Analytical Solutions of Microplastic Particles Dispersion Using a Lotka–Volterra Predator–Prey Model with Time-Varying Intraspecies Coefficients

Round 1

Reviewer 1 Report

This is an interesting paper to retrieve the Lotka-Volterra model with time-varying intraspecific coefficients for interpreting ecological quantities referring to microplastics dispersion in predator-prey system.  The manuscript has been well written.  I have a few of comments to let authors improve the manuscript.

1.     The title should include “predator and prey, or predator-prey system”, which is crucial to this manuscript.

2.     In Abstract and Introduction, the objective, or research quation of this study should be clearly described.

3.     In Materials and Methods, after introducing Huang et al. (2020) LVM model with three paragraphs, authors should provide the methodology and analysis approach of your own equations in detail. Some of them might have been introduced in Results part.

4.     In Results, some methodological approaches by modifying Huang et al. can be moved up to Materials and Methods. In Results section, you can focus on your results’ pattern.

5.     Discussion can (1) be extended to compare your results and Huang et al. (2020) results, with emphasizing your new discovery and findings, (2) cite more references for such comparison and discussion.  This section, only two references are cited, which is far away enough.

6.     In conclusion, you may want to offer some new research questions relating to this paper and Huang et al. (2020) to expand or deepen potential directions.

Author Response

This is an interesting paper to retrieve the Lotka-Volterra model with time-varying intraspecific coefficients for interpreting ecological quantities referring to microplastics dispersion in predator-prey system. The manuscript has been well written. I have a few of comments to let authors improve the manuscript.

We thank the Reviewer for the thorough reading of our manuscript and to the issues raised, that certainly contributed to a better version of the text. Apart many typo corrections to mention, the modifications are in bold face in the text.

1. The title should include ``predator and prey, or predator-prey system'', which is crucial to this manuscript.
       
       OK. We have included the term ``predator-prey'' in the title.
     
2. In Abstract and Introduction, the objective, or research quation of this study should be clearly described.
       
       OK. We have improved the description of our objectives in these sections. We added the following sentence in the end of the sixth paragraph of introduction.
       
       We rewrite the Huang et al. four equations model, reducing it to a two equations equivalent one and include the time-varying intraspecies coefficients. This approach clarify the model parameters meanings and allows us to solve analytically three special cases of ecological relationships. These special cases are based on possible extreme effects of predatory performance reduction caused by exposure to MP particles and are mathematically characterized by the decoupling of the differential equations of the model, for which we also perform numerical simulations. We also propose a second-order differential equation as a possible next step to address this model.
3. In Materials and Methods, after introducing Huang et al. (2020) LVM model with three paragraphs, authors should provide the methodology and analysis approach of your own equations in detail.  Some of them might have been introduced in Results part.
       
       OK. Our detailed methodological approach has improved. The detailed description has been moved from Results to Materials and Methods. We believe this improves the legibility of the manuscript.
      
4. In Results, some methodological approaches by modifying Huang et al. can be moved up to Materials and Methods. In Results section, you can focus on your results’ pattern.
       
       OK. Please, see item 3. In Results, we have focused only on the special ecological regimes.
5. Discussion can (a) be extended to compare your results and Huang et al. (2020) results, with emphasizing your new discovery and findings,
           
           OK, we added in the end of discussion section.
   It is important to emphasize that the underlying structure of the model is exponential and the three special situations that we studied are limit cases, which lead to very different predictions from the original study and result in the extremely large numbers of organisms shown in the Figs. 1, 2 and 3.
           
           Please, see response to main comment of Reviewer 2.  
           
            (b) cite more references for such comparison and discussion.
           This section, only two references are cited, which is far away enough.}
           
           OK. We have added two more citations (18 and 19) and more references in this section.
       
   6. In conclusion, you may want to offer some new research questions relating to this paper and Huang et al. (2020) to expand or deepen potential directions.
       
       Indeed, after the submission of this manuscript we have noticed that the model can be simplified even more. Since it points a very sharp aspect of the model we preferred to address in a future manuscript.

Reviewer 2 Report

The manuscript is generally well-structured. However, it needs revisions and/or clarifications before publication. I have one major issue and several minor issues for this manuscript.

The major issue is about the large numbers obtained in all Figures 1, 2, 3. In Figure 1, for example, the curve shows extremely large numbers 10^19 and 10^14. These numbers do not reflect the previous results of Huang et al. Please explain and do recheck if these are correct. It seems strange that these numbers are extremely large!

Minor issues are as follows:

1. The word PYTHON should be Python. Please revise.

2. Please state explicitly in the manuscript, what algorithms (or what methods) in the Python software that you use for solving the systems of ODEs and/or your equations.

3. Please make sure that the complete name and the complete affiliation in the Acknowledgments are all correct.

4. Reference number [30]: please write the year it is published.

Thank you.

Author Response

The manuscript is generally well-structured. However, it needs revisions and/or clarifications before publication. I have one major issue and several minor issues for this manuscript.

We thank the Reviewer for the thorough reading of our manuscript and to the issues raised, that certainly contributed to a better version of the text. Apart many typo corrections to mention, the modifications are in bold face in the text.

The major issue is about the large numbers obtained in all Figures 1, 2, 3. In Figure 1, for example, the curve shows extremely large numbers $10^{19}$ and $10^{14}$. These numbers do not reflect the previous results of Huang et al. Please explain and do recheck if these are correct. It seems strange that these numbers are extremely large!

OK. We have rechecked our results and in the cases addressed by Huang et. al., we retrieve exactly the same values as them.  It is important to emphasize that the underlying structure of the model is exponential and the Figures 1, 2, and 3 refer to limiting situations, which the exponential behavior dominates, leading to large numbers. These comments have been included in the section Discussion. This is a very nice contribution to a better understanding of the manuscript. We have added the following comment in the end of the last paragraph of discussion section. 

It is important to emphasize that the underlying structure of the model is exponential and the three special situations that we studied are limit cases, which lead to very different predictions from the original study and result in the extremely large numbers of organisms shown in the Figs. 1, 2 and 3.

Minor issues are as follows:

    1. The word PYTHON should be Python. Please revise.   OK.
    
    2. Please state explicitly in the manuscript, what algorithms (or what methods) in the Python software that you use for solving the systems of ODEs and/or your equations.
    
    OK. We have included at the beginning of the section Results the Python function, package, and library that we used to implement the model.
     
   Implementing the model of Eqs.~(10) using the SciPy Python library, with the function odeint from Scipy.integrate package, and considering the five scenarios described in the previous section, we successfully reproduced the phase-portraits and the population dynamics graphs of Huang et al.
    
    3. Please make sure that the complete name and the complete affiliation in the Acknowledgments are all correct. OK.
    
    4. Reference number [30]: please write the year it is published. OK