Round  1

Reviewer 1 Report

The present study deals with investigating two different indexes for biodiversity. There are two main remarks of mine regarding the study: 1) The formulas and mathematics are quite simple and this was admitted by the authors. 2) Nevertheless the comparison of two indexes can have some potential impact for problems in ecology. Mainly because of these two I would consider this paper to be evaluated more on its ecological background rather than its mathematical achievements. Hence, I would not recommend it for publication in mathematics and advice authors to submit it to more ecologically-oriented journal. My recommendation can be also supported by their references: majority of them are from ecological field. 

One of my minor remarks would be that although the authors say that they review previous findings, I found their review to be a bit narrow. They city mainly their own previous works, and they don't do any parallels with studies of diversity in microbiomes. However, there were plenty of recent works there and often they dealt with similar problems. For example, see Bashan et al Liu 2016 Nature (https://www.nature.com/articles/nature18301) 

Author Response

The present study deals with investigating two different indexes for biodiversity. There are two main remarks of mine regarding the study:
1) The formulas and mathematics are quite simple and this was admitted by the authors. 2) Nevertheless the comparison of two indexes
can have some potential impact for problems in ecology. Mainly because of these two I would consider this paper to be evaluated more on
its ecological background rather than its mathematical achievements. Hence, I would not recommend it for publication in mathematics and
advice authors to submit it to more ecologically-oriented journal. My recommendation can be also supported by their references: majority of them are from ecological field.

Answer: Our submission is to the special issue of Mathematics, devoted to the application of mathematical methods to ecology.
In this sense, our paper merely demonstrates how mathematical methods, namely special indices, can be accomodated to the analysis of fundamental ecological problems . We hope that our approach attracts attention of mathematicians who are interested the theory of similarity and dissimilarity indices.

One of my minor remarks would be that although the authors say that they review previous findings, I found their review to be a bit narrow.

Answer: actually, we do not say that we wish to "review previous findings" -  that would be far beyond the scope of the manuscript - and also
far beyond the scope of the journal.

They city mainly their own previous works, and they don't do any parallels with studies of diversity in microbiomes. However, there were plenty of recent works there and often they dealt with similar problems.
For example, see Bashan et al Liu 2016 Nature (https://www.nature.com/articles/nature18301)

Answer: The paper suggested deals with a completely different problem: not because their study object is microbiological, but due to the
different mathemtical approach and objective. They have the finding that "These results fundamentally improve our understanding of the processes that shape human microbial ecosystems,
and pave the way to designing general microbiome-based therapies." We think that the we provide a sufficiently wide picture on the subject mater we are discussing.

Reviewer 2 Report

In this paper authors define taxonomic, phylogenetic and functional beta diversity of species assemblages based on the generalized Jaccard dissimilarity index. This coefficient does not give equal weight to species, because traditional site dissimilarities are lowered by taking into account the taxonomic, phylogenetic or functional similarity of differential species in one site to the species in the other. These, together with the traditional, taxon- (species-) based beta diversity are decomposed into two additive fractions, one due to taxonomic, phylogenetic or functional excess and the other to replacement. In addition to numerical results, taxonomic, phylogenetic and functional community structure is visualized by 2D simplex or ternary plots. Redundancy with respect to taxon-based structure is expressed in terms of centroid distances between point clouds in these diagrams. The approach is illustrated by examples coming from vegetation surveys representing different ecological conditions. They found that beta diversity decreases in the following order: taxon-based, taxonomic (Linnaean), phylogenetic and functional. Therefore, they put forward the Beta-redundancy Hypothesis suggesting that this ordering may be most often the case in ecological communities, and discuss potential reasons and possible exceptions to this supposed rule. Whereas the pattern of change in diversity may be indicative of fundamental features of the particular community being studied, the effect of the choice of functional traits – a more or less subjective element of the framework – remains to be investigated.

The subject of the paper is interesting. I do not have deep knowledge about the subject so I can not evaluate its novelty. If the editor finds the subject of the paper fit to the scope of this journal, I suggest accepting.

Author Response

Thanks for your support.

Reviewer 3 Report

1.    The Introduction section is too long. It should be halved.

2.    Authors must specify which is the clear motivation for weight forms from (4a) and (4b).

3.    Why do the coefficients S, D and R have the expressions from (10) - (12)?

4.    In the manuscript there are many references to relations and results from some papers already published, for instance, in the form Ricotta et al.. So, I think the authors need to emphasize more clearly the contribution of the manuscript from a scientific point of view.

5.    References are not uniformly written. In some references the name of the  journal  is written in full and in others it is abbreviated.

