https://doi.org/10.1140/epjp/s13360-025-07253-5
Regular Article
Dynamics of a predator–prey system incorporating Smith growth and group defense in prey, and intraspecies competition among predators
1
Department of Mathematics, Raja N.L. Khan Women’s College (Autonomous), Midnapore, India
2
Department of Mathematics, Guru Ghasidas Vishwavidyalaya, Bilaspur, India
a
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Received:
25
September
2025
Accepted:
22
December
2025
Published online:
22
January
2026
Abstract
The investigation of group defense phenomena in prey–predator models is of significant ecological importance because defensive behaviors can substantially influence the population stability and ecosystem resilience. Thus, we investigate the impact of group defense mechanism in a prey-predator model in the context of Smith-type growth in prey and intraspecies competition among predators to understand the deeper insights of complex dynamics. Smith growth dynamics is incorporated to examine the impact of resource limitation or crowding effect and toxicant environment on prey species. The model exhibits rich dynamics, including the existence of multiple interior equilibrium points and bifurcations, which enhances the level of complexity in the model. To examine the local behavior of the model in the vicinity of the equilibrium points, we have conducted a local stability study. Subsequently, we explore the study of both codimension-one and codimension-two bifurcations to demonstrate the behavioral transition of the model due to the variations of some sensitive parameters. Thus, we analyze the dynamical changes in the model where the system exhibits some major bifurcations including transcritical, Hopf, generalized Hopf, saddle-node, Bogdanov–Takens, and cusp bifurcations with respect to the parameter responsible for group defense phenomena. Various threshold values of the corresponding parameters are derived regarding these bifurcations to determine qualitative changes in the ecosystem. Our study indicates that the stronger crowding effect employs a stabilizing impact on the dynamics. One notable ecological observation is that the model exhibits bubbling phenomena in the context of intraspecies competition, indicating the transition of stable coexistence to oscillatory dynamics. Finally, numerical simulations are performed for the dynamical model to validate theoretical studies, which comprise sensitivity analysis of the parameters, some graphical illustrations, and numerical computations.
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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.
