The Role of Alternative Food Quality in Predator Persistence: Stability and Forward Bifurcation Analysis of  a Predator-Prey Model

Resmawan Resmawan; Binti Mualifatul Rosydah; Rizka Putri Handayani

doi:10.55657/rmns.v5i2.299

Authors

Resmawan Resmawan Department of Mathematics, Universitas Negeri Gorontalo, Bone Bolango 96554, Indonesia
Binti Mualifatul Rosydah Study Program of Safety Engineering, Politeknik Perkapalan Negeri Surabaya, Surabaya 60111, Indonesia
Rizka Putri Handayani Department of Mathematics, Universitas Brawijaya, Malang 65145, Indonesia

DOI:

https://doi.org/10.55657/rmns.v5i2.299

Keywords:

Alternative food quality, Forward bifurcation, Local stability, Predator–prey model

Abstract

This study develops a two-dimensional predator-prey model incorporating logistic prey growth, prey group defense, alternative food, and ecological costs to the predator. The novelty of the model lies in the introduction of an alternative food quality parameter, which distinguishes the availability of alternative food from its biological effectiveness in supporting predator growth. The analysis focuses on the existence of equilibrium points, local stability, and forward bifurcation. The analytical results show that the model admits a trivial equilibrium, a predator-free equilibrium, and an interior equilibrium under certain existence conditions. The trivial equilibrium is always unstable, whereas the predator-free equilibrium is locally asymptotically stable when the quality of alternative food is below a critical value and loses its stability when the quality parameter exceeds this threshold. At the critical value, the system undergoes a forward bifurcation characterized by a stability transition from the predator-free equilibrium to a positive interior equilibrium. Numerical simulations confirm the analytical findings and illustrate three main scenarios: predator extinction when the alternative food quality is low, stable coexistence when the alternative food quality is high, and a stability transition through a bifurcation diagram as the alternative food quality parameter crosses its critical value. These results indicate that alternative food quality acts as a threshold parameter determining predator persistence and the transition from predator extinction to stable coexistence.

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