Dynamic perspectives on biodiversity quantification: beyond conventional metrics

Simpson’s index

Simpson’s index (Simpson, 1949), introduced by Edward Hugh Simpson in 1949, is a biodiversity assessment method that incorporates both species abundance and richness when evaluating biodiversity. (1)${\displaystyle D=\sum n\left(n-1\right)/N\left(N-1\right).}$

Shannon’s diversity index

The Shannon measure (Shannon & Weaver, 1949), introduced by Claude Shannon and Warren Weaver in 1949, is another widely utilized method for assessing biodiversity. Like Simpson’s index, Shannon’s diversity considers both species richness and evenness when assessing biodiversity. (2)${\displaystyle H=-\sum _{i=1}^{s}PilnPi.}$

Brillouin’s diversity index

The Brillouin index, developed by L. Brillouin in 1962 (Brillouin, 2013), is a biodiversity measure that primarily focuses on species richness. It is defined using the formula below. (3)${\displaystyle H_B=ln\left(N!\right)-\sum ln\left(n_i!\right)/N.}$

Margalef’s diversity index

Margalef López, a Spanish ecologist, introduced a biodiversity measure in 1958 (Margalef, 1973) that emphasizes species richness. Margalef’s diversity index is particularly suitable for small sample sizes and is formulated using the following formula. (4)${\displaystyle R=\left(S-1\right)/lnN.}$

Menhinick’s index

Menhinick’s diversity index, formulated by Menhinick (1964), is based on species richness. It is calculated using the formula provided below.

Menhinick’s Index (5)${\displaystyle M=S/\sqrt{N}.}$

McIntosh diversity index

Robert P. McIntosh introduced a diversity measure in 1967 (McIntosh, 1967), which considers both species richness and evenness. McIntosh’s diversity measure utilizes the Euclidean distance measure and is calculated using the mathematical formula provided below (6)${\displaystyle D_m=\left(N-U\right)/\left(N-\sqrt{N}\right).}$

Berger-Parker’s index

Berger-Parker’s diversity index, introduced by Berger & Parker (1970), estimates the proportion of relative species abundance. It is calculated using the formula shown below (7)${\displaystyle d=N_{max}/Nord=1-N_{max}/N.}$

In the formula, N_max represents the number of individuals in the most abundant species, and N represents the total number of individuals in the sample. This diversity index focuses on species richness when assessing biodiversity. Caruso et al. (2006) experimented and found that this index is effective in monitoring biodiversity of distributed soils.

In addition to these measures, several other measures also exist such as Hurlbert’s index (Hurlbert, 1971), Fager’s Indices (Fager, 1972), Hill numbers (Hill, 1973), Keefe and Bergersen’s Index (Keefe & Bergersen, 1977). Rao’s Diversity and dissimilarity coefficients (Rao, 1982).

While diversity indices are crucial tools in ecological research, each comes with its own set of disadvantages. Simpson’s diversity index, though simple to calculate and interpret, tends to ignore rare species and operates within a limited range, which can be less intuitive. Shannon’s diversity index, despite being sensitive to both rare and common species and widely used, suffers from complex calculation and sensitivity to sample size, necessitating larger datasets for accuracy. Brillouin’s index, ideal for small, complete samples, is complex to compute and less commonly used, hindering cross-study comparisons. Margalef’s richness index, although easy to compute and effective with large samples, fails to account for species evenness and is highly sensitive to sample size. Menhinick’s index, similarly simple, also ignores evenness and is less commonly utilized, impacting comparative analyses. The McIntosh index, which considers both richness and evenness, is more robust against sample size variations but is complex to calculate and less intuitive. Berger-Parker’s index, focusing on dominance, is straightforward and useful for identifying dominant species but overlooks rare species and may mislead interpretations of diversity. Hill’s diversity index, despite its comprehensive approach, involves mathematical complexity, sensitivity to sample size, and can be difficult to interpret and apply correctly, particularly for non-specialists. Pielou’s Equitability, while normalizing evenness, inherits the Shannon index’s limitations and assumes equal sampling effort, making it contextually specific and occasionally challenging to interpret. Beta diversity indices, which are invaluable for understanding species composition variations across habitats, come with several disadvantages. One major drawback is their sensitivity to sample size and sampling effort, which can lead to biased or inconsistent results. Many indices, such as the Bray-Curtis Dissimilarity and Morisita-Horn index, require detailed species abundance data, making them resource-intensive and complex to calculate. Indices such as Jaccard and Sørensen only consider species presence or absence, ignoring abundance, which limits their informativeness in communities where abundance data are critical. Additionally, these indices may be less sensitive to rare species, potentially overlooking important ecological nuances. Hence, these combined limitations underscore the importance of developing a new mathematical model.

