Modelling Functional Shifts in Two-Species Hypercycles

Round 1

Reviewer 1 Report

The authors have responded to the Wyszcze comments and accept the article as submitted.

Reviewer 2 Report

The authors' replies and revision have satisfactorily addressed my comments with regard to the earlier version of the manuscript. I think the findings of this study are highly relevant to the  community dynamics modelling community. 

Reviewer 3 Report

The authors did not make any significant changes to the work. They consider a model consisting of two ODEs. For this system, they study the existence and stability of equilibrium states. This analysis turns out to be sufficient: solutions always converge to one of the equilibrium states. So the authors did not even have to solve the problem of finding and studying periodic solutions. Such an analysis is trivial and of no mathematical interest.

At the same time, the interpretation of the obtained results may be of some applied interest. Therefore, I recommend to submit an article to the another journal of purely applied orientation, but not in the mathematical journal.

This manuscript is a resubmission of an earlier submission. The following is a list of the peer review reports and author responses from that submission.

Round 1

Reviewer 1 Report

The article considers the two-dimensional system of ordinary differential equations. Authors studied equilibrium states and its stability. All methods are well-known. From the point of view of mathematics, the article contains nothing new. It can be recommended for publishing in another journal of purely applied orientation, but not in the mathematical journal.

Reviewer 2 Report

Reviewer’s report for “Modelling functional shifts in two-species hypercycles”

In this article, the authors design a two-member hypercycle model considering that one species performs a functional shift, and analyze the behavior of the hypercycle under different scenarios. They first introduce the model without functional shift to characterize the dynamics under obligate and facultative cooperation only. They then consider two more cases involving cross-catalysis namely, a model describing the dynamics of ribozymes where a fraction of the population of one replicator degrades the other molecular species while the other fraction still receives catalytic aid; and a system in which a given fraction of the population predates on the cooperating species while the rest of the population still receives aid. They characterize the key bifurcation parameters determining extinction, survival, and coexistence of species. An important result emerging from their analysis is the finding that predation, regardless of the fraction that benefits from it, does not significantly change dynamics with respect to the degradative case and as such, does not alter the dynamics and bifurcations. They discuss the dynamical implications of early replicators.

I enjoyed reading this manuscript. It addresses an interesting subject and shows how the usual model of community dynamics can be extended to incorporate functional shifts.  Overall, this is a well written paper. I do have questions with regard to the model

Comments and suggestions to the authors

[1] L119: The reason for bounding parameter values to the unit interval should be indicated.

[2] Why is the strength of density-dependent regulation represented by the carrying capacity c_0 assumed to be the same for the two species? This is a strong assumption. The same observation applies to the parameter epsilon.

[3] What is the implication of setting c_0 to zero?

[4] epsilon (with no index) and epsilon_i,j have different meanings, which may be  confusing. I am wondering why the authors did chose to use the same symbol?  I would suggest using different symbols.

[5] non-negativity constraints are required: episllon>=0, episllon_i,j >=0, beta>=0, gamma>=0, eta_k>=0, etc.

Reviewer 3 Report

The article is correctly drafted and describes the subject in a very interesting way, but it needs a few improvements:

Correct the font in Figure 1.

In the article the investigated problem is presented in a correct way, the results and mathematical considerations are presented clearly. The structure of the article is correct.