Dynamical behavior of stochastic fractional-order predator–prey system with nonlinear functional response and Allee effect Dynamical behavior of stochastic

• Published in: Journal of applied mathematics & computing Vol. 71; no. 3; pp. 4043 - 4065
• Main Author: Ali, Ishtiaq
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.06.2025
• Subjects: Computational Mathematics and Numerical Analysis; Mathematical and Computational Engineering; Mathematics; Mathematics and Statistics; Mathematics of Computing; Original Research; Theory of Computation; 37A50; Allee effects; Numerical simulations; Nonlinear functional response; 26A33; 37H30; Stability analysis; Stochastic fractional-order predator–prey model
• ISSN: 1598-5865; 1865-2085
• DOI: 10.1007/s12190-025-02396-1

To enhance the ability of predator–prey models to simulate complex biological interactions in uncertain environments and nonlinear predation dynamics, it is important to incorporate the Allee effects, a stochastic term and nonlinear functional response. The inclusion of these components leads to rich dynamics including multiple equilibria, limit cycles, or chaotic behavior. They also improve the realism of the model by more accurately simulating real ecosystems, where even minor adjustments can cause the system to surpass tipping points, causing abrupt changes in population survival or extinction. The resulting integrated model is ecologically significant for managing ecosystems by predicting the effects of environmental changes on predator–prey interactions, identifying critical population thresholds and assessing extinction risks, and understanding resilience by examining how populations respond to disturbances. This study aims to investigate the impact of environmental noise and Allee effects subject to nonlinear functional response of Holling type 1V on system of fractional-order predator–prey equation. The necessary and sufficient conditions for local stability equilibria are provided for both deterministic and stochastic predator–prey models. We first investigated the dynamics of deterministic version of predator–prey system of fractional order, followed by the stochastic equation. A numerical method based on Grünwald–Letnikov method is used to find the approximate solution of fractional order deterministic equation, while the same method combined with Euler–Maruyama method is used for the stochastic version. The theoretical results were validated using several numerical simulations.