具有非高斯稳定Lévy噪声的随机Rosenzweig-MacArthur捕食模型的分岔与混沌行为
Bifurcation and chaotic behaviour in stochastic Rosenzweig-MacArthur prey-predator model with non-Gaussian stable Lévy noise
摘要：
对α-稳定Lévy运动驱动的随机Rosenzweig-MacArthur模型进行了动力学分析.我们分析了平衡点的存在性，并提供了一个清晰的说明，他们的稳定性。结果表明，该非线性模型最多有三个平衡点。如果共存平衡点存在，它是渐近稳定的，吸引所有附近的轨迹。绘制相图，以获得有用的见解捕食者-捕食者相互作用的动力学基础。具体地说，我们提出了一个跨临界分岔曲线，系统分叉。稳态概率密度由非局域Fokker-Planck方程描述，并通过一些数值模拟得到证实。本文应用蒙特卡罗方法，利用统计数据，对随机系统的参数变化进行了大量的模拟。对于任意接近原点的初始条件，噪声项中的参数变化会导致未来路径或轨迹的显著不同，这反映了互惠相互作用的两种群食饵-捕食者系统在随机影响下的混沌行为.
英文摘要：
We perform dynamical analysis on a stochastic Rosenzweig-MacArthur model driven by α-stable Lévy motion. We analyze the existence of the equilibrium points, and provide a clear illustration of their stability. It is shown that the nonlinear model has at most three equilibrium points. If the coexistence equilibrium exists, it is asymptotically stable attracting all nearby trajectories. The phase portraits are drawn to gain useful insights into the dynamical underpinnings of prey-predator interaction. Specifically, we present a transcritical bifurcation curve at which system bifurcates. The stationary probability density is characterized by the non-local Fokker-Planck equation and confirmed by some numerical simulations. By applying Monte Carlo method and using statistical data, we plot a substantial number of simulated trajectories for stochastic system as parameter varies. For initial conditions that are arbitrarily close to the origin, parameter changes in noise terms can lead to significantly different future paths or trajectories with variations, which reflect chaotic behaviour in mutualistically interacting two-species prey-predator system subject to stochastic influence.
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