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ISSN:2164-6457 (print)
ISSN:2164-6473 (online)
Journal of Applied Nonlinear Dynamics
Miguel A. F. Sanjuan (editor), Albert C.J. Luo (editor)
Miguel A. F. Sanjuan (editor)

Department of Physics, Universidad Rey Juan Carlos, 28933 Mostoles, Madrid, Spain

Email: miguel.sanjuan@urjc.es

Albert C.J. Luo (editor)

Department of Mechanical and Industrial Engineering, Southern Illinois University Ed-wardsville, IL 62026-1805, USA

Fax: +1 618 650 2555 Email: aluo@siue.edu

Dynamical Analysis of a Delayed and Stochastic Predator-Prey System with Allee Effect, Holling type IV functional response, and Supplemental Food

Journal of Applied Nonlinear Dynamics 15(4) (2026) 885--900 | DOI:10.5890/JAND.2026.12.006

B. Ramesh$^{1,2}$, K. Ramesh$^{2}$

$^{1}$ Department of Mathematics, AVN Institute of Engineering and Technology, Ibrahimpatnam, Hyderabad- 501510, Telangana, India

$^{2}$ School of Engineering, Department of Mathematics, Anurag University, Venkatapur, Hyderabad-500088, Telangana, India

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Abstract

The purpose of this study is to explore the dynamical behaviour of a predator-prey environment that incorporates Allee effects in prey, group defence, and supplemental food for the predator, while taking into account both delayed and stochastic factors. Two nonlinear differential equations make up the deterministic model. In this model, the population of prey is characterised by Allee effects and group defence, while the predator is able to reap the benefits of an alternative food supply. The model also takes into account the temporal delay in the reproduction of predators, which allows it to capture the influence that the availability of prey in the past has on the expansion of predators. By incorporating Gaussian white noise disturbances into the prey and predator populations, the system is expanded to a stochastic framework, which takes environmental fluctuations into consideration. Our analysis consists of a study on the stability of equilibrium points, the derivation of Hopf bifurcation conditions due to delay, and an investigation into the effects that noise has on the persistence and extinction of populations. The current work provides a thorough integration of these mechanisms into a single framework, in contrast to earlier research that looked at discrete elements like the Allee effect, temporal delay, or stochasticity in isolation. To verify the theoretical findings and illustrate the intricate interaction between deterministic, delayed, and stochastic elements in influencing predator-prey dynamics, numerical simulations are utilised. The results offer important new information for managing ecosystems and conserving biodiversity in settings that are prone to natural oscillations.

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