Bifurcation Analysis of a Discrete Basener–Ross Population Model: Exploring Multiple Scenarios
Round 1
Reviewer 1 Report
Comments and Suggestions for Authors I find the results intriguing and mathematically sound. The novelty of the findings is notable, and the presentation is appropriate. I recommend this paper for publication in this esteemed journal, with some suggested revisions outlined below: 1- Improve and elaborate on the abstract to more accurately represent the paper's content. 2- Enhance the introduction to convey the importance and originality of the study. 3- Augment the introduction with recent advancements in the field and their real-world implications, citing relevant recent publications. 4- Evaluate the strengths and weaknesses of the theory applied. 5- Emphasize the key findings of the study in the conclusion section
Author Response
Reviewer #1: I find the results intriguing and mathematically sound. The novelty of the
findings is notable, and the presentation is appropriate. I recommend this paper for publication
in this esteemed journal, with some suggested revisions outlined below
Dear Reviewer #1,
We are very grateful to you for your comments and suggestions which we have taken into consideration
and incorporated into the revised paper. The revisions are highlighted in red in the
revised manuscript. Below are the point-by-point responses to your comments and suggestions:
1. Comment: Improve and elaborate on the abstract to more accurately represent the paper’s
content.
Response: Thank you for the comment. According to the Reviewer’s advice, we have rewritten,
improved, and elaborated the abstract to more accurately represent the paper content.
2. Comment: Enhance the introduction to convey the importance and originality of the study.
Response: Thank you for the comment. Furthermore, we have revised the paper’s title and
removed the section on chaos control for two specific reasons. Firstly, the revised paper was
becoming overly tall. Secondly, to thoroughly discuss all aspects of controlling chaos and to
apply them in real-world scenarios, a new paper is necessary. Therefore, in the near future,
we plan to focus on a new paper solely dedicated to controlling chaos within the model. So,
the new title is transformed into "Bifurcation Analysis of a Discrete Basener-Ross Population
Model: Exploring Multiple Scenarios." According to the Reviewer’s advice, we have
enhanced the introduction section to convey the importance and originality of the study and
updated Refs, we added the following paragraph:
Our research conducts a comprehensive study of bifurcation analysis and chaos control for
1
the discrete-time Basener-Ross model (1.4). According to Schaffer and Kot [20], it is critical
to understand the periodic or chaotic dynamics that arise in ecological models. Their
findings suggest that, far from being chaotic and disorderly, the chaotic trajectory structure
could actually include crucial information about an ecosystem dynamics. Bifurcation of equilibria
is a well-known cause of an ecosystem complicated dynamics. In [21], the authors
explored numerically the bifurcation behaviors of codimension one for model (1.4). Here, we
examine analytically the codimension-one bifurcations, including flip and Neimark-Sacker
bifurcations, and extend our study to codimension-two bifurcations characterized by 1:2, 1:3,
and 1:4 resonances, emphasizing the model complex dynamical structure. For analyzing codimension
one bifurcations, we will use the center manifold theorem and bifurcation theory.
This method is most effective for low-dimensional models. However, as the dimension of
the model increases, the complexity of constructing and analyzing the center manifold grows
significantly. Besides, its utility is limited for higher co-dimension bifurcations. Next, we
will examine the bifurcation behaviors of co-dimension two for the current model using the
normal form approach and bifurcation theory. This approach does not need a transition into
Jordan form and the computation of the model center manifold approximation. It is sufficient
to compute the critical non-degeneracy coefficients to verify the existence of various bifurcation
forms. Numerous studies have focused on bifurcation and chaotic behaviors in both
discrete-time and continuous-time models. While numerous co-dimension one bifurcation
literature have been considered, as shown in [4-7], only a small number of co-dimension two
bifurcations literature are theoretically feasible [22-25]. To the best of our knowledge, there is
very little literature on the topic of bifurcation behaviors of discrete-time Basener-Ross model
as a function of two parameters that used both theoretical and numerical approaches, including
continuation, invariant manifolds, maximal Lyapunov exponents, and normal forms. Our
goal is to meet this need. Not only that, numerical simulations are used to verify our theoretical
results and describe other model behaviors like bifurcations of higher iterations (such the
third and fourth iterations).
