Dynamics Analysis of a Discrete-Time Commensalism Model with Additive Allee for the Host Species
Round 1
Reviewer 1 Report
Comments and Suggestions for AuthorsI found this article extremely interesting and informative. The references are appropriate and, while self-referencing is indeed present, it serves the purpose of delivering context and is not excessive. I found some places in which some text editing may improve readability, as in eqn (1) lines 52 and 53, stating initial conditions and range of t, for example, $t in [0, \infty)$, same thing for eqn (2), (3) and (4). Line 115 could read "...when $\tau$ tends to $n+1$, results in..." and put a number to that equation. The equation between 117 and 118 has no number, and it should also state the range for n, for example, $n \in \{0, 1, \dots\}$ .
The paragraph between Lines 135 and 138 should be rewritten, for the sake of clarity.
Now, in the proof of theorem 2, there is a comma at the end of the equation for \Delta V(y). Here, I'd like to see an explanation of how $\Delta V(y)$ is computed since I could not follow the step between $V$ and $\Delta V$.
Line 228 has a double +.
Figure 3 (page 11) shows a remarkable similarity with Figure 1 from reference 38, so the authors may wish to comment on that.
Table 3 on page 27, please change the caption.
Figure 12, page 29, is unreadable. Please redraw or remove it. In fact, it seems that it is not referenced in the text; see lines 484 to 490, so if the authors think that it is essential, an explanation would fit in between lines 484 to 490, but then a different representation would be needed.
Author Response
Please see the attachment.
Author Response File: 
Author Response.pdf
Reviewer 2 Report
Comments and Suggestions for AuthorsNo comments
Comments for author File: Comments.pdf
Author Response
Dear reviewer, thank you for your review on the manuscript.
Reviewer 3 Report
Comments and Suggestions for AuthorsPopulation dynamics models are studied in case of the so-called commensalism model (the interaction between species is beneficial for one of them and neutral for another one). This case, compared with symbiotic or parasitic interaction, is much less studied. In addition, authors study the interaction in case of Allee effect applied to the host species. The fixed points and their investigations are described analytically and, then, numerically. Although, the applied mathematical techniques (including numerical methods) are quite standard, the obtained results are definitely new and, therefore, the research must be interesting for readers. As a further extension of the research, I would suggest to study separatrices of saddles and thus obtain homoclinic and/or heteroclinic cycles. The Hopf bifurcation can also be observed for the studied systems. Although I have no major remarks, there is quite a long list of minor remarks:
1) Abstract, line 3. ‘.’ à ‘,’
2) It looks like there is a misprint in Eq. (1). There should be b_1 instead of b_1 x.
3) Equations (1) and (2). Perhaps, it should be noticed that all the parameters must be positive.
4) Page 3. There is a variable n that is the integer part of \tau. However, the notion [\tau] is also used. Is there a reason to apply both the notions.
5) Page 3, line 93. What does the sentence ‘the same as the previous definition’ mean?
6) Page 3, line 115. ‘It’ à’it’.
7) Page 4, line 134. ‘;’ à ‘.’
8) Page 4. A new paragraph starts after Eq. (7). Probably, this is a misprint.
9) Page 4, line 34-35. Something is wrong with capital letters here.
10) Page 5, name of Section 2.2. ‘point’ à ‘points’.
11) Page 6, the beginning of the proof of Theorem 2. The function V(y) is called Lyapunov function. But in this case, one must prove explicitly that the function is positively definite.
12) Page 6, equation for \Delta V and further ones. Probably, there should be ‘\exp(2 …’ instead of ‘\exp 2…’.
13) Page 7, proof of Theorem 3. I do not understand the phrase ‘Note that the map (6) have the following map’.
14) Page 10, line 228. There is a double ‘+’ in the equation.
15) Page 12, line 255. ‘has been analysis’ à ‘has been analyzed’.
16) Page 18, line 357. What is ‘p507’?
17) Page 21, line 399. What is ‘\Theta’?
18) Page 28, line 476. ‘Lyapunov’ or ‘Lyapunov’? There are both spellings of this name in the text.
19) Summary. It is a bit strange that a survey is given here. Usually, this is done in Introduction.
All things considered, I recommend the manuscript for publication, provided the above remarks are taken into account.
See the remarks above.
Author Response
Please see the attachment.
Author Response File: Author Response.pdf
