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Extending the concept of entropy-negentropy for the assessment of ecological dominance and diversity at alpha, beta and gamma levels
Extending the concept of entropy-negentropy for the assessment of ecological dominance and...
Kumar, Vinod; Thukral, Ashwani Kumar; Sharma, Anket; Bhardwaj, Renu
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 firstname.lastname@example.org 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