3. Comment: Augment the introduction with recent advancements in the field and their realworld
implications, citing relevant recent publications.
Response: Thank you for the comment. According to the Reviewer’s advice, we have rewritten
and augmented the introduction section with recent advancements in the field and their
real-world implications, citing relevant recent publications.
4. Comment: Evaluate the strengths and weaknesses of the theory applied.
Response: Thank you for the comment. According to the Reviewer’s advice, we have outlined
the strengths and weaknesses of the applied theory and included the following part in
the introduction section:
For analyzing co-dimension one bifurcations, we will use the center manifold theorem and
bifurcation theory. This method is most effective for low-dimensional models. However, as
the dimension of the model increases, the complexity of constructing and analyzing the center
manifold grows significantly. Besides, its utility is limited for higher co-dimension bifurcations.
Next, we will examine the bifurcation behaviors of co-dimension two for the current
model using the normal form approach and bifurcation theory. This approach does not need a
transition into Jordan form and the computation of the model center manifold approximation.
It is sufficient to compute the critical non-degeneracy coefficients to verify the existence of
various bifurcation forms.
5. Comment: Emphasize the key findings of the study in the conclusion section.
Response: Thank you for the comment. According to the Reviewer’s advice, we have up-
2
dated the conclusion section as follows
In this study, we have conducted a comprehensive analysis of the dynamical behavior of a
discrete population model for human population and its resource, represented by model (1.4).
We have done an elaborate analysis of the existence and stability of model fixed points. For
the positive fixed point, we have examined how it undergoes several co-dimension one bifurcations,
including flip bifurcation, Neimark-Sacker bifurcation, and the emergence of chaotic
attractors. Moreover, we have provided a thorough examination of codimension two bifurcations
related to 1:2, 1:3, and 1:4 resonance and their characteristics. To achieve this objective,
a successful technique was implemented. Specifically, we have analyzed the dynamics of
model (1.4) using the normal form approach. The process of normalizing a model involves
distilling it to its essential components. Utilizing the normal form, we identified the criteria
that govern the occurrence of subcritical or supercritical bifurcations. To further support the
complexity of model (1.4) dynamics, we have numerically computed the maximal Lyapunov
exponents. Finally, numerical simulations were conducted to validate our analytical findings,
ensuring their consistency with the actual behavior of the system.
The authors in [19] contend that the renewable rate of resources could serve as either a stabilizing
or destabilizing factor in model (1.3). However, our theoretical analysis indicates
that the impact of the renewable rate may vary, potentially stabilizing or destabilizing model
(1.4), contingent upon the values of other model parameters. Even with substantial increases
in renewable rate of resource, the human population may not gain advantages, as this would
result in corresponding rise in population due to the availability of harvested resources. As
the human population surges, resource depletion may lead to extinction, which would have
dire consequences for the human population. Besides, according to the Poincaré-Bendixon
theorem [34], the two-dimensional BR continuous-time model (1.3) exhibits either stable coexistence
or oscillations. The continuous-time BR system (1.3) exhibits no additional complicated
dynamics or multistability. Our discrete-time model (1.4) exhibited complex dynamical
behaviors, including periodicity, quasiperiodicity, and chaos. The bifurcation diagram demonstrated
the presence of periodic bubbling and periodic windows leading to chaotic behavior.
The maximal Lyapunov exponents validated the existence of non-periodic dynamics in the
system. These behaviors highlight that when population growth and resource growth coexist,
they give rise to highly intricate patterns.
Fractional-order predator-prey models incorporate memory effects and hereditary features,
making them more suitable for simulating real-world phenomena where previous states affect
current dynamics. For fractional-order version of model (1.4), fractional derivative memory
effects can significantly affect dynamics, bifurcation thresholds, stability features, and oscillatory
behaviors. It would be interesting to tackle such kind of modeling in future work.
