the outcome in this case, there is no benefit to be add: added with earlier gen of the probabilities. In this case, the entropy is highest at value of 1 bit. Intermediary values falls between these cases. For example, if pi = ¼, Binary entropy and negentropy functions there is tranquil a measure of ambiguity on the result, Binary entropy function is the Shannon’s entropy of but one can still foresee the result appropriately more a 2-class variable, with the probabilities of the two frequently than not, so the ambiguity measure or states being p and q, such that q = 1 – p (Guedes entropy, is less than 1 full bit. (Figure 2) gives binary et al., 2016). The entropy of this variable is given as, information plot between the information and prob- ability of a two-class variable for Simpson’s domi- H ¼ ðp ln pþ q ln qÞ nance (C) and diversity (C’). Figure 1. Binary information plots for Shannon’s entropy and negentropy. GEOLOGY, ECOLOGY, AND LANDSCAPES 5 Figure 2. Binary information plots for Simpson’s dominance and diversity. 2 2 organized actions of the control subject eradicates C ¼ p þ q ambiguity (entropy) and it is the quantitative informa- tion measure. The removing of the ambiguity is articu- 0 2 2 C ¼ 1 p þ q lated through the alterations of conditions that executed on the system and its entropy. The protective Simpson’s C gives a convex energy curve, like that of of system structure strength is a fight against entropy, Shannon’s negentropy, whereas Simpson’s C’ gives which reveals aging process in technical systems. The a concave information curve, like Shannon’s entropy. requisite to upsurge the negentropy is the requirement (Figure 3) shows that entropy and negentropy; and to recompense the entropy as much as the entropy can diversity and dominance are negatively linearly corre- be reduced itself. The problem solution of the removal lated, and thus can be used as complementary com- ambiguity will permit to plan actions to decrease munity attributes. The diversity decreases with the entropy established on the negentropy principle of increase of dominance as shown by (Figure 3). Brillouin’s information about the structural system Dominance will be maximum when diversity alterations. A negative change in entropy refers to approaches towards unity, although diversity of com- increasing orderness thereby increasing the diversity munity consisting of more than one species will be of community. The possibility of such an occurrence maximum if all the species have an equal number of seems to break the laws of thermodynamics, but that individuals. The negentropy characterising with Figure 3. Negative correlation between binary Shannon’s entropy (H’) and negentropy (E’); and Simpson’s diversity (C’) and dominance (C) . 6 V. KUMAR ET AL. law refers to an isolated system, not each individual instead of probability (0–1). It is seen that Brillouin’s part of the system. For example, when our cells con- diversity and dominance give typical concave curve vert ADP and phosphate into ATP, the product is for diversity and convex for dominance. From (Figure more ordered than the reaction and a negative change 5), it was found that if Brillouin’s diversity increases in entropy has happened. The entire system, however, then Brillouin’s dominance decreases and both are has become more disordered because the cell has to inversely related with each other. In initial phase the convert glucose, an ordered compound, into water and Brillouin’s diversity was increased but dominance carbon dioxide. The entire system, therefore, obeys decreases and after that at the end phase Brillouin’s the second law of thermodynamics (Martin et al., diversity decreases while Brillouin’s dominance 2013). increases. (Figure 4) gives the binary information plot In order to understand the behavior of diversity between the probability, and Simpson’s diversity (con- and dominance attributes under different situations, cave) and dominance (convex) indices. Since data were created using hypothetical communities Brillouin’s index uses the number of individuals of (Tables 3–5). (Table 3) presents 