K Ramesh, Kumar G Ranjith, Khan Aziz, Abdeljawad Thabet
Department of Mathematics, Anurag University, Venkatapur, Hyderabad, Telangana, India.
Department of Mathematics and General Sciences, Prince Sultan University, Riyadh, Saudi Arabia.
PLoS One. 2025 Jan 3;20(1):e0305179. doi: 10.1371/journal.pone.0305179. eCollection 2025.
This study proposes and analyses a revised predator-prey model that accounts for a twofold Allee impact on the rate of prey population expansion. Employing the Caputo fractional-order derivative, we account for memory impact on the suggested model. We proceed to examine the significant mathematical aspects of the suggested model, including the uniqueness, non-negativity, boundedness, and existence of solutions to the noninteger order system. Additionally, all potential equilibrium points for the strong and weak Allee effect are examined under Matignon's condition, along with the current state of conditions and local stability analysis. Analytical results are also provided for the necessary circumstances for the Hopf bifurcation initiated by the fractional derivative order to occur. We also demonstrated the global asymptotic stability for the positive equilibrium point in both the strong and weak Allee effect cases by selecting an appropriate Lyapunov function. This study's innovation is its comparative investigation of the stability of the strong and weak Allee effects. To conclude, numerical simulations validate the theoretical findings and provide a means to investigate the system's more dynamical behaviours.
本研究提出并分析了一个修正的捕食者 - 猎物模型，该模型考虑了阿利效应（Allee effect）对猎物种群扩张速率的双重影响。采用卡普托分数阶导数，我们考虑了记忆对所提出模型的影响。我们进而研究了所提出模型的重要数学方面，包括非整数阶系统解的唯一性、非负性、有界性和存在性。此外，在马蒂尼翁条件下，研究了强阿利效应和弱阿利效应的所有潜在平衡点，以及条件的当前状态和局部稳定性分析。还给出了分数阶导数引发霍普夫分岔发生的必要条件的分析结果。我们还通过选择合适的李雅普诺夫函数，证明了在强阿利效应和弱阿利效应情况下正平衡点的全局渐近稳定性。本研究的创新之处在于对强阿利效应和弱阿利效应稳定性的比较研究。最后，数值模拟验证了理论结果，并提供了一种研究系统更多动态行为的方法。