Contemporary research continues to advance our understanding of these dynamics by employing
more sophisticated mathematical models and advanced data analysis techniques. Factors
such as ecological changes and climatic variations on the ecosystem are also being considered
in these investigations. The implications of these findings extend to various fields, including
economics, biology, and ecology, where they can find practical applications.
We hope that the revised paper now meets your expectations. However, if any further points
need to be clarified, we will be glad to do. We would like to once again express our most sincere
thanks to you for your report which we believe has helped to significantly improve the overall
presentation of our paper.
Sincerely yours,
A. A. Elsadany, A. M. Yousef , S. A. Ghazwani , A. S. Zaki
Author Response File: 
Author Response.pdf
Reviewer 2 Report
Comments and Suggestions for Authors1. What is the innovation of the author's research? The reviewer believes that the author's current research is not very new. And some conclusions have already been reported.
2. The author needs to use the maximum Lyapunov exponent and chaos graph to analyze the causes and critical values of chaos.
3. Does the author need to explain the reasons for bifurcation?
4. The author needs to compare with previous results, especially for continuous dynamical systems.
Author Response
Responses to Reviewer#2 comments
Manuscript ID: computation-3337847
Manuscript title: Bifurcation Analysis of a Discrete Basener-Ross Population Model: Exploring
Multiple Scenarios
Authors: A. A. Elsadany, A. M. Yousef , S. A. Ghazwani , A. S. Zaki
Reviewer #2: The main issues are listed as follows:
1. What is the innovation of the author’s research? The reviewer believes that the author’s current
research is not very new. And some conclusions have already been reported.
2. The author needs to use the maximum Lyapunov exponent and chaos graph to analyze the
causes and critical values of chaos.
3. Does the author need to explain the reasons for bifurcation?
4. The author needs to compare with previous results, especially for continuous dynamical systems.
Dear Reviewer #2,
We are very grateful to you for your comments and suggestions which we have taken into consideration
and incorporated into the revised paper. The revisions are highlighted in red in the
revised manuscript. Below are the point-by-point responses to your comments and suggestions:
1. Comment: What is the innovation of the author’s research? The reviewer believes that the
author’s current research is not very new. And some conclusions have already been reported.
Response: Thank you for the comment. Furthermore, we have revised the paper’s title and
removed the section on chaos control for two specific reasons. Firstly, the revised paper was
becoming overly tall. Secondly, to thoroughly discuss all aspects of controlling chaos and to
apply them in real-world scenarios, a new paper is necessary. Therefore, in the near future,
we plan to focus on a new paper solely dedicated to controlling chaos within the model. So,
the new title is transformed into "Bifurcation Analysis of a Discrete Basener-Ross Population
Model: Exploring Multiple Scenarios.". According to the Reviewer’s advice, the introduction
section have been updated as follows:
Our research conducts a comprehensive study of bifurcation analysis and chaos control for
the discrete-time Basener-Ross model (1.4). According to Schaffer and Kot [20], it is critical
to understand the periodic or chaotic dynamics that arise in ecological models. Their
findings suggest that, far from being chaotic and disorderly, the chaotic trajectory structure
could actually include crucial information about an ecosystem dynamics. Bifurcation of equilibria
is a well-known cause of an ecosystem complicated dynamics. In [21], the authors
explored numerically the bifurcation behaviors of codimension one for model (1.4). Here, we
examine analytically the codimension-one bifurcations, including flip and Neimark-Sacker
bifurcations, and extend our study to codimension-two bifurcations characterized by 1:2, 1:3,
and 1:4 resonances, emphasizing the model complex dynamical structure. For analyzing codimension
one bifurcations, we will use the center manifold theorem and bifurcation theory.
