Extending the concept of entropy-negentropy for the assessment of ecological dominance and diversity at alpha, beta and gamma levels

Figures & data

Table 1. Some beta diversity indices from PAST software (Hammer et al., 2001).

Whittaker (1960)	$\frac{K}{\alpha }-1$
Harrison et al. (1992)	$\frac{\frac{K}{\alpha }-1}{N-1}$
Cody (1975)	$\frac{g\left(S\right)+l\left(S\right)}{2}$
Wilson-Shmida (1984)	$\frac{g\left(S\right)+l\left(S\right)}{2\alpha }$
Mourelle and Ezcurra (1997)	$\frac{g\left(S\right)+l\left(S\right)}{2\alpha \left(N-1\right)}$
Harrison et al. (1992)	$\frac{\frac{K}{{\alpha }_{max}}-1}{N-1}$
Williams (1996)	$1-\frac{{\alpha }_{max}}{K}$

Table 2. Negentropy measures of four common indices used in ecological diversity and dominance.

Measure	Entropy as Diversity	Negentropy as Dominance	Maximum entropy
Shannon’s	$H{ }^{\mathrm{\prime }}=-\sum _{i=1}^K{p}_{i}ln{p}_i$	$E{ }^{\mathrm{\prime }}=lnK-\sum _{i=1}^K{p}_{i}ln{p}_i$	$lnK$
Simpson’s	$C{ }^{\mathrm{\prime }}=1-\sum _{i=1}^K{p}_i^2$	$C=\sum _{i=1}^K{p}_i^2$	1
Simpson’s inverse	$D^{\prime }=\frac{1}{{\sum }_{i=1}^K{p}_i^2}$	$D=K-\frac{1}{{\sum }_{i=1}^K{p}_i^2}$	K
Brillouin’s	$H=ln\frac{N!}{n_1!n_2!\dots n_K!}$	$E=\frac{lnN!}{N}-ln\frac{N!}{n_1!n_2!\dots n_K!}$	$\frac{lnN!}{N}$

Figure 1. Binary information plots for Shannon’s entropy and negentropy.

Figure 2. Binary information plots for Simpson’s dominance and diversity.

Figure 3. Negative correlation between binary Shannon’s entropy (H’) and negentropy (E’); and Simpson’s diversity (C’) and dominance (C) .

Figure 4. Binary information plots for Simpson’s inverse dominance and diversity.

Figure 5. Binary information plots for Brillouin’s dominance and diversity.

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’max -H’)	0	0	0	0	0	0
Div + Dom = H’max	0	0	0	0	0	1.61
	Simpson’s diversity, dominance
Div = (1-∑ pi^2)	0	0	0	0	0	0.8
Dom = (∑pi^2)	1	1	1	1	1	0.2
Div + Dom = C’max	1	1	1	1	1	1
	Simpson’s inverse diversity, dominance
Div = (1/∑pi^2)	1	1	1	1	1	5
Dom = (K-1/∑pi^2)	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 = (Hmax-H)	1.51	1.51	1.51	1.51	1.51	1.51
Div + Dom = Hmax	1.51	1.51	1.51	1.51	1.51	2.97
	α		γ		βadd. = γ-α
	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

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’max -H’)	0	0	0	0	0	0
Div + Dom = H’max	0	0	0	0	0	0
	Simpson’s diversity, dominance
Div = (1-∑ pi^2)	0	0	0	0	0	0
Dom = (∑pi^2)	1	1	1	1	1	1
Div + Dom = C’max	1	1	1	1	1	1
	Simpson’s inverse diversity, dominance
Div = (1/∑pi^2)	1	1	1	1	1	1
Dom = (K-1/∑pi^2)	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 = (Hmax-H)	1.51	1.51	1.51	1.51	1.51	2.97
Div + Dom = Hmax	1.51	1.51	1.51	1.51	1.51	2.97
	α		γ		βadd. = γ-α
	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

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’max – H’)	0.42	0.73	0.62	0.44	0.48	0.02
Div + Dom = H’max	1.61	1.61	1.61	1.61	1.61	1.61
	Simpson’s diversity, dominance
Div = (1-∑pi^2)	0.61	0.51	0.52	0.67	0.58	0.79
Dom = (∑pi^2)	0.39	0.49	0.48	0.33	0.42	0.21
Div + Dom = C’max	1	1	1	1	1	1
	Simpson’s inverse diversity, dominance
Div = (1/∑pi^2)	2.56	2.04	2.08	3.03	2.38	4.76
Dom = (K-1/∑pi^2)	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 =  (Hmax-H)	1.66	2.68	2.67	2.97	2.07	3.51
Div + Dom = Hmax	2.68	3.49	3.59	4.10	3.09	5.08
	α		γ		βadd. = γ-α
			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

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’max – H’)	0.49	1.01	0.38	0	0.48	0.28
Div + Dom = H’max	1.39	1.39	1.10	0	1.61	1.61
	Simpson’s diversity, dominance
Div = (1-∑pi^2)	0.50	0.16	0.43	0	0.58	0.70
Dom = (∑pi^2)	0.50	0.84	0.57	1	0.42	0.30
Div + Dom = C’max	1	1	1	1	1	1
	Simpson’s inverse diversity, dominance
Div = (1/∑pi^2)	2	1.19	1.75	1	2.38	3.33
Dom = (K-1/∑pi^2)	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 = (Hmax-H)	1.77	2.81	2.83	2.99	2.07	3.38
Div + Dom = Hmax	2.55	3.13	3.50	2.99	3.09	4.67
	α		γ		βadd. = γ-α
	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

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’max – H’)	0.15	0.83		0.11	0.61
Div + Dom = H’max	1.95	1.79		2.20	2.77
	Simpson’s diversity, dominance
Div = (1-∑pi^2)	0.81	0.43		0.86	0.80
Dom = (∑pi^2)	0.19	0.57		0.14	0.20
Div + Dom = C’max	1	1		1	1
	Simpson’s inverse diversity, dominance
Div = (1/∑pi^2)	5.14	1.75		7.32	4.91
Dom = (K-1/∑pi^2)	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 = (Hmax-H)	1.35	3.09		1.33	2.51
Div + Dom = Hmax	2.95	3.98		3.21	4.56
	α		γ		βadd. = γ-α
	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’_max: Maximum energy, H’_max: Maximum entropy, E’: Negentropy, H’: Entropy.

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