Addendum to "Persistence and extinction in an Elk-Wolf prey-predator system with refuge and inter-regional movement. Appl. Math. Comput. 514 (2026) 129834"

TL;DR

This paper revisits and clarifies the mathematical analysis of an elk-wolf predator-prey model with refuge and movement, ensuring the stability, boundedness, and bifurcation conditions are rigorously established.

Contribution

It provides rigorous re-derivations and clarifications of stability, boundedness, and bifurcation conditions for the elk-wolf model, supporting future predator-prey research.

Findings

Re-evaluated local and global stability analyses.

Clarified the construction of Lyapunov functions.

Re-derived Hopf bifurcation conditions with proper criteria.

Abstract

The elk - wolf model with movements between refuge and open habitat was put forward in Maji et al. (Appl. Math. Comput. 514 (2026)), which is rigorously re-examined in this remark. We re-evaluate the local and global stability analyses, especially the construction of the Lyapunov function, and provide mathematical clarifications on boundedness, model formulation, and the existence of equilibria. The sensitivity and numerical results are re-examined for consistency and re-producibility, and the Hopf bifurcation conditions are rederived using the proper transversality criteria. The purpose of this note is to support future studies of predator-prey systems based on refuges by offering mathematically consistent conditions.