This method is most effective for low-dimensional models. However, as the dimension of
the model increases, the complexity of constructing and analyzing the center manifold grows
significantly. Besides, its utility is limited for higher co-dimension bifurcations. Next, we
will examine the bifurcation behaviors of co-dimension two for the current model using the
4
normal form approach and bifurcation theory. This approach does not need a transition into
Jordan form and the computation of the model center manifold approximation. It is sufficient
to compute the critical non-degeneracy coefficients to verify the existence of various bifurcation
forms. Numerous studies have focused on bifurcation and chaotic behaviors in both
discrete-time and continuous-time models. While numerous co-dimension one bifurcation
literature have been considered, as shown in [4-7], only a small number of co-dimension two
bifurcations literature are theoretically feasible [22-25]. To the best of our knowledge, there is
very little literature on the topic of bifurcation behaviors of discrete-time Basener-Ross model
as a function of two parameters that used both theoretical and numerical approaches, including
continuation, invariant manifolds, maximal Lyapunov exponents, and normal forms. Not
only that, numerical simulations are used to verify our theoretical results and describe other
model behaviors like bifurcations of higher iterations (such the third and fourth iterations).
2. Comment: The author needs to use the maximum Lyapunov exponent and chaos graph to
analyze the causes and critical values of chaos.
Response: Thank you for the comment. According to the Reviewer’s advice, the maximum
Lyapunov exponent and chaos graph have been used to analyze the causes and critical values
of chaos.
3. Comment: Does the author need to explain the reasons for bifurcation?
Response: Thank you for the comment. According to the Reviewer’s advice, we add the
following remarks
Remark 1. Flip bifurcation as a biological phenomena that happens when the population
size fluctuates with periods 2, 4, 8, : : :, until it becomes completely chaotic.
Remark 2. From a sustainability perspective, a stable invariant curve arises from the coexistence
fixed point E2 once a surpasses the critical value A
1?B . This indicates a stable and
reproducing cohabitation between the human population and resources. On the other hand,
ecological imbalance will result from human population and their resources instability if the
invariant curve bifurcates from E2 is unstable as a approaches the critical value A
1?B .
Remark 3. The existence of a 1:2 strong resonance signifies that model (1.4) is acutely responsive
to variations in bifurcation parameters, influencing its intricate dynamics. The nondegenerate
Neimark-Sacker bifurcation has important biological consequences, causing periodic
or quasi-periodic fluctuations in the population-resource system as the bifurcation parameters
(a,h) move around the ( ˜ a,˜h) region. These fluctuations can lead to long-period variations,
significant population surges, and even chaotic behavior in the population-resource
system. This arises from periodic oscillations with periods of 2, 4, and 8, or due to the presence
of a homoclinic structure.
Remark 4. In the 1:3 resonance scenario, the meeting of stable and unstable manifolds
creates an infinite number of orbits with a period of three, leading to a homoclinic tangency.
This reveals that a Period-3 cycle can cause chaos. Biologically, this means that
the population-resource system may experience periodic or quasi-periodic fluctuations due
to the non-degenerate Neimark-Sacker bifurcation. The Period-3 fluctuations, resulting from
a saddle cycle of period three, can generate chaotic sets. These chaotic sets contribute to
long-term fluctuations, population explosions, and overall chaos, all due to the presence of
the homoclinic structure.
Remark 5. The presence of 1:4 resonance signifies the existence of a nondegenerate Neimark-
Sacker bifurcation, allowing for the formation of an invariant cycle with a period of 4 in a
5
specific parameter range. In biological systems, the nondegenerate Neimark-Sacker bifurcation
can lead to periodic or quasiperiodic fluctuations in the population-resource system.
Moreover, the presence of an invariant cycle with a period of 4 indicates that a stable state of
the population-resource system would transition into a state that repeats (almost) after every
four time intervals.
4. Comment: The author needs to compare with previous results, especially for continuous dynamical
systems.
Response: Thank you for the comment. According to the Reviewer’s advice, the conclusion
section have been updated as follows:
In this study, we have conducted a comprehensive analysis of the dynamical behavior of a
discrete population model for human population and its resource, represented by model (1.4).
