Global Stability of Delayed Ecosystem via Impulsive Differential Inequality and Minimax Principle

Round 1

Reviewer 1 Report

General comments:

the paper studies stability properties of an infinite dimensional system with a time delay and impulsive actions. The considered differential equations are nonlinear but impulsive action is linear. The are many works devoted to abstract nonlinear infinite dimensional systems with impulses. Different stability conditions (typically dwell-time conditions) are established in the literature, also in the case when the impulsive action is nonlinear. The author does not mention these works and provides no comparison of the results with the known ones. This should be done including an appropriate literature review.

Another problem is that the paper is not self contained. Even for the used notation and meaning of the used variables the author refers to other papers. This make the paper hard to read and should be improved.

Other problems are related to the mathematical correctness. For example I do not see from which spaces the initial functions are chosen. In which sense the solutions should be understood? What about well posedness? This also depends on the relation between the size of delay and the time sequence t_k.

Next problem is that the author derives exponential stability for a nonlinear system, which is not natural and usually leads to very restrictive conditions on the system.

Finally, the stability conditions need interpretation. What do they mean for the system and for the sequence t_k? Is there any dwell-time condition among them?

Specific questions:

1. Why impulsive actions are introduced, what is the motivation for them? How they can be realized in the considered ecosystem practically?

2. What is the meaning of condition (2.3). Can you give an interpretation?

3. The condition in Lemma 2.2. that 0<\theta_i<1 should be dropped because it is a part of A2.

4. Assumption A1 is not really an assumption because it can be always satisfied.

5. Some properties of t_k should be required in advance (monotone? no accumulation?)

6. As far as I see t_k^+=t_k, hence this extra notation is not necessary, but t_k^- is not defined.

7. The proof of Theorem 1 is not rigorous. For example, the prove the existence of stationary solutions begins with assumption, that they exist. It is not clear, where from (3.4) comes.

8. There are other unclear places, some times due to unclear English.

Finally I conclude that the paper not ready for publication and its contribution needs to be clarified in view of the existing literature (not only [6,7]).

 

 

Author Response

Please see the attachment.

Author Response File: Author Response.pdf

Reviewer 2 Report

The paper tries to show a way to improve classic algorithm, reducing the role of impulsive control, when dealing with delayed feedback Gilpin-Ayala competition model with impulsive disturbance.

Author should avoid splitting maths expressions when changing line.

Introduction

• Short, compact, but very interesting. This introduction is almost an abstract, stayed in the border line, but the cohesion of subjects mentioned, and respective references works. Just the last sentence needs a little bit improvement, to avoid repetition – the same sentence is used in abstract – and to give some structure to how things will be done along paper to achieve the main goal.

Preparatory knowledge

• “… and the meaning of symbols and variables is the same as that of [7].” The author should say that this is a continuation of previous work, or a new result, or something else to avoid the danger of self-reference. Also, this way to put things is not the best one, since the readers cannot have access to the previous version, thus, symbols and variables will be very hard to understand on this paper.
• (PS) condition should be presented in this paper due to its importance.
• The presentation of the second system in (2.4) is enough.
• “Considering the impulse disturbance on (2.6), one can get the following system.” Where is the impulse disturbance? Accordingly, to that it is written, (2.8) and (2.6) are the same system, written in a different way.

Main results

• “Next, the author claim that Y satisfies the (PS) condition.” It is impossible to see if the claim will be valid, since the (PS) condition is not in the paper, and it is not a usual result on the field.
• Theorem 3.2. is a nice result, of course if Theorem is also valid.
• Remark 1 should be part of conclusions section as remark 2 also.

Conclusions

• Should reorganized to accommodate remark 1 and 2, since both strengthen what it is said.

Author Response

Please see the attachment.

Author Response File: Author Response.pdf