Hierarchy Establishment from Nonlinear Social Interactions and Metabolic Costs: An Application to Harpegnathos saltator

Round 1

Reviewer 1 Report

The system described is continuous but not Lipschitz, due to the terms max/min(C_w - C_g ), so uniqueness of solutions is not evident. You need to investigate the evolution of the system near C_w=C_g to check uniqueness.

The proof of Theorem 3.2 is very repetitive and can be reduced a great deal. Many details can be skipped. 

The statement of this theorem can also be simplified, for example there is no need to use trigonometric functions to state the result. 

The claim about stability of the equilibrium at the beginning of sec. 4.2 is not supported by analysis.  It may be beneficial to study the linearization of the system around the critical point to obtain asymptotic stability, if the is any. 

Author Response

1. Comment: The system described is continuous but not Lipschitz, due to the terms max/min(C_w - C_g ), so uniqueness of solutions is not evident. You need to investigate the evolution of the system near C_w=C_g to check uniqueness.

 

Response:  Thanks for the comments. Per your suggestion, we have changed the proof of Theorem 3.1 regarding positive invariance of the system by investigating the evolution of the system near C_w=C_g to check uniqueness. The changes were made on pages 16-17 of the manuscript.

 

2. Comment: The proof of Theorem 3.2 is very repetitive and can be reduced a great deal. Many details can be skipped. The statement of this theorem can also be simplified, for example there is no need to use trigonometric functions to state the result.

 

Response: Thanks for the comments. The statement and proof of theorem 3.2 have been simplified. The changes to the statement of theorem 3.2 were made on page 9, and the changes to the proof of this theorem were made on pages 17-20 of the manuscript.

 

3. Comment: The claim about stability of the equilibrium at the beginning of sec. 4.2 is not supported by analysis. It may be beneficial to study the linearization of the system around the critical point to obtain asymptotic stability, if there is any.

 

Response: Thanks for the comments. The claim at the beginning of sec. 4.2, on page 13 of the manuscript, has been changed as “ Our simulations suggest that the system (9)-(10) always has a unique interior equilibrium (N∗, Ng∗) which appears to be globally stable”.

Author Response File: Author Response.docx

Reviewer 2 Report

This work addresses the mathematical modelling of a particular ant's society considered as a hierarchical structure. The model is formulated within the classic way of populations dynamics with the nonlinear functional response and fractional power-law dependences, which are typical for metabolic systems. The proposed ODE system is properly analysed and supported with simulations using the parameters taken from sources providing real experimental data. 

Thus, I recommend accepting this work. 

Some minor comments, which do not require additional reviewing and may be taken into account during the manuscript's editing:
1) although variables are mentioned in the caption to Fig. 2, it would be better to put variable names to axes;
2) Parameters in Table 1 should be supplied with dimensionality, especially taking into account that rate variables include time dimension (per day?)

One more very optional suggestion, just for curiosity: curves in Fig. 1 have hyperbolic form, if to plot this figure in double-logarithmic coordinates, maybe there will be a linearized geometry when experimental points are placed between linear bounds? If so, it may be more illustrative. 

Author Response

Response to Referee #2—changes written in blue

This work addresses the mathematical modelling of a particular ant's society considered as a hierarchical structure. The model is formulated within the classic way of populations dynamics with the nonlinear functional response and fractional power-law dependences, which are typical for metabolic systems. The proposed ODE system is properly analysed and supported with simulations using the parameters taken from sources providing real experimental data. 

Thus, I recommend accepting this work. 

Some minor comments, which do not require additional reviewing and may be taken into account during the manuscript's editing;

Reviewer 2:

1. Comment: Although variables are mentioned in the caption to Fig. 2, it would be better to put variable names to axes.

 

Response:  Thanks for the comments. Variable names have been added to Fig. 2 axes, on page 11 of the manuscript.

2. Comment: Parameters in Table 1 should be supplied with dimensionality, especially taking into account that rate variables include time dimension (per day?)

 

Response:  Thanks for the comments.  Dimensionality has been specified for parameters in Table 1, on page 6 of the manuscript.

 

 

 

Author Response File: Author Response.docx

Reviewer 3 Report

I accept the paper as it is, and thank the authors for thier work.

Author Response

Thanks for accepting the paper as it is

Author Response File: Author Response.docx