We have done an elaborate analysis of the existence and stability of model fixed points. For
the positive fixed point, we have examined how it undergoes several co-dimension one bifurcations,
including flip bifurcation, Neimark-Sacker bifurcation, and the emergence of chaotic
attractors. Moreover, we have provided a thorough examination of codimension two bifurcations
related to 1:2, 1:3, and 1:4 resonance and their characteristics. To achieve this objective,
a successful technique was implemented. Specifically, we have analyzed the dynamics of
model (1.4) using the normal form approach. The process of normalizing a model involves
distilling it to its essential components. Utilizing the normal form, we identified the criteria
that govern the occurrence of subcritical or supercritical bifurcations. To further support the
complexity of model (1.4) dynamics, we have numerically computed the maximal Lyapunov
exponents. Additionally, numerical simulations were conducted to validate our analytical
findings, ensuring their consistency with the actual behavior of the system. Finally, model
(1.4) chaotic dynamics have been stabilized and controlled using the OGY feedback control
technique.
The authors in [19] contend that the renewable rate of resources could serve as either a stabilizing
or destabilizing factor in model (1.3). However, our theoretical analysis indicates
that the impact of the renewable rate may vary, potentially stabilizing or destabilizing model
(1.4), contingent upon the values of other model parameters. Even with substantial increases
in renewable rate of resource, the human population may not gain advantages, as this would
result in corresponding rise in population due to the availability of harvested resources. As
the human population surges, resource depletion may lead to extinction, which would have
dire consequences for the human population. Besides, according to the Poincaré-Bendixon
theorem [34], the two-dimensional BR continuous-time model (1.3) exhibits either stable coexistence
or oscillations. The continuous-time BR system (1.3) exhibits no additional complicated
dynamics or multistability. Our discrete-time model (1.4) exhibited complex dynamical
behaviors, including periodicity, quasiperiodicity, and chaos. The bifurcation diagram demonstrated
the presence of periodic bubbling and periodic windows leading to chaotic behavior.
The maximal Lyapunov exponents validated the existence of non-periodic dynamics in the
system. These behaviors highlight that when population growth and resource growth coexist,
they give rise to highly intricate patterns.
Fractional-order predator-prey models incorporate memory effects and hereditary features,
making them more suitable for simulating real-world phenomena where previous states affect
current dynamics. For fractional-order version of model (1.4), fractional derivative memory
effects can significantly affect dynamics, bifurcation thresholds, stability features, and oscillatory
behaviors. It would be interesting to tackle such kind of modeling in future work.
6
Contemporary research continues to advance our understanding of these dynamics by employing
more sophisticated mathematical models and advanced data analysis techniques. Factors
such as ecological changes and climatic variations on the ecosystem are also being considered
in these investigations. The implications of these findings extend to various fields, including
economics, biology, and ecology, where they can find practical applications.
We hope that the revised paper now meets your expectations. However, if any further points
need to be clarified, we will be glad to do. We would like to once again express our most sincere
thanks to you for your report which we believe has helped to significantly improve the overall
presentation of our paper.
Sincerely yours,
A. A. Elsadany, A. M. Yousef , S. A. Ghazwani , A. S. Zaki
Author Response File: Author Response.pdf
Reviewer 3 Report
Comments and Suggestions for AuthorsPlease see the attached file.
Comments for author File: Comments.pdf
Please see the attached file.
Author Response
Responses to Reviewer#3 comments
Manuscript ID: computation-3337847
Manuscript title: Bifurcation Analysis and Chaos Control of a Discrete Basener-Ross Population
Model: Exploring Multiple Scenarios
Authors: A. A. Elsadany, A. M. Yousef , S. A. Ghazwani , A. S. Zaki
Reviewer #3: The authors investigates the dynamics and control of chaos in a discrete Basener-
Ross model through multiple bifurcation analysis techniques. The analysis focuses on the existence
and stability of fixed points for this model. In addition, the authors explore the multiple
bifurcation scenarios through the criteria for the occurrence of several types of codimension one
bifurcation, such as Flip and Neimark-Sacker bifurcations, are determined using a centre manifold
theorem and bifurcation theory. Moreover, the bifurcation diagrams, Maximal Lyapunov
exponent, and phase portraits show that the model exhibits a wide range of exciting complex
dynamical behaviors, including limit cycle, periodic solutions, chaos, and codimension-1 bifurcations.
