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Diffusion-Driven Instability and Bifurcation in the Predator–Prey System with Allee Effect in Prey and Predator Harvesting

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Abstract

Modelling of predator–prey interactions has vast history in ecological theory and these models have been developed in large scale since the initial Lotka–Volterra model in order to explain the processes of predation, cooperation and diffusion more realistically. Introducing Allee effect and diffusion in predator–prey models are very important facet of these evolutions, where per capita growth rate of species increases with increase in population density and species move from region with low population density to region with high population density. In this paper, dynamics of a predator–prey model considering strong Allee effect, diffusion and Holling type-I function response is studied. Existence and stability of equilibrium points are discussed. Analysis of the model reveals that by applying some conditions on the parameters, diffusion can induce Turing instability. Furthermore, conditions for the existence of Hopf bifurcation are obtained for both the ODE and PDE models. Bifurcation direction and the stability of the bifurcating periodic solution are determined by center manifold theorem and the normal form theory. Numerical simulations are performed to support our mathematical findings. By this study, we see that if Allee threshold crosses a critical value, the interior equilibrium point losses its stability. Therefore, system undergoes Hopf bifurcation and periodic solution arises. Here, we note that Allee effect plays an important role in the dynamics of predator–prey model. We also observe that diffusion destabilized the model which was stable in the absence of diffusion.

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This manuscript has associated data in a data repository. [Authors comment: The data that support the findings of this study are available within the article.]

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Acknowledgements

The second author gratefully acknowledges the financial support provided by the University Grant Commission (UGC), New Delhi, India through the Combined Research Entrance Test (CRET).

Funding

The funding is provided by the University Grant Commission (UGC), New Delhi, India through the Combined Research Entrance Test (CRET).

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  1. Department of Mathematics, University of Allahabad, Prayagraj, 211002, India

    Sapna Devi & Reda Fatma

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  1. Sapna Devi
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  2. Reda Fatma
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Contributions

SD: Developed the theoretical formalism, supervised the research paper and helped in different stages of analytical finding, numerical computations and writing of the manuscript. RF: Formulated the mathematical model, analysed the mathematical model, performed the analytical calculations and numerical simulations, and wrote the manuscript.

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Correspondence to Reda Fatma.

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Devi, S., Fatma, R. Diffusion-Driven Instability and Bifurcation in the Predator–Prey System with Allee Effect in Prey and Predator Harvesting. Int. J. Appl. Comput. Math 10, 39 (2024). https://doi.org/10.1007/s40819-023-01673-6

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