5 communities, each different species instead of probability, we have plotted community having a unique species not present in any binary information function for Brillouin’s diversity other community. There is an increase in diversity due and dominance as defined in (Table 2) and (Figure 5), to Shannon’s, Simpson’s inverse information and using the number of individuals (n) on a 0–10 scale, Brillouin’s entropies in the pooled community. But Figure 4. Binary information plots for Simpson’s inverse dominance and diversity. Figure 5. Binary information plots for Brillouin’s dominance and diversity. GEOLOGY, ECOLOGY, AND LANDSCAPES 7 Table 3. Dominance and diversity indices in five single species communities consisting of different species with equal number of individuals. Number of individuals in five α communities (C1 . . . C5) γ-Community Species C1 C2 C3 C4 C5 C1 . . . C5 I 10 - - - - 10 II - 10 - - - 10 III - - 10 - - 10 IV - - - 10 - 10 V - - - - 10 10 Total ind. 10 10 10 10 10 50 No. of Spp. 1 1 1 1 1 5 Shannon’s diversity, dominance (negentropy) Div = (H’) 0 0 0 0 0 1.61 Dom = (H’ -H’) 0 0 0 0 0 0 max Div + Dom = H’ 0 0 0 0 0 1.61 max Simpson’s diversity, dominance Div = (1-∑ p ) 0 0 0 0 0 0.8 Dom = (∑p ) 1 1 1 1 1 0.2 Div + Dom = C’ 1 1 1 1 1 1 max Simpson’s inverse diversity, dominance Div = (1/∑p ) 1 1 1 1 1 5 Dom = (K-1/∑p ) 0 0 0 0 0 0 Div + Dom = K 1 1 1 1 1 5 Brillouin’s diversity, dominance Div = (H) 0 0 0 0 0 1.46 Dom = (H -H) 1.51 1.51 1.51 1.51 1.51 1.51 max Div + Dom = H 1.51 1.51 1.51 1.51 1.51 2.97 max α γ β = γ-α add. Shannon’s α, γ, β Diversity 0 1.61 1.61 Dominance 0 0 0 Div + Dom 0 1.61 1.61 Simpson’s α, γ, β Diversity 0 0.8 0.8 Dominance 1 0.2 −0.8 Div + Dom 1 1 0 Simpson’s inverse α, γ, β Diversity 1 5 4 Dominance 0 0 0 Div + Dom 1 5 4 Brillouin’s α, γ, β Diversity 0 1.46 1.46 Dominance 1.51 1.51 0 Div + Dom 1.51 2.97 1.46 as expected, the negentropy-based dominance indices flowing downstream from Beas 31.51 N, 75.28 E – did not change in Shannon’s, Simpson’s inverse and Goindwal Sahib 31.68 N, 75.13 E (about 36 km) and Brillouin’s models. then to Harike wetland 31.17 N, 75.20 E (about (Table 4) consisting of the same species in indivi- 41 km). The distances between the three sites along dual and pooled communities shows that Shannon’s the river from Beas to Goindwal Sahib and then to and Simpson’s inverse indices due to entropy and Beas, were 36 km and 41 km, respectively. Harike negentropy do not change in pooled community. If wetland a man-made lake situated at the confluence the number of species are equal in all the commu- of two rivers: Sutlej and Beas, and is Ramsar site. nities, the sum of β diversity and β dominance is zero River Beas is a relatively less polluted river and har- for Shannon’s, Simpson’s and Simpson’s inverse bors a threatened species of dolphin, Platanista indices (Table 5). (Table 6) is an example of the most minor. The quadrats of size 100 cm × 100 cm were common situation, where both the number of species laid down arbitrarily at each site. The methodology and number of individuals per species are different in applied was adapted from Kumar et al. (2018). The different communities. The sum of entropy and order of diversity indices used was Harike > Beas > negentropy for all the dominance and diversity indices Goindwal Sahib. For dominance, the order was is additive. Goindwal Sahib > Beas > Harike. The combined results of diversity and dominance indices showed the trend as Harike > Beas > Goindwal Sahib for Beta diversity along river Beas, India – A case Shannon’s diversity and Simpson’s inverse diversity, study however, for Simpson’s diversity the value remains (Table 7) gives number of individuals of species pre- constant. For Brillouin’s diversity index of domi- sent in the river-side vegetation along river Beas, nance, the trend obtained is as: Harike > Goindwal 8 V. KUMAR ET AL. Table 4. Dominance and diversity indices in five single species communities consisting of the same species with equal number of individuals. Number of individuals in five α communities (C6 . . . C10) γ-Community