It is shown that the model contains codimension-2 bifurcations with 1:2 resonance, 1:3
resonance and 1:4 resonance. The obtained results are new and enrich the bifurcation theory of
differential equation theory to some degree.
Dear Reviewer #3,
We are very grateful to you for your comments and suggestions which we have taken into consideration
and incorporated into the revised paper. The revisions are highlighted in red in the
revised manuscript. Below are the point-by-point responses to your comments and suggestions:
1. Comment: Many scholars deal with the bifurcation and chaos control for discrete dynamical
models. Compared with these works, what is the advantage of the discrete dynamical models?
Response: Thank you for the comment. Furthermore, we have revised the paper’s title and
removed the section on chaos control for two specific reasons. Firstly, the revised paper was
becoming overly tall. Secondly, to thoroughly discuss all aspects of controlling chaos and to
apply them in real-world scenarios, a new paper is necessary. Therefore, in the near future,
we plan to focus on a new paper solely dedicated to controlling chaos within the model. So,
the new title is transformed into "Bifurcation Analysis of a Discrete Basener-Ross Population
Model: Exploring Multiple Scenarios.". According to the Reviewer’s advice, the introduction
section have been updated as follows:
Our research conducts a comprehensive study of bifurcation analysis and chaos control for
the discrete-time Basener-Ross model (1.4). According to Schaffer and Kot [19], it is critical
to understand the periodic or chaotic dynamics that arise in ecological models. Their
7
findings suggest that, far from being chaotic and disorderly, the chaotic trajectory structure
could actually include crucial information about an ecosystem dynamics. Bifurcation of equilibria
is a well-known cause of an ecosystem complicated dynamics. In [21], the authors
explored numerically the bifurcation behaviors of codimension one for model (1.4). Here, we
examine analytically the codimension-one bifurcations, including flip and Neimark-Sacker
bifurcations, and extend our study to codimension-two bifurcations characterized by 1:2, 1:3,
and 1:4 resonances, emphasizing the model complex dynamical structure. For analyzing codimension
one bifurcations, we will use the center manifold theorem and bifurcation theory.
This method is most effective for low-dimensional models. However, as the dimension of
the model increases, the complexity of constructing and analyzing the center manifold grows
significantly. Besides, its utility is limited for higher co-dimension bifurcations. Next, we
will examine the bifurcation behaviors of co-dimension two for the current model using the
normal form approach and bifurcation theory. This approach does not need a transition into
Jordan form and the computation of the model center manifold approximation. It is sufficient
to compute the critical non-degeneracy coefficients to verify the existence of various bifurcation
forms. Numerous studies have focused on bifurcation and chaotic behaviors in both
discrete-time and continuous-time models. While numerous co-dimension one bifurcation
literature have been considered, as shown in [4–7], only a small number of co-dimension two
bifurcations literature are theoretically feasible [22–24]. To the best of our knowledge, there is
very little literature on the topic of bifurcation behaviors of discrete-time Basener-Ross model
as a function of two parameters that used both theoretical and numerical approaches, including
continuation, invariant manifolds, maximal Lyapunov exponents, and normal forms. Our
goal is to meet this need. Not only that, numerical simulations are used to verify our theoretical
results and describe other model behaviors like bifurcations of higher iterations (such the
third and fourth iterations).
2. Comment: In fact, there are many works on Codimension-one bifurcation ; Codimensiontwo
bifurcation, what is the novelty of the proposed research methods in this article?
Response: Thank you for the comment. As mentioned in comment 1, we have updated the
introduction section based on the Reviewer’s advice.
3. Comment: Grammatical issues shall be checked carefully.
Response: Thank you for the comment. We have carefully checked the entire manuscript for
grammatical issues, following the Reviewer’s advice.
