Hopf Bifurcation Analysis of a Diffusive Nutrient–Phytoplankton Model with Time Delay
Round 1
Reviewer 1 Report
The topic of research is relevant since the results can be used in the study of the phytoplankton population. The authors propose a dynamic model of phytoplankton with a delay with a diffusion component and study its qualitative properties: Hopf bifurcation and diffusion stability. These studies are of great practical importance. However, the following remarks may be made to the work:
- Literature analysis is poorly presented, the most recent source, according to the authors, dates back to 2014. At least there are recent works, for example, Mathematical analysis of a nutrient–plankton system with delay (2016) and I think that there are more works on this topic.
- In system (1.3) there are no explanations for the quantities ΔP and ΔN .
- Most of the theorems and lemmas are given without proof (Lemma 2.1, Theorem 2.1., Theorem 2.3, Theorem 3.1).
- The article lacks phase trajectories and bifurcation diagrams, especially since the authors study the Hopf bifurcation.
- Section 4 presents a numerical analysis, but the authors do not indicate how the system (4.1) was solved numerically or analytically.
- There are also no examples of diffusion resistance.
- The list of references needs to be refreshed, there are sources of literature that are fresher than the authors give (see remark 1).
Based on the above, the article needs to be improved.
Author Response
1.Literature analysis is poorly presented, the most recent source, according to the authors, dates back to 2014. At least there are recent works, for example, Mathematical analysis of a nutrient–plankton system with delay (2016) and I think that there are more works on this topic.
Reply: Thanks for your comment and recommend the literature. We have modified the manuscript and added some new references (page: 2 and 23).
2.In system (1.3) there are no explanations for the quantities ΔP and ΔN .
Reply: Thanks for your comment. We have modified the manuscript and added the explanation (page: 2).
3.Most of the theorems and lemmas are given without proof (Lemma 2.1, Theorem 2.1., Theorem 2.3, Theorem 3.1).
Reply: Thanks for pointing out this. We have added the proof for Theorem 2.1 (page: 6), Theorem 2.3 (page: 9). The Lemma 2.1 is just a result in (or a part of) the reference [18], and has been proofed. So we omit the proof. Similarly, the Theorem 3.1 is also a conclusion of the reference [19], which is a standardized procedure. But we added a note to indicate the origin of Lemma 2.1 and Theorem 3.1.
4.The article lacks phase trajectories and bifurcation diagrams, especially since the authors study the Hopf bifurcation.
Reply: Thanks for your suggestion. We have rewritten the numerical simulations section and added the bifurcation diagrams (page: 18-21). We considered the effect of predator’s diffusion coefficient, release rate of toxic chemicals and time delay on the stability of coexisting equilibrium. Unfortunately, we couldn't draw the phase diagram since the N(x,t) and P(x,t) are binary functions. We will try to solve this problem after further research.
5.Section 4 presents a numerical analysis, but the authors do not indicate how the system (4.1) was solved numerically or analytically.
Reply: Thanks for pointing out this. We have added this (pages:18 and 19). Especially, the numerical simulation of the systems with $\tau=0$ is implemented by pdepe function in Matlab, and $\tau>0$ is implemented by the finite-difference methods.
6.There are also no examples of diffusion resistance.
Reply: Thanks for your suggestion. We have rewritten the numerical simulations section and added the effect of diffusion (page: 19 and 20). Especially, when d2> d^∗_2, then the Turing instability of (N∗,P∗) may occur. When d2 > d^∗_2, then a > a− holds which implies (N∗,P∗) is locally asymptotically stable.
7.The list of references needs to be refreshed, there are sources of literature that are fresher than the authors give (see remark 1).
Reply: Thanks for your comment and recommend the literature. We have modified the manuscript and added some new references (page: 2 and 23).
Reviewer 2 Report
Please, find comments/suggestions in the attached pdf file.
