摘要
生态系统的动态行为研究是生态学与数学交叉领域的热点问题，传统整数阶微分方程在描述生态过程时存在局限性，而分数阶微分方程因其记忆性和遗传性特征为生态建模提供了新思路。本研究以分数阶微分方程为基础，构建了包含捕食-被捕食关系的非线性生态模型，并通过稳定性理论分析了系统平衡点的存在性与稳定性。采用Lyapunov直接法和数值模拟技术，探讨了分数阶阶数对系统动力学行为的影响，揭示了分数阶参数如何调控生态系统的长期演化趋势。研究表明，分数阶阶数不仅影响系统的稳定区域，还可能引发振荡收敛或混沌现象，这为理解生态系统复杂性提供了新的视角。本研究的主要贡献在于首次将分数阶微分方程应用于捕食-被捕食模型的稳定性分析，提出了基于分数阶特性的生态动力学机制解释，为生态建模与预测提供了更精确的数学工具。关键词 分数阶微分方程；捕食-被捕食模型；稳定性分析；生态动力学；Lyapunov直接法
Abstract
The study of ecosystem dynamics is a hotspot in the interdisciplinary field of ecology and mathematics. Conventional integer-order differential equations have limitations in describing ecological processes, whereas fractional-order differential equations, characterized by their memory and hereditary properties, offer new insights for ecological modeling. This study establishes a nonlinear ecological model incorporating predator-prey interactions based on fractional-order differential equations and investigates the existence and stability of system equilibrium points through stability theory. By employing the Lyapunov direct method and numerical simulation techniques, the impact of the fractional-order parameter on the system's dynamical behavior is explored, revealing how this parameter regulates the long-term evolutionary trends of ecosystems. The findings indicate that the fractional order not only affects the stability region of the system but may also induce oscillatory convergence or chaotic phenomena, providing a novel perspective for understanding ecological complexity. The primary contribution of this research lies in its pioneering application of fractional-order differential equations to the stability analysis of predator-prey models, proposing explanations of ecological dynamical mechanisms grounded in fractional-order characteristics, and offering more precise mathematical tools for ecological modeling and prediction.
Keywords Fractional Order Differential Equation;Predator-Prey Model;Stability Analysis;Ecological Dynamics;Lyapunov Direct Method
目录
摘要 I
Abstract II
1 绪论 1
1.1 基于分数阶微分方程的生态模型研究背景 1
1.2 分数阶微分方程在生态稳定性分析中的意义 1
2 分数阶微分方程理论基础 3
2.1 分数阶微分方程的基本概念 3
2.2 分数阶微分方程的数学特性分析 3
2.3 生态模型中分数阶微分方程的应用框架 4
3 生态模型的稳定性条件分析 5
3.1 稳定性定义与判别准则 5
3.2 分数阶系统稳定性的影响因素探讨 5
3.3 典型生态模型的稳定性条件验证 6
4 案例分析与数值模拟 7
4.1 生态模型的具体案例构建 7
4.2 分数阶微分方程的数值求解方法 7
4.3 模拟结果与稳定性分析对比 8
结论 9
参考文献 10
致谢 11