Scenario’s description

In the process of evaluation of biodiversity, we initially identified a few scenario’s which closely resemble the real-time situations and identified sub cases that encompass changes in species populations over specific time periods.

Scenarios:

1. Species may disappear completely.

2. Species count may increase/ decrease significantly.

3. New species may be added.

4. No change in species.

Scenario’s and associated subcases

Subsequently, we considered all possible sub-cases within each of these scenario’s. These sub-cases encompassed various events, including sudden increase, decrease or disappearance in species populations at particular moments.We have considered species namely S1, S2, S3, S4 and time periods T1 & T2.

Scenario 1: Species may disappear completely.

Subcases

1. The species (S3) present in Time T1 vanished by the time T2 arrived.

2. The species (S3) that was highly abundant in Time T1 vanished entirely by Time T2.

3. Every species present in Time T1 vanished by the time T2 arrived.

4. Certain species that appeared in T1 completely disappeared in T2, while all other species exhibited a significant increase in population size.

Scenario 2: Species count may increase/ decrease significantly.

Subcases

1. All species are high in number in both time periods T1 and T2.

2. All species are less in number in both time periods T1 and T2.

3. Species S2, which was abundant in time period T1, experienced a notable decrease in abundance during time period T2.

4. Species S2, initially scarce in number during time period T1, exhibited a substantial increase in abundance during time period T2.

5. The species that were abundant during time period T1 experienced a notable decrease in population size during time period T2.

6. The species that were initially scarce in number during T1 exhibited a substantial increase in population size during T2.

7. Certain species present in T1 underwent a proportional increase in abundance by the time T2 occurred.

8. Certain species present in T1 experienced a proportional decrease in abundance by the time T2 occurred.

9. Every species present in T1 experienced a proportional increase in abundance by the time T2 occurred.

10. Every species present in T1 underwent a proportional decrease in abundance by the time T2 arrived.

Scenario 3: New species may be added.

Subcases

1. During T2, new species that were not present in T1 have been introduced.

2. At T1, no species were present, yet they emerged in notably large numbers by T2.

3. Several species absent in T1 have emerged in significant numbers during T2.

Scenario 4: No change in species.

Subcases

1. No alterations in species composition occurred during both time periods T1 and T2.

Crafting synthetic data

Table 1 depicts the generated synthetic data tailored to scenario’s and use cases closely resembling natural settings (Díaz et al., 2020). In generating this synthetic data, we have considered different species namely S1, S2, S3, S4 and their associated counts for two different time periods T1 & T2.

Design of proposed mathematical measure

These limitations emphasize the need for robust measures to assess biodiversity. In the computation of biodiversity, it is very much essential to calculate the evenness and richness separately unlike Simpson and Shannon.

We formulated our mathematical model which is robust enough in computing the impact of biodiversity when there is a transition of species across different time periods without inclining to any factors such as species dominance and sample size. Further this mathematical model computes evenness and richness separately so that it will be very effective in differentiating the changes of species.

In line with these views, we have proposed a new mathematical model for computing biodiversity as follows. Firstly, we have calculated the deviation of each individual species in the time T1. Based on these values we have calculated the average deviation of all species. To derive the evenness, we subtracted the evenness whole value with the average deviation. The generic evenness formula is as follows: (8)${\displaystyle Evenness=\left(N_S-\sum _{i=1}^{N_S}\sqrt{{\left(1/N_S-a_i/N\right)}^2}\right)/N_S}$

where,

a_i = No. of species of ‘a’ in time period T_i

N_S = No. of different species

N = Total count of different species in T1

Now, we calculate the biodiversity value using the formula shown in Eq. (9) and by utilizing the above computed evenness value in the Eq. (8). (9)${\displaystyle Biodiversity=\frac{\sum _{i=1}^{N_S}\frac{a_i}{T_i}\ast Evenness}{N_S}}$

where,

a_i = No. of species of ‘a’ in time period T_i

T_i = Total count of the ith individual species

Repeat the above process for different time periods. Through this repetition process for different time periods (e.g.,: T1, T2), we obtained two different biodiversity values (biodiversity at time T1 & T2). The difference of these two values indicates the change in biodiversity.