4. Comment: How to obtain model (1.4) from (1.3)? please give the discrete process.
Response: Thank you for the comment. Based on the Reviewer’s recommendations, we have
incorporated the following sentence into the introduction section:
In this work, using the forward Euler approach, we get the discrete-time type of model (1.3)
as follows:
Pn+1 = Pn+aPn
1?
Pn
Rn
;
Rn+1 = Rn+cRn
1?
Rn
K
?hPn:
(1.4)
Here, we consider the integral step size to be equal to 1.
5. Comment: Can you deal with the Neimark-Sacker bifurcation of fractional-order version of
model (1.4)? Please add it as future research direction in Conclusion.
Response: Thank you for the comment. According to the Reviewer’s advice, we have updated
the conclusion section as follows:
8
Fractional-order predator-prey models incorporate memory effects and hereditary features,
making them more suitable for simulating real-world phenomena where previous states affect
current dynamics. For fractional-order version of model (1.4), fractional derivative memory
effects can significantly affect dynamics, bifurcation thresholds, stability features, and oscillatory
behaviors. It would be interesting to tackle such kind of modeling in future work.
6. Comment: What is the practical application of Neimark-Sacker bifurcation, Codimensionone
bifurcation ; Codimension-two bifurcation in biology?
Response: Thank you for the comment. According to the Reviewer’s advice, we add the
following remarks:
Remark 1. Flip bifurcation as a biological phenomena that happens when the population
size fluctuates with periods 2, 4, 8, : : :, until it becomes completely chaotic.
Remark 2. From a sustainability perspective, a stable invariant curve arises from the coexistence
fixed point E2 once a surpasses the critical value A
1?B . This indicates a stable and
reproducing cohabitation between the human population and resources. On the other hand,
ecological imbalance will result from human population and their resources instability if the
invariant curve bifurcates from E2 is unstable as a approaches the critical value A
1?B .
Remark 3. The existence of a 1:2 strong resonance signifies that model (1.4) is acutely responsive
to variations in bifurcation parameters, influencing its intricate dynamics. The nondegenerate
Neimark-Sacker bifurcation has important biological consequences, causing periodic
or quasi-periodic fluctuations in the population-resource system as the bifurcation parameters
(a,h) move around the ( ˜ a,˜h) region. These fluctuations can lead to long-period variations,
significant population surges, and even chaotic behavior in the population-resource
system. This arises from periodic oscillations with periods of 2, 4, and 8, or due to the presence
of a homoclinic structure.
Remark 4. In the 1:3 resonance scenario, the meeting of stable and unstable manifolds
creates an infinite number of orbits with a period of three, leading to a homoclinic tangency.
This reveals that a Period-3 cycle can cause chaos. Biologically, this means that
the population-resource system may experience periodic or quasi-periodic fluctuations due
to the non-degenerate Neimark-Sacker bifurcation. The Period-3 fluctuations, resulting from
a saddle cycle of period three, can generate chaotic sets. These chaotic sets contribute to
long-term fluctuations, population explosions, and overall chaos, all due to the presence of
the homoclinic structure.
Remark 5. The presence of 1:4 resonance signifies the existence of a nondegenerate Neimark-
Sacker bifurcation, allowing for the formation of an invariant cycle with a period of 4 in a
specific parameter range. In biological systems, the nondegenerate Neimark-Sacker bifurcation
can lead to periodic or quasiperiodic fluctuations in the population-resource system.
Moreover, the presence of an invariant cycle with a period of 4 indicates that a stable state of
the population-resource system would transition into a state that repeats (almost) after every
four time intervals.
We hope that the revised paper now meets your expectations. However, if any further points
need to be clarified, we will be glad to do. We would like to once again express our most sincere
thanks to you for your report which we believe has helped to significantly improve the overall
presentation of our paper.
Sincerely yours,
A. A. Elsadany, A. M. Yousef , S. A. Ghazwani , A. S. Zaki
Author Response File: Author Response.pdf
Round 2
Reviewer 3 Report
Comments and Suggestions for AuthorsThe problems have been solved.