6.    Authors  must strengthen the References section with some articles which use similar techniques, for instance

-       Contributions on uniqueness in thermoelastodynamics on bodies with voids, Cienc. Mat.(Havana) 16 (2), 101-109, 1998

-       A temporally evolutionary equation in elasticity of micropolar bodies with voids, U.P.B. Sci. Bull., Series A-Applied Mathematics Physics, vol. 60 (3-4), 3-12, 1998

-       Weak Solutions in Elasticity of Dipolar Porous Materials, Math Probl Eng, Vol. 2008, Art. No. 158908, pp. 1-8, 2008

Author Response

1.    The Introduction section is too long. It should be halved.

Answer. The Introduction is only 2 and a half manuscript pages and contains information absolutely necessary
for understadning the next sections of the paper.

2.    Authors must specify which is the clear motivation for weight forms from (4a) and (4b).

Answer: These two equations express the same thing Izsak and Price expressed only verbally - so we feel that
the half-sentence "Equation (3) can be written as" is sifficient here.

3.    Why do the coefficients S, D and R have the expressions from (10) - (12)?

Answer: These are generalizations of the forms appearing in Equation (9) such that new terms appear in the
numerators, as defined in equation (5)

4.    In the manuscript there are many references to relations and results from some papers already published, for instance, in the form Ricotta et al..
So, I think the authors need to emphasize more clearly the contribution of the manuscript from a scientific point of view.

Answer: Ricotta et al. (in fact, ourselves) introduced the basic formula, the generalization of the Jaccard index. The novelty in this paper is its application to simplex analysis in order to decompose taxonomic, phylogenetic and functional diversity. we hope that the following passages from the introduction are sufficiently clear: "We show, further, that these generalized dissimilarities may be partitioned into additive fractions  that have similar interpretation as those of species-based dissimilarities, and extend the functional approach suggested by Ricotta et al. [17] to taxonomic and phylogenetic data. Following the framework originally suggested by Podani and Schmera [6], all pairwise fractions in a given set
of plots (sites etc.) are then considered for illustrating community structure graphically, in a ternary plot. Simple artificial examples and three sets of actual vegetation data will be used to show how the original, species-based method modifies when the taxonomic, phylogenetic and functional distinctness of species is also incorporated into the analysis."

5.    References are not uniformly written. In some references the name of the  journal  is written in full and in others it is abbreviated.

Answer: Thank you. We have now standardized the references. There is only one journal name that remains in full: an old journal which has no abbreviated name.

6.    Authors  must strengthen the References section with some articles which use similar techniques, for instance

-       Contributions on uniqueness in thermoelastodynamics on bodies with voids, Cienc. Mat.(Havana) 16 (2), 101-109, 1998

-       A temporally evolutionary equation in elasticity of micropolar bodies with voids, U.P.B. Sci. Bull., Series A-Applied Mathematics Physics, vol. 60 (3-4), 3-12, 1998

-       Weak Solutions in Elasticity of Dipolar Porous Materials, Math Probl Eng, Vol. 2008, Art. No. 158908, pp. 1-8, 2008

Answer: These references are irrelevant to our study. For instance, the last paper hes the following abstract:
"The main aim of our study is to use some general results from the general theory of elliptic equations in order to obtain some qualitative
results in a concrete and very applicative situation. In fact, we will prove the existence and uniqueness of the generalized solutions for the
boundary value problems in elasticity of initially stressed bodies with voids (porous materials)."

Round  2

Reviewer 1 Report

First of all, I am sorry but I don't remember seeing that the present study is for the special issue of mathematics for applications in ecology. I will review the manuscript based on this ground. In this respect, I would classify the study as specialized with interest to researchers working in that field, and with rather limited interest of a general readers. One of the reason is that the manuscript contains a specific terminology even starting from the first sentence and it does not facilitate the  reading for someone who is not familiar with it. Although the manuscript is technically sound. Based on this, I would recommend the manuscript to be considered for publication in the special issue.

Altjough it was emphasized by the authors that many studies in human microbiom field are similar but deal with a different aspect of complexity, I wish to see this reflected in the manuscript. Every individual can be regarded as a "site", bacteria in a gut as "species", so the question how to characterize dissimilarity or universality between different individuals naturally arises. The authors can give few sentences on this matter in their manuscript.

Technical remark:

- all variables in the text and in formulas should be written in italic

Author Response

Yes, it is indeed true that communities can be conceived in many different ways, also in microbiology. There are quite many areas which could also be mentioned in this regard - but our discussion is valid only to cases when there is taxonomy, phylogeny, functionality of the attributes. Thus, we

modified one sentence in the last section of the paper to include microbiology as well:

Of course, further studies representing animals, woody vegetation, deserts, aquatic and microbial communities etc. are needed to confirm or reject the above hypothesis.

The symbols are now given in italics in the text and the formulae.

Reviewer 3 Report

No more comments

Author Response

Thank you.