Review of Manuscript: "Complex Dynamics of A Discrete Time One Prey Two Predator System with Prey Refuge"

The manuscript aims to explore the dynamics of a discrete one prey-two predator system with a prey refuge. However, it suffers from several critical flaws that significantly undermine its scientific contribution.

Key Flaws:

The model presented in the manuscript is incorrect as it lacks the parameters, which represent how efficiently the predator converts prey into its own offspring. Using the same parameter a or b in both the prey equation and the predator equation for the predator response function terms is unrealistic and overly restrictive, leading to fundamental flaws in the model itself. Nondimensionalization cannot be applied in this case because it has already been performed – we are working with a unit carrying capacity of the environment, which means using a dimensionless state variable for the prey. Another way to convert the system into the form presented by the authors is to change the time scale; however, this approach encounters the issue of two efficiency parameters, which cannot both be omitted without loss of generality.

Theorems 7 and 8 are entirely redundant and, moreover, indicate a misunderstanding of terminology. Theorem 7: Equilibria are  always cycles of period 1, so there is no reason to state this as a theorem. Moreover, it is mentioned at the beginning of section 3.1. The calculation of equilibria is also a trivial task. Theorem 8: A fixed point x always satisfies F(x)=x, and thus F^{n}(x)=x. However, the period is defined as the smallest such n. Additionally, the term "prime period" is used completely incorrectly before the theorem.

The manuscript incorrectly refers to Hopf bifurcation in the context of a discrete-time model; the correct term for the bifurcation described is Neimark-Sacker bifurcation.

Although the text mentions Jury conditions and conditions for the occurrence of Neimark-Sacker (misleadingly Hopf) and flip bifurcations, these are not actually utilized. For the endemic equilibrium E, no calculations are provided to describe the parameter regions where Neimark-Sacker or flip bifurcations occur. Such calculations can be performed using, for example, an appropriate Gröbner basis for the system of the given conditions. Algorithms for these calculations are already standardly implemented in software like Maple, Matlab, R, etc. The Gröbner basis approach to exclude state variables to get proper descriptions of bifurcation manifolds is explained in Hajnová, V., & Přibylová, L. (2019). Bifurcation manifolds in predator-prey models computed by Gröbner basis method. Mathematical Biosciences. doi:10.1016/j.mbs.2019.03.008. ​In its current form, the theoretical part of the article does not provide any new results.

Numerical calculations have been performed; however, some of the figures are illegible (2a-c). Additionally, there is no link to the source codes, which is essential for reproducibility and is a common practice in the current era of open science.

Conclusion:

Due to the significant flaws and fundamental errors in the model, terminology, and presentation, as well as the lack of novel results, the manuscript does not meet the standards for publication in its current form.

Recommendation: Rejection