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Qualitative Analysis of a Nonautonomous Delayed Stochastic Predator–Prey Model with Beddington–DeAngelis Functional Response
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Department of Basic Teaching, Dianchi College, Kunming 650228, China
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School of Mathematical Sciences, Sichuan Normal University, Chengdu 610066, China
3
College of Applied Mathematics, Chengdu University of Information Technology, Chengdu 610225, China
*
Author to whom correspondence should be addressed.
Biology 2025, 14(8), 1078; https://doi.org/10.3390/biology14081078
Submission received: 23 June 2025
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Revised: 4 August 2025
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Accepted: 12 August 2025
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Published: 18 August 2025
(This article belongs to the Section Ecology)
Simple Summary
This study creates a new ecological model to investigate how multiple predator and prey species interact under unpredictable environmental changes. Unlike previous models, it incorporates realistic factors such as delayed species reactions and specific hunting patterns. The researchers aimed to demonstrate long-term survival of all species (permanence), confirm global stability of population dynamics, and verify these outcomes through simulations. Key findings reveal that populations stabilize within healthy ranges when specific ecological conditions are met. However, if these conditions deteriorate—for example, when top predators find there is insufficient prey—their populations may decline toward extinction. This model provides a mathematical framework for anticipating and managing ecosystems affected by environmental randomness, supporting efforts to conserve biodiversity and maintain ecological harmony. By clarifying how response delays and species interactions influence survival, the research aids sustainable strategies protecting wildlife in shifting climates. Understanding these dynamics helps policymakers design interventions to prevent imbalances, ensuring healthier ecosystems for future generations.
Abstract
In this paper, we propose a novel stochastic multi-species predator–prey model that integrates time delays and the Beddington–DeAngelis functional response, marking a significant advancement in ecological modeling under environmental stochasticity. This model explicitly accounts for environmental fluctuations by perturbing intrinsic growth and death rates, offering a more realistic and nuanced portrayal of complex predator–prey interactions. Our primary objectives are to establish the existence and uniqueness of a global positive solution for any positive initial value, derive sufficient conditions for the uniform persistence (permanent coexistence) of all species, and investigate the global stability of the system’s solutions. By constructing sophisticated Lyapunov functions tailored to the model’s stochastic and delayed characteristics, we derive rigorous criteria that ensure the global stability of solutions, which constitutes a notable achievement considering the system’s inherent complexity. Additionally, we formulate sufficient conditions for uniform persistence, providing a theoretical foundation for understanding the long-term survival and stability of multi-species ecosystems under random environmental disturbances and historical effects. To validate our theoretical findings, we conduct extensive numerical simulations using the Milstein method. These simulations not only corroborate our analytical results but also elucidate the dynamic behaviors of the model, demonstrating system permanence within bounded densities, the convergence of solutions under varying initial conditions, and the extinction of the top predator when theoretical conditions are not met. This analysis highlights the necessity of the derived criteria for maintaining ecological balance. The analytical framework developed here lays a solid mathematical foundation for understanding the periodic and long-term behaviors of complex food chains under environmental stochasticity, with significant implications for the conservation and management of biological populations.