Species C6 C7 C8 C9 C10 C6 . . . C10 I 10 10 10 10 10 50 II - - - - - - III - - - - - - IV - - - - - - V - - - - - - Total ind. 10 10 10 10 10 50 No. of Spp. 1 1 1 1 1 1 Shannon’s diversity, dominance (negentropy) Div = (H’) 0 0 0 0 0 0 Dom = (H’ -H’) 0 0 0 0 0 0 max Div + Dom = H’ 0 0 0 0 0 0 max Simpson’s diversity, dominance Div = (1-∑ p ) 0 0 0 0 0 0 Dom = (∑p ) 1 1 1 1 1 1 Div + Dom = C’ 1 1 1 1 1 1 max Simpson’s inverse diversity, dominance Div = (1/∑p ) 1 1 1 1 1 1 Dom = (K-1/∑p ) 0 0 0 0 0 0 Div + Dom = K 1 1 1 1 1 1 Brillouin’s diversity, dominance Div = (H) 0 0 0 0 0 0 Dom = (H -H) 1.51 1.51 1.51 1.51 1.51 2.97 max Div + Dom = H 1.51 1.51 1.51 1.51 1.51 2.97 max α γ β = γ-α add. Shannon’s α, γ, β Diversity 0 0 0 Dominance 0 0 0 Div + Dom 0 0 0 Simpson’s α, γ, β Diversity 0 0 0 Dominance 1 1 0 Div + Dom 1 1 0 Simpson’s inverse α, γ, β Diversity 1 1 0 Dominance 0 0 0 Div + Dom 1 1 0 Brillouin’s α, γ, β Diversity 0 0 0 Dominance 1.51 2.97 1.46 Div + Dom 1.51 2.97 1.46 Sahib > Beas. The results clearly indicate that diver- Palaghianu (2014) developed BIODIV software and sity indices are better in contrast to dominance. compared diversity measures, and suggested that Furthermore, the application of both in combination Shannon’s and Simpson’s indices are better measures better elucidated the ecological information. (Table for diversity evaluation and opined that Simpson’s 7) shows the method of calculation, and relations indices for dominance and diversity evaluation have among alpha, beta and gamma dominance and diver- more flexibility for ecological interpretation. Justus sity indices. (2011) also opined that various diversity indices employed for the study of communities, Shannon’s and Simpson’s indices are more popular. But based Discussion on certain criteria, the author proved that Simpson’s We have attempted to relate the well-studied indices indices perform better than that of Shannon’s index. of diversity and less studied dominance indices Brillouin’s index is also extensively used as an index of through energy-entropy relations (Figure 6). (Figure diversity. Therefore, we have taken Shannon’s, 6) showed the different stages for conversion of energy Simpson’s and Brillouin’s indices to develop domi- into entropy. When energy is utilised then entropy is nance indices for our analysis at alpha, beta and equals to H’ = E’max – E’ = H’max – E’, and when gamma levels. Most of the β-diversity indices used in system shows negentropy then energy equals to literature are based on the presence or absence of E’ = E’max – H’ = H’max – H’. When entropy shows species in the samples. But we have used entropy- order, in that case the maximum energy is equals to negentropy relations which are based on both the the sum of negentropy and entropy, and when entropy number of species and number of individuals per shows disordered, then maximium entropy is equal to sample in our study. Koleff et al. (2003) listed 24 beta the sum of negentropy and entropy. diversity indices using data on the presence or absence GEOLOGY, ECOLOGY, AND LANDSCAPES 9 of species in quadrats, and recommended that the E ¼ H E max obs: indices which exhibit homogeneity property under all circumstances perform better. Attempts to measure It is seen from the study that negentropy can be inter-habitat spatial diversity between two habitats used as a measure of dominance of a community. separated by a distance (L), were made by Margalef Entropy measures the uncertainty of a system, whereas (1958) with the equation (Sherwin & Prat, 2019), negentropy measures its certainty (De Vries & Geva, 2009). H ðH þ H Þ=2 pooled 1 2 Jost (2007) stated that diversity is essentially parti- Heterogeneity ¼ tioned into alpha, beta and gamma components. The where H is the Brillouin’s entropy. H is the average three diversities are related either by additive information per individual and is given as, lawðγ ¼ αþ βÞ, or by Whittaker’s multiplicative � � lawðγ ¼ αβÞ. When the diversity indices are converted 1 N! to number equivalents, or effective number of species, H ¼ ln N N !N ! . . . N ! 