The disparity in values can be understood as provided in Table 4.

To understand which individual species have made an impact on biodiversity can be calculated by using the formula given in Eq. (10). (10)${\displaystyle Biodiversity\left(k\right)=\frac{\frac{a_k}{T_i}\ast Evenness+\sum _{\genfrac{}{}{0ex}{}{i=1}{i\ne k}}^{N_S}\frac{a_i}{T_i}\ast Evenness}{N_S}.}$

Further the effectiveness of the proposed mathematical model will be evaluated in the next section by considering the same synthetic data which we considered for evaluation in Table 1.

Insensitive to the sample changes

One of the significant drawbacks of traditional methods lies in the insensitivity of both Shannon and Simpson’s measures towards sample size (Soetaert & Heip, 1990), posing a challenge in accurately assessing biodiversity. In order to be sensitive, any measure should be able to detect minor or major changes in the species. When we tried different species combinations and experimented to evaluate the sensitivity, proposed measure was effective in the following use cases.

•   A species that was abundant in time period T1 experienced a notable decrease in population size during time period T2.

•   A species that was initially scarce in number during time T1 underwent a significant increase in population size during time T2.

•   Certain species present in T1 exhibited a proportional increase in abundance during T2.

•   All species present in T1 underwent a proportional decrease in abundance during Time T2.

•   All species that were abundant during time period T1 exhibited a notable decrease in population size during time period T2.

The proposed measure can detect species changes in all circumstance. For instance, it is sensitive to sudden increase or decrease of single species in any of the time periods. Also proposed measure effectively recognises even the increase or decrease of multiple species in the given time periods.

Inclination to the dominance of few species

Another significant drawback of the traditional method is inclining to the more dominant or abundant species which is resulting in an underestimation of diversity. To eliminate dominance bias, the proposed measure aims to be free from any influence of dominant species. To validate scenarios involving dominance bias, we have examined various combinations of dominance conditions. These included use cases such as significant increase in all species populations between time periods, proportional increase across all species, non-proportional increase across all species, substantial increase in particular species while others remain nominal, few species disappearances between time periods while all other species exhibiting substantial population growth, and the significant addition of new species in any of the time periods. When tested with all these use cases proposed measure traded-off well in the following use cases when compared to the traditional methods.

•    All species that were initially scarce in number during T1 experienced a substantial increase in population size during T2.

•    Every species present in T1 experienced a proportional increase in abundance during Time T2.

•   A species that was initially scarce in number during T1 underwent a significant increase in population size during T2.

•    Certain species that appeared in T1 completely disappeared by T2, while all other species exhibited significantly large populations.

Ignoring rare species identity

Ignoring the presence of rare species is another flaw that exists in traditional methods. It is very important to consider the presence of rare species as it plays crucial role in maintaining the ecosystem diversity. However, traditional methods ignored the identity of rare species as they focused on the abundance of species. Hence these methods present challenges in detecting changes in biodiversity. The proposed measure was modelled to overcome this limitation of underestimation of rare species.

The proposed measure is examined by taking various scenarios into consideration such as addition of new species in any of the time periods, complete disappearance of any species in particular time periods, changes in the species abundance such as sudden increase in all species exists in time period T1 to T2 and sudden disappearance in all species from one time period to other. When we tried different species combinations and experimented to evaluate the identity of the rare species proposed measure was effective in the following use cases.

•    Certain species absent in T1 have emerged in large numbers during T2.

•    Any species that was highly abundant in Time T1 vanished entirely by Time T2.

•    Every species present in Time T1 vanished by the time T2 arrived.

The proposed measure for identifying rare species has demonstrated effectiveness across various scenarios, including the addition of new species, complete disappearance of certain species, and changes in species abundance over time. Through rigorous experimentation and evaluation of different species combinations, we observed compelling evidence supporting the efficacy of the proposed measure in multiple use cases to identify the rare species.

The outcomes across all scenarios consistently demonstrate the effectiveness of the proposed measure in rare species identification. Whether dealing with dominant species, varying sample sizes, or the identity of rare cases, the measure proves reliable and versatile. Its ability to accurately identify new species introductions, sudden disappearances, and broad-scale changes in species abundance underscores its importance in biodiversity conservation efforts. By providing a robust framework that transcends ecological complexities, the proposed measure significantly contributes to advancing conservation strategies, ensuring the preservation of precious biodiversity for generations to come.