Asymptotic stability of deterministic and stochastic prey-predator models with prey herd immigration

Jawdat Alebraheem; Jawdat Alebraheem

doi:10.3934/math.2025214

AIMS Mathematics

2025, Volume 10, Issue 3: 4620-4640. doi: 10.3934/math.2025214

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Asymptotic stability of deterministic and stochastic prey-predator models with prey herd immigration

Jawdat Alebraheem ^,

Department of Mathematics, College of Sciences, Majmaah University, Al-Majmaah 11952, Saudi Arabia

Received: 06 December 2024 Revised: 05 February 2025 Accepted: 17 February 2025 Published: 03 March 2025
MSC : 34D05, 37H30, 60H10, 92-10

This paper aimed to study the influence of herd immigration in prey species on the stability of the prey–predator interaction, in which prey immigration is modeled as a herd movement for defensive purposes. A stochastic version of the model was formulated to incorporate the influence of random noises. Positivity and boundedness are discussed for both deterministic and stochastic models, which validate the model biologically. For the deterministic model, the local asymptotic stability of the feasible equilibrium points is discussed, and the Hopf bifurcation is exhibited with respect to an immigration factor. Using a suitable Lyapunov function, sufficient conditions for global asymptotic stability are established for deterministic and stochastic models. Numerical simulations are carried out to verify and clarify our analytical findings. It is demonstrated that increasing prey herd immigration rates stabilizes the systems. Numerical simulations of the stochastic system reveal that population density fluctuations grow more consistently as prey herd immigration increases; these simulations also exhibit diverse dynamics, including quasi-steady states and quasi-limit cycles. It is concluded that the immigration of prey herds improves the survival of both species in deterministic and stochastic systems. Thus, it may be beneficial for prey to immigrate in groups to support unstable systems.
- immigration,
- herd immigration,
- prey–predator system,
- asymptotic stability
Citation: Jawdat Alebraheem. Asymptotic stability of deterministic and stochastic prey-predator models with prey herd immigration[J]. AIMS Mathematics, 2025, 10(3): 4620-4640. doi: 10.3934/math.2025214

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Abstract

This paper aimed to study the influence of herd immigration in prey species on the stability of the prey–predator interaction, in which prey immigration is modeled as a herd movement for defensive purposes. A stochastic version of the model was formulated to incorporate the influence of random noises. Positivity and boundedness are discussed for both deterministic and stochastic models, which validate the model biologically. For the deterministic model, the local asymptotic stability of the feasible equilibrium points is discussed, and the Hopf bifurcation is exhibited with respect to an immigration factor. Using a suitable Lyapunov function, sufficient conditions for global asymptotic stability are established for deterministic and stochastic models. Numerical simulations are carried out to verify and clarify our analytical findings. It is demonstrated that increasing prey herd immigration rates stabilizes the systems. Numerical simulations of the stochastic system reveal that population density fluctuations grow more consistently as prey herd immigration increases; these simulations also exhibit diverse dynamics, including quasi-steady states and quasi-limit cycles. It is concluded that the immigration of prey herds improves the survival of both species in deterministic and stochastic systems. Thus, it may be beneficial for prey to immigrate in groups to support unstable systems.

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