Some Finite Difference Methods to Model Biofilm Growth and Decay: Classical and Non-Standard
Round 1
Reviewer 1 Report
The authors study numerically a nonlinear advection-diffusion-reaction model for biofilm formation. They present two new non-standard finite difference schemes to approximate numerically the solution of the mathematical model.
I have only one minor comment: It would be useful if the authors describe at least the variables of the models they present in the Introduction: what are \eta_{bc} and \et_{bf} in eq (1)? Are "u" and "v" in model (4) the same as those in model (2)?
Author Response
Response:
Comment from reviewer 1:
The authors study numerically a nonlinear advection-diffusion-reaction model for biofilm formation. They present two new non-standard finite difference schemes to approximate numerically the solution of the mathematical model.
I have only one minor comment: It would be useful if the authors describe at least the variables of the models they present in the Introduction: what are \eta_{bc} and \et_{bf} in eq (1)? Are "u" and "v" in model (4) the same as those in model (2)?
Response from authors:
We thank you for your comment. We have described all the variables in Eq. (1) as suggested. u and v in model (4) are the same as u and v in model (2). Kindly refer to page 4, line 108 of the revised manuscript.
We have revised the paper after taking into consideration all the points raised by Reviewer 1 and Reviewer 2. We read the paper few times to make sure that typos have been corrected.
Reviewer 2 Report
In this work, the authors present two novel non-standard finite difference schemes to give an approximate solution to the mathematical model of biofilm formation. The novelty of this work is to design a novel structure preserving (positivity and boundedness) finite-difference scheme for the full and limiting cases of Eq. (2). We will test the performance of our scheme against earlier known work in literature and with classical finite difference scheme. I accept publication this paper after minor comments:
- Reduce the abstract Section to be more focused on the results.
- Motivation Section should be improved to be more fitted.
- More details around the mathematical parts should be listed along the paper.
- The authors proposed many Figures, more explanations should be reported, especially, Figures 7-11.
- The authors proposed a new approach to give an approximate solution to the mathematical model of biofilm formation, why the authors did not make a comparison using some competitive models in the literature.
- What are the results when putting a random noise part in the main equation?.
- Abbreviations Section should be added in the Appendix.
- The authors should add recent references in the introduction Section.
- Please, add more explanations around Theorem 1.
- What is the difference between Equations 9 and 10.
Author Response
The authors would like to thank Reviewer 1 and Reviewer 2 for all the points raised which enabled them to considerably improve the presentation of the paper. We updated the introduction, conclusion and other parts of the paper. Kindly find below the responses from the authors.
1.Feedback from reviewer: Reduce the abstract Section to be more focused on the results.
Response from authors:
The abstract section now contains the results of the study. Kindly see page 1, line 13-15.
2. Feedback from Reviewer:
Motivation Section should be improved to be more fitted.
Response from authors:
We thank the reviewer for pointing this out. We have improved the motivation section of the manuscript. Kindly see page 4, lines 89-92 of the revised manuscript.
3. Feedback from Reviewer:
More details around the mathematical parts should be listed along the paper.
Response from authors:
Detailed information is now given around the mathematical part of the revised manuscript. Kindly see Equation (1) for example on page 2.
4. Feedback from Reviewer:
The authors proposed many Figures, more explanations should be reported, especially, Figures 7-11.
Response from authors:
We thank the reviewer for this feedback. We have added more explanation on the results obtained. Kindly refer to page 27 (line 270-272, line 279 – 281 and line 282-283) of the revised manuscript.
5. Feedback from Reviewer:
The authors proposed a new approach to give an approximate solution to the mathematical model of biofilm formation, why the authors did not make a comparison using some competitive models in the literature.
Response from authors:
We have compared results of our proposed schemes with those from classical finite difference scheme and the scheme of Sun et al. (2017). In Figure 4, we showed the performance of the scheme presented in Sun et al. (2017). Due to limitation in number of pages in the manuscript, we have compared results from our proposed scheme with results from two other schemes using tables and figures.
6. Feedback from Reviewer:
What are the results when putting a random noise part in the main equation?.
Response from authors:
In our future study, we intend to investigate the effects of putting a random noise part in the equation using a stochastic rather than a deterministic method. Kindly see the last sentence (page 27, line 308-309) in the revised manuscript.
7. Feedback from Reviewer:
Abbreviations Section should be added in the Appendix.
Response from authors:
We have added the abbreviation section as advised. Kindly see page 28, line 324-325 of the revised manuscript.
8. Feedback from Reviewer:
The authors should add recent references in the introduction Section.
Response from authors:
The introductory section now contains recent published works in page 4, line 81-86. We have added the references [10], [22] and [23].
9. Feedback from Reviewer:
Please, add more explanations around Theorem 1.
Response from authors:
We have supplied more information for Theorem 1. Kindly see page 5, line 119-120.
10. Feedback from Reviewer:
What is the difference between Equations 9 and 10.
Response from authors:
We thank you for your comment. We have now collapsed Equations 9 and 10 into one equation, since it is just our numerical discretization for space and time. Kindly see page 6 of the revised manuscript.