1 2 K the multiplicative law is better. The best computa- tional model for Simpson’s dominance and diversity where, N is the total number of individuals of all the is additive model (γ = α + β). Simpson’s β dominance species, N is the number of individuals of different and β diversity are calculated using the relation (γ – α), species, and K is the number of species. Buzas and which can be applied to both single species and multi- Hayek (1996); Buzas & Hayek (2005) decomposed species communities. The sum of Simpson’s β dom- diversity (H) into two components, maximum and inance and diversity is zero. The sum of Simpson’s additive residual (E). The residual component is dominance and diversity for α and beta diversities is 1. a measure of evenness and an inverse measure of For communities having equal number of species, the richness. Table 5. Dominance and diversity indices in five communities consisting of equal number of species with different number of individuals. Number of individuals in five α communities (C11 . . . C15) γ-Community Species C11 C12 C13 C14 C15 C11 . . . C15 I 2 54 3 51 2 112 II 8 2 23 45 7 85 III 21 1 61 2 12 97 IV 1 27 2 1 34 65 V 5 2 6 61 2 76 Total ind. 37 86 95 160 57 435 No. of Spp. 5 5 5 5 5 5 Shannon’s diversity, dominance (negentropy) Div = (H’) 1.18 0.88 0.99 1.17 1.13 1.59 Dom = (H’ H’) 0.42 0.73 0.62 0.44 0.48 0.02 max – Div + Dom = H’ 1.61 1.61 1.61 1.61 1.61 1.61 max Simpson’s diversity, dominance Div = (1-∑p ) 0.61 0.51 0.52 0.67 0.58 0.79 Dom = (∑p ) 0.39 0.49 0.48 0.33 0.42 0.21 Div + Dom = C’ 1 1 1 1 1 1 max Simpson’s inverse diversity, dominance Div = (1/∑p ) 2.56 2.04 2.08 3.03 2.38 4.76 Dom = (K-1/∑p ) 2.44 2.96 2.92 1.97 2.62 0.24 Div + Dom = K 5 5 5 5 5 5 Brillouin’s diversity, dominance Div = (H) 1.02 0.81 0.92 1.13 1.02 1.57 Dom = 1.66 2.68 2.67 2.97 2.07 3.51 (H -H) max Div + Dom = H 2.68 3.49 3.59 4.10 3.09 5.08 max α γ β = γ-α add. Shannon’s α, γ, β Diversity 1.07 1.59 0.52 Dominance 0.54 0.02 −0.52 Div + Dom 1.61 1.61 0 Simpson’s α, γ, β Diversity 0.58 0.79 0.21 Dominance 0.42 0.21 −0.21 Div + Dom 1 1 0 Simpson’s inverse α, γ, β Diversity 2.42 4.76 2.34 Dominance 2.58 0.24 −2.34 Div + Dom 5.00 5.00 0 Brillouin’s α, γ, β Diversity 0.98 1.57 0.59 Dominance 2.41 3.51 1.10 Div + Dom 3.39 5.08 1.69 10 V. KUMAR ET AL. Table 6. Dominance and diversity indices in five communities consisting of different number of species with different number of individuals. Number of individuals in five α communities (C16 . . . C20) γ-Community Species C16 C17 C18 C19 C20 C16 . . . C20 I 2 54 3 51 2 112 II 8 2 23 - 7 40 III 21 1 61 - 12 95 IV 1 - - - 34 35 V - 2 - - 2 4 Total ind. 32 59 87 51 57 286 No. of Spp. 4 4 3 1 5 5 Shannon’s diversity, dominance (negentropy) Div = (H’) 0.90 0.38 0.72 0 1.13 1.33 Dom = (H’ H’) 0.49 1.01 0.38 0 0.48 0.28 max – Div + Dom = H’ 1.39 1.39 1.10 0 1.61 1.61 max Simpson’s diversity, dominance Div = (1-∑p ) 0.50 0.16 0.43 0 0.58 0.70 Dom = (∑p ) 0.50 0.84 0.57 1 0.42 0.30 Div + Dom = C’ 1 1 1 1 1 1 max Simpson’s inverse diversity, dominance Div = (1/∑p ) 2 1.19 1.75 1 2.38 3.33 Dom = (K-1/∑p ) 2 2.81 1.25 0 2.62 1.67 Div + Dom = K 4 4 3 1 5 5 Brillouin’s diversity, dominance Div = H) 0.78 0.32 0.67 0 1.02 1.29 Dom = (H -H) 1.77 2.81 2.83 2.99 2.07 3.38 max Div + Dom = H 2.55 3.13 3.50 2.99 3.09 4.67 max α γ β = γ-α add. Shannon’s α, γ, β Diversity 0.63 1.33 0.70 Dominance 0.47 0.28 −0.19 Div + Dom 1.10 1.61 0.51 Simpson’s α, γ, β Diversity 0.65 0.87 0.22 Dominance 0.35 0.13 −0.22 Div + Dom 1 1 0 Simpson’s inverse α, γ, β Diversity 1.66 3.33 1.67 Dominance 1.74 1.67 −0.07 Div + Dom 3.4 5 1.6 Brillouin’s α, γ, β Diversity 0.56 1.29 0.73 Dominance 2.49 3.38 0.89 Div + Dom 3.05 4.67 1.62 sum of β (dominance) and β (diversity) is zero. convex energy curves for dominance. Alpha negentro- add add Beck et al. (2013) hypothesised that rare species are pic index of dominance, α(E’), may be defined as the less represented in small sample and underestimate average dominance of M communities: their importance, thus affecting the robustness of the diversity evaluation. It is also seen that Harike wetland 0 0 αðEÞ ¼ E has higher diversity than Beas and Goindwal. Harike j¼1 wetland has more favorable conditions than the other th where E’ is negentropy of the j community. sides. Therefore, larger samples can produce better Negentropic β index of dominance, β(E’) of interpretation. It is therefore evident negentropy can a landscape, may be defined as, be used as an index of dominance for a diversity index based on entropy of the community. We can extend 0 0 0 βðEÞ ¼ γðEÞ αðEÞ the proposed negentropic dominance indices for com- munity analysis at α, β and γ levels to provide better γ(E’), is gamma negentropic dominance index of interpretation of the community structure. the landscape consisting of M communities. Shannon’s and Simpson’s indices are more consistent at beta and gamma levels. The results of a case study Conclusions on the Beas river indicate that on the basis of diversity We have proposed negentropic indices of dominance indices Harike recorded maximum diversity followed for Shannon’s entropy, Simpson’s information, by Beas and Goindwal Sahib. Based upon dominance, Simpson’s inverse diversity and Brillouin’s entropy. the Goindwal Sahib recorded maximum dominance of Binary entropy functions give concave diversity species, while Beas and Harike found low dominance. curves, whereas binary negentropy functions give However, the combined results of dominance and GEOLOGY, ECOLOGY, AND LANDSCAPES 11 Table 7. Dominance and diversity indices along River Beas downstream from Beas to Harike, India. Data on number of individuals from six quadrats (1 sq. m each) from each community. Number of individuals in three α communities γ-Community Species Beas Goindwal Harike Riverine Landscape Ageratum conyzoides L. 17 5 22 Alternanthera philoxeroides (Mart.) Grisb. 6 6 Ampelopteris prolifera (Retz.) Copel 6 6 Argemone mexicana L. 4 4 Cannabis sativa L. 3 3 Chenopodium ambrosoides L. 5 15 20 Erigeron bonariensis L. 8 8 Fumaria parviflora Lam. 10 10 Oxalis corniculata L. 5 5 Parthenium hysterophorus L. 6 6 12 Polygonum barbatum L. 3 7 10 Polygonum plebeium R.Br. 4 4 Rumex dentatus L. 10 8 18 Saccharum bengalense Retz. 6 9 15 Sesbania bispinosa (Jacq.) W.F.Wight. 106 106 Tamarix dioica Roxb. ex Roth 6 6 Total number of individuals 49 142 64 255 No. of Spp. 7 6 9 16 Shannon’s diversity, dominance (negentropy) Div = (H’) 1.80 0.96 2.09 2.16 Dom = (H’ H’) 0.15 0.83 0.11 0.61 max – Div + Dom = H’ 1.95 1.79 2.20 2.77 max Simpson’s diversity, dominance Div = (1-∑p ) 0.81 0.43 0.86 0.80 Dom = (∑p ) 0.19 0.57 0.14 0.20 Div + Dom = C’ 1 1 1 1 max Simpson’s inverse diversity, dominance Div = (1/∑p ) 5.14 1.75 7.32 4.91 Dom = (K-1/∑p ) 1.86 4.25 1.68 11.09 Div + Dom = K 7 6 9 16 Brillouin’s diversity, dominance Div = (H) 1.60 0.89 1.88 2.05 Dom = (H -H) 1.35 3.09 1.33 2.51 max Div + Dom = H 2.95 3.98 3.21 4.56 max α γ β = γ-α add. Shannon’s α, γ, β Diversity 1.62 2.16 0.54 Dominance 0.40 0.61 0.21 Div + Dom 1.98 2.77 0.79 Simpson’s α, γ, β Diversity 0.70 0.80 0.10 Dominance 0.30 0.20 −0.10 Div + Dom 1.0 1.0 0 Simpson’s inverse α, γ, β Diversity 4.73 4.91 0.18 Dominance 2.60 11.09 8.49 Div + Dom 7.33 16 8.76 Brillouin’s α, γ, β Diversity 1.46 2.05 0.59 Dominance 1.92 2.51 0.59 Div + Dom 3.38 4.56 1.18 Figure 6. Conversion of energy to entropy. E’ : Maximum energy, H’ : Maximum entropy, E’: Negentropy, H’: Entropy. max max 12 V. KUMAR ET AL. diversity indices followed the trend as Harike > Beas > Buzas, M. A., & Hayek, L. C. (2005). On richness and evenness within and between communities. Goindwal Sahib for Shannon’s diversity and Paleobiology, 31(2), 199–220. https://doi.org/10.1666/ Simpson’s inverse diversity, nevertheless, for 0094-8373(2005)031[0199:ORAEWA]2.0.CO;2 Simpson’s diversity the value remains constant. For Cody, M. L. 1975. 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Geology Ecology and Landscapes – Taylor & Francis
Published: Jan 2, 2023
Keywords: Entropy; negentropy; dominance; Shannon’s diversity; Simpson’s dominance; Brillouin’s information; binary entropy plot; beta diversity; community; phytosociology
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