Department of Mathematics, Ariel University, Ariel 40700, Israel.
School of Mathematics and Information Science, Shaanxi Normal University, Xi'an 710062, China.
Math Med Biol. 2024 Mar 15;41(1):19-34. doi: 10.1093/imammb/dqae001.
Stochastically perturbed models, where the white noise type stochastic perturbations are proportional to the current system state, the most realistically describe real-life biosystems. However, such models essentially have no equilibrium states apart from one at the origin. This feature makes analysis of such models extremely difficult. Probably, the best result that can be found for such models is finding of accurate estimations of a region in the model phase space that serves as an attractor for model trajectories. In this paper, we consider a classical stochastically perturbed Lotka-Volterra model of competing or symbiotic populations, where the white noise type perturbations are proportional to the current system state. Using the direct Lyapunov method in a combination with a recently developed technique, we establish global asymptotic properties of this model. In order to do this, we, firstly, construct a Lyapunov function that is applicable to the both competing (and globally stable) and symbiotic deterministic Lotka-Volterra models. Then, applying this Lyapunov function to the stochastically perturbed model, we show that solutions with positive initial conditions converge to a certain compact region in the model phase space and oscillate around this region thereafter. The direct Lyapunov method allows to find estimates for this region. We also show that if the magnitude of the noise exceeds a certain critical level, then some or all species extinct via process of the stochastic stabilization ('stabilization by noise'). The approach applied in this paper allows to obtain necessary conditions for the extinction. Sufficient conditions for the extinction (that for this model occurs via the process that is known as the 'stochastic stabilization', or the 'stabilization by noise') are found applying the Khasminskii-type Lyapunov functions.
随机摄动模型中，白噪声类型的随机摄动与当前系统状态成正比，最能真实地描述现实生活中的生物系统。然而，这样的模型除了原点之外，实际上没有平衡状态。这个特点使得对这类模型的分析变得极其困难。可能，对于这类模型能找到的最好结果是找到模型相空间中一个作为模型轨迹吸引子的区域的精确估计。在本文中，我们考虑了一个经典的随机摄动Lotka-Volterra 竞争或共生种群模型，其中白噪声类型的摄动与当前系统状态成正比。我们使用直接 Lyapunov 方法与最近开发的技术相结合，建立了该模型的全局渐近性质。为了做到这一点，我们首先构建了一个 Lyapunov 函数，该函数适用于竞争（和全局稳定）和共生确定性 Lotka-Volterra 模型。然后，将这个 Lyapunov 函数应用于随机摄动模型，我们证明了具有正初始条件的解会收敛到模型相空间中的一个确定的紧致区域，并在此区域内振荡。直接 Lyapunov 方法可以找到这个区域的估计值。我们还表明，如果噪声的幅度超过某个临界水平，那么某些或所有物种将通过随机稳定化过程（即“噪声稳定化”）灭绝。本文应用的方法允许获得灭绝的必要条件。灭绝的充分条件（对于这个模型来说，是通过称为“随机稳定化”或“噪声稳定化”的过程发生的）是通过 Khasminskii 型 Lyapunov 函数找到的。
