https://doi.org/10.1140/epjp/s13360-025-07123-0
Regular Article
Influence of Allee effect and cooperative hunting on the dynamics of a modified Hastings–Powell model in deterministic, discrete, and stochastic environments
1
Department of Mathematics, University of Kalyani, 741235, Kalyani, India
2
Department of Mathematics, School of Computer Science and Artificial Intelligence, SR University, 506371, Warangal, Telangana, India
3
Sultan Moulay Slimane University, Faculté Polydisciplinaire Khouribga, BP: 145 Khouribga principale, 25000, Khouribga, Morocco
4
Department of Mathematics and Computing, Indian Institute of Technology (Indian School of Mines) Dhanbad, 826004, Dhanbad, Jharkhand, India
5
Department of Basic Science and Humanities, Indian Institute of Information Technology, 813210, Bhagalpur, India
a
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Received:
17
July
2025
Accepted:
26
November
2025
Published online:
7
December
2025
Abstract
In this study, we explore a modified Hastings–Powell food chain model that incorporates the Allee effect in prey growth and hunting cooperation among middle predators. We analyze the system’s equilibria along with stability, revealing that species persistence and extinction depend critically on the parameters. The continuous system exhibits rich dynamical behaviors, including Hopf as well as saddle-node bifurcations, chaotic dynamics, and bistability. Our findings indicate that lower prey growth rates or Allee thresholds lead to population extinction, while their moderate values yield initially stable coexistence followed by chaotic fluctuations. At higher values, populations ultimately collapse. Similarly, increased hunting cooperation destabilizes the system, causing transitions from stable coexistence to irregular oscillations and extinction. We further examine the discrete-time version of the model, uncovering additional complex dynamics not observed in the continuous system. Our analysis reveals a period-doubling route to chaos, highlighting the system’s sensitivity to parameter variations and initial conditions. To manage these chaotic dynamics, we employ hybrid control methods that effectively suppress bifurcations and stabilize the system under specific conditions. Moreover, we investigate the impacts of environmental noise, demonstrating their critical role in species persistence and extinction. We observe that lower noise intensities permit stable coexistence, while slight increments destabilize the system, leading to species extinction. These results emphasize the fragility of ecological systems under stochastic fluctuations. Overall, our finding underscores the interplay of deterministic and stochastic factors in shaping ecological dynamics and provides valuable strategies for managing complex ecosystems.
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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.
