https://doi.org/10.1140/epjp/s13360-024-05628-8
Regular Article
Global dynamics of a prey–predator model with component Allee effect for predator reproduction and Crowley–Martin-type functional response: the role of strong Allee effect
1
Department of Mathematics, VIT Bhopal University, Bhopal-Indore Highway, 466114, Sehore, Madhya Pradesh, India
2
Department of Mathematics, National Institute of Technology Rourkela, 769008, Rourkela, Odisha, India
Received:
19
June
2024
Accepted:
7
September
2024
Published online:
9
October
2024
Authors have detected the importance of the Allee effect and have highlighted significant changes to system dynamics in the ecological environment. In this paper, using the theory of dynamical systems, we explore our investigation of a two-dimensional prey–predator model into two aspects: (i) we modify a competent Allee effect and the self-limitation term for the predator model system by incorporating the Crowley–Martin-type functional response, and (ii) we extend this modified model system by adding a strong Allee effect in prey growth. We report the behavior of the model system under the Crowley–Martin-type functional response and the impact of a strong Allee effect. We examine that an initial condition with high prey and low predator intensity always settles to predator extinction in the absence of a strong Allee effect. In the presence of a strong Allee effect, an initial condition with low prey and predator intensities leads the system to total extinction, while high prey and low predator intensity allows the model system to settle at predator extinction and high prey concentration. The addition of a strong Allee effect in the model system enriches the boundary equilibrium point. In attractor examination, we demonstrated that the model system without the Allee effect has attractors between boundary equilibrium and coexistence equilibrium, and between coexistence equilibria. The addition of a strong Allee effect produces attractors between coexistence equilibrium as well as between boundary equilibrium points. We deduce that both model systems experience enriched coexistence equilibria in a small parametric region. Modifying the model system without the Allee effect produces three attractors (one predator-free equilibrium point and two coexistence equilibrium points). The model system with a strong Allee effect gives four attractors (two predator-free equilibrium points and two coexistence equilibrium points). Our comprehensive bifurcation analysis reports both local and global bifurcations for both types of model systems. We explore bifurcation analysis through codimension-one and codimension-two bifurcations extensively by choosing the Allee effect as one of the key parameters. In this context, our modified model system exhibits saddle-node bifurcation, Hopf bifurcation, Bogdanov–Takens bifurcation and generalized Hopf bifurcation. We derive the sensitivity index of model system parameters for both types of model systems. We validated our analytical findings with the help of numerical simulations.
The authors Udai Kumar and Ankur Kanaujiya have contributed equally to this work.
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© The Author(s), under exclusive licence to Società Italiana di Fisica and Springer-Verlag GmbH Germany, part of Springer Nature 2024. 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.