Comments for author File: 
Comments.pdf
Author Response
- Page 3. The sentence “Then system(1.3) becomes the following form” should be written as “The corresponding problem has the following form”. Moreover, is the any particular reason for the absence of a space in phases such as “system(1.3)” throughout the manuscript?
Reply: Thanks for pointing out this error. We have corrected this (pages:3). There is no reason for the absence of a space, just an error.
2.Page 3, equation (1.4), last sentence. θ should be replaced by t, since θ is an already used parameter and should not be mistaken with the time variable t.
Reply: Thanks for pointing out this. We have corrected it (pages:3).
3.Page 3. Is there any particular reason for expressing point (N0, P0) as E(N0, P0)? Moreover, this point should be written as (N∗, P∗) instead, because N0 and P0 are used for the initial data.
Reply: Thanks for pointing out this error. We have corrected this (pages:3). The positive equilibrium has been uniformly denoted as (N∗, P∗).
4.Page 3. Many full stops are missing from the manuscript, such as after the equation inside Lemma 2.1.
Reply: Thanks for pointing out this. We have revised the manuscript to void these mistakes .
5.Page 4. Instead of “In the following, we just assume system (1.4) has a positive equilibrium, and denote it as E(N0, P0)”, it should be written that “In what follows, we assume that 0 < ac − de ≤ bθ and we study the stability of the constant solution (N, P) ≡ (N∗, P∗) of problem (1.3) for (N0, P0) ≡ (N∗, P∗)”.
Reply: Thanks for pointing out this. We have modified this (pages 4).
6.Page 4. P2(t) = . . . should be written as N2(t) = . . . .
Reply: Thanks for pointing out this. We have modified this (pages 4).
- Page 4. Equation (2.1) should be written as “ ˙N = (DΔ+L)N”. Moreover, “L(ϕt) = L1ϕ(0)+L2ϕ(−τ )” should be written as “Lϕ(·) = L1ϕ(·) + L2ϕ(· − τ )”, and also “ϕ(t) = (ϕ1(t), ϕ2(t))T , ϕ(t)(·) = (ϕ1(t + ·), ϕ2(t + ·))T ” should be erased.
Reply: Thanks for pointing out this. We have modified this (pages 4).
- Page 4, equation (2.2). Instead of D = cβP0(P20−μ^2)/(μ2+P20 )^2 , it should be written that D = c(-α+2βP0(P20−μ^2)/(μ2+P20 )^2)
Reply: Thanks for your comment. In fact, D is the value at the positive equilibrium (N∗, P∗), and we have simplified D by the relationship between N∗ and P∗ (−α + N∗−βP∗/(µ^2+P∗^2)= 0). So D can also be rewritten as D = cβP*(P*^2−μ^2)/(μ^2+P*^2 )^2.
9.Page 5. Is there any particular reason for the enumeration (i.e. (2.6)) of equation (H)? The command “\tag” should be used instead.
Reply: Thanks for your comment. The hypothesis (H) is to guarantee the $T_0<0$, $D_0>0$ hold, that means the equilibrium (N∗,P∗) is locally asymptotically stable when d1= d2= 0, τ = 0. This is also a necessary condition for Turing instability. We have added the proof of Theorem 2.1, then it can show the reason for hypothesis (H) (page 6).
- Page 5. Theorem 2.1 should be moved right under equation (H), since only this equation and not any of what follows is needed for its proof.
Reply: Sorry, we didn't write this part clearly. We have added the proof of Theorem 2.1. Then, it is clear to show the necessary for the hypothesis (H) (page 6).
- Page 4. The discriminant of Dn should not be called as Δ, because Δ is used for the Laplacian operator.
Reply: Thanks for pointing out this. We have modified it (page:6). We denote it as Γ.
Round 2
Reviewer 1 Report
The authors took into account the comments in the new version of the article, so the article can be published.
Reviewer 2 Report
correct those please...
Comments for author File: Comments.pdf
