On Large and Small Data Blow-Up Solutions in the Trojan Y Chromosome Model
, Matthew A. Beauregard 2, Thomas Griffin 3, Landrey Bobo 4 and Rana D. Parshad 3,*Round 1
Reviewer 1 Report
In the current manuscript, the authors investigate a three species TYC reaction diffusion model and, by restricting the parameters and initial data regimes, they still obtain the biologically meaningful solutions for this model. As a result, the authors also discuss the practical relevance of their results to biological control and possible restrictions.
Assuming that all computations are correct, the main results seem to be true and present some interest for the researchers working in the fields of ecology or differential equations. In fact, the topic of the paper is quite interesting and the paper is well organized. Therefore, I recommend this manuscript to be published in Axioms after making the following minor revisions:
1. Page 1, in the first line of the Abstract section, ``is'' should be ``are".
2. Page 1, in the eleventh line of the Abstract section, ``a'' should be ``an".
3. Page 3, in the fifth line of the first paragraph, ``always'' should be consistent with other word formats.
4. Page 4, in the last line of the Negative Solutions section, ``not'' should be consistent with other word formats.
5. Page 6, in the fifth line of the second paragraph, ``is assume" should be preceded by ``assumes".
6. Page 7, in the first line of the Remark 3 section, ``is " should be preceded by ``are".
7. Page 7, in the fourth line of the Remark 3 section, there should be a space between ``blow " and ``up".
8. Page 8, ``specified " should not be preceded by a space in the fifth line.
9. The data given below figure 2, figure 3, figure 4, figure 5 and figure 7 are incomplete.
10. Page 13, in the tenth line of the Discussion and Conclusion section, the value of ``f(0)=m(0)'' given is incomplete.
11. Page 14, in the second line of the first paragraph and the second line of the second paragraph, ``-'' should be deleted.
12. For reference, the name format of the authors should be consistent, for example:
(1). For reference 2, there should be a space between the author's first name and last name.
(2). For reference 4, the second and third author names should be in the same format.
(3). For reference 6 and reference 18, there should be ``and'' between names.
(4). For reference 11, the authors's last name and first name should be in the same order.
(5). For reference 31, there shouldn't be a comma before ``and''.
13. For reference 17, the page number format is incorrect and ``2020'' should not be in parentheses.
14. For reference 18, the year should be at the end.
Author Response
please see attached letter
Author Response File: 
Author Response.pdf
Reviewer 2 Report
The paper deals with a mathematical model of the Trojan Y Chromosome Strategy (TYC), which is used as a genetic biological method for controlling invasive populations with an XX-XY sex determinism. The detrimental impact of invasive species on native biota is well-known. Traditionally, control methods rely on chemical treatment, which harms the environment, including species we want to protect. Therefore, instead of using poisons, it is better to use biological methods, where a species is released to control the invasive population by predation, competition, disease, or manipulating the mating system. However, the practical application of such control methods also carries potential risks and should have a scientific basis. This, in particular, requires developing and applying mathematical and computer modeling methods. Therefore, the research is essential and relevant.
The article considers two mathematical models: "the classical model", which is an ODE system, and "the PDE model", based on a second-order parabolic PDEs system. For "the PDE model", the authors proved statements about the properties of solutions; in particular, they obtained the conditions of positivity (and negativity), and the conditions of blow-up in finite time. A computational experiment was performed for "the classical model", which allowed one to obtain additional information about the properties of solutions.
With a generally good impression of the article, we have several significant comments and questions.
- We have not found a clear justification in the article that "the PDE model" is correct. The principles that guided the authors of the model are omitted. Moreover, it is unclear whether the authors developed the model or it is known. In the first case, you should describe the construction of the model, in the second case, compare your results with the known ones. Besides, the choice of the initial-boundary condition for "the PDE model" is also not justified. For example, what is the meaning of Neumann boundary conditions in this case from the point of view of biology?
- In our opinion, the authors emphasize considering the biological problem without having sufficient grounds for this. As you know, any model is not identical to a real object and has limits of applicability. In this context, formulations like "The male population is negative" look unconvincing. Obviously, here we can only speak about the negative solution to the differential equation, which shows that the model seems to be inapplicable in this case. We have not found any examples of using the model to study specific invasive populations in the article. Thus, the work is not directly related to biology. The biological objects should be mentioned only in motivation and discussion, and the central part should be about mathematical objects (differential equations) and their properties.
- The paper presents strictly proven mathematical statements, the correctness of which does not raise questions. But are they really new? The extensive reference list does not include the classical monograph by Ladyzenskaja, O.A.; Solonnikov, V.A.; Ural’ceva N.N. Linear and Quasi-Linear Equations of Parabolic Type, 1988, in which such statements were considered. The authors should clearly explain the difference between their results and the known ones.
- The computational experiment for "the PDE model" is incomplete and illustrative. It is impossible to draw any meaningful conclusions from these calculations. The vast majority of calculations are made for "the classical model". Thus, there is a gap between the theoretical and computational parts of the paper.
There are also less essential shortcomings that also need to be addressed.
- The introduction is replete with group references. The number of works that are mentioned at the same time reaches 17. If the same thing is written in these works, why are they all given? If they differ, then how?
- There are carelessness and typos in the article. For example, in the formulation of Lemma 2.2 there is a reference to "the system (2.1)", but there is no such system in the work.
If the authors give convincing answers to the questions and correct the comments, the article can be re-reviewed.
Author Response
please see attached letter
Author Response File: Author Response.pdf
Round 2
Reviewer 2 Report
The authors have done a good job, and the paper has become much better. They have fixed our comments and given convincing answers the questions. Now we can recommend accepting the paper. The last thing should be done: please, use MDPI template to format the paper.
Author Response
The manuscript has been edited in MDPI template
