https://doi.org/10.1140/epjp/s13360-025-06615-3
Regular Article
Hydra effect and stability analysis of a delayed predator–prey model with Allee effect in the predator
Department of Mathematics, Indian Institute of Engineering Science and Technology, 711103, Shibpur, Howrah, West Bengal, India
Received:
9
April
2025
Accepted:
3
July
2025
Published online:
18
July
2025
We investigate a delayed predator–prey model incorporating a Holling type II functional response and the Allee effect in the predator population. A discrete time delay is introduced in the predator’s growth process. We establish the existence, uniqueness, positivity, and boundedness of solutions for both delayed and non-delayed systems. The model admits a maximum of four interior equilibria, in addition to the trivial and boundary equilibria. A geometric analysis reveals conditions for the existence of a twofold interior equilibrium. The trivial equilibrium is always a saddle point, while the boundary equilibrium is a stable node or focus, independent of the delay. For interior equilibria, analytical stability conditions are challenging to derive; instead, we utilize the Graphical Jacobian method to assess local stability. This approach reveals bistability, where certain equilibria transition between stability and instability, and back again, as parameters vary. The model exhibits rich dynamical behaviors such as bistability, the hydra effect, saddle-node bifurcation, Hopf bifurcation, and the bubbling effect. Analytical results confirm the global asymptotic stability of the interior and boundary equilibria under specific parametric conditions. Notably, two Hopf bifurcations occur, where oscillations emerge with increasing amplitude and subsequently diminish-a signature of the bubbling effect. Bifurcation analysis demonstrates that increasing the bifurcation parameter causes transitions between stable and unstable dynamics. The coexistence of multiple nonlinear phenomena in a single delayed predator–prey model with an Allee effect in the predator highlights the novelty and significance of our study.
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© The Author(s), under exclusive licence to Società Italiana di Fisica and Springer-Verlag GmbH Germany, part of Springer Nature 2025
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
