https://doi.org/10.1140/epjp/s13360-026-07977-y
Regular Article
Influence of Allee effect in prey, hunting cooperation among predators, and a competing predator on predator–prey dynamics with a modified Holling type-II functional response
1
Department of Mathematics, Indian Institute of Technology Patna, 801103, Patna, India
2
Institute of Environmental Sciences, The Robert H. Smith Faculty of Agriculture, Food and Environment, The Hebrew University of Jerusalem, Rehovot, Israel
a
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Received:
13
January
2026
Accepted:
15
June
2026
Published online:
6
July
2026
Abstract
Species survival is affected by ecological factors such as the Allee effect in prey, hunting cooperation among predators, and interspecific competition by a competitive predator. These mechanisms have a major impact on ecological stability, extinction risk, and species survival. To investigate these dynamics, we develop a predator–prey model that incorporates a strong Allee effect in the prey population, hunting cooperation among predator population, and interspecific competition by a competitive predator. Here, we consider that the population density of the competitive predator is constant. The trophic interaction is described using a modified Holling type-II functional response to capture more realistic foraging behavior of the predator. Mathematical analysis ensures positivity and boundedness of the solution and identifies equilibrium points with their stability properties. The system exhibits rich dynamical behaviors, including transcritical, saddle-node, Hopf, limit point of cycles, Bogdanov–Takens (codimension two), homoclinic bifurcations, and a codimension-two SN–TC (saddle-node–transcritical) point. Numerical simulations demonstrate how changes in ecological parameters, such as the Allee threshold, predator’s cooperation strength, and wasted time due to interspecific competition between predators, drive switches between extinction, prey-only survival, co-existence, and oscillatory dynamics. Our findings reveal that the combined effects of Allee effect, hunting cooperation among predators, and interspecific competition give rise to bistability and tristability, providing novel insights into species persistence, co-existence, and extinction.
Focus Point on Biodiversity and Ecosystem Dynamics: The Impact of Climate Change on Complex Ecological Networks. Guest Editor: A. K. Misra
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© The Author(s), under exclusive licence to Società Italiana di Fisica and Springer-Verlag GmbH Germany, part of Springer Nature 2026
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.
