Department of Mathematics, Tufts University, Bromfield-Pearson Hall, 503 Boston Avenue, Medford, MA, 02155, USA.
Department of Mathematics, University of Alabama, 345 Gordon Palmer Hall, Box 870350, Tuscaloosa, AL, 35487-0350, USA.
J Math Biol. 2021 May 7;82(6):56. doi: 10.1007/s00285-021-01606-1.
We analyze a general theory for coexistence and extinction of ecological communities that are influenced by stochastic temporal environmental fluctuations. The results apply to discrete time (stochastic difference equations), continuous time (stochastic differential equations), compact and non-compact state spaces and degenerate or non-degenerate noise. In addition, we can also include in the dynamics auxiliary variables that model environmental fluctuations, population structure, eco-environmental feedbacks or other internal or external factors. We are able to significantly generalize the recent discrete time results by Benaim and Schreiber (J Math Biol 79:393-431, 2019) to non-compact state spaces, and we provide stronger persistence and extinction results. The continuous time results by Hening and Nguyen (Ann Appl Probab 28(3):1893-1942, 2018a) are strengthened to include degenerate noise and auxiliary variables. Using the general theory, we work out several examples. In discrete time, we classify the dynamics when there are one or two species, and look at the Ricker model, Log-normally distributed offspring models, lottery models, discrete Lotka-Volterra models as well as models of perennial and annual organisms. For the continuous time setting we explore models with a resource variable, stochastic replicator models, and three dimensional Lotka-Volterra models.
我们分析了一个受随机时间环境波动影响的生态群落共存和灭绝的一般理论。该结果适用于离散时间（随机差分方程）、连续时间（随机微分方程）、紧和非紧状态空间以及退化或非退化噪声。此外，我们还可以在动力学中包含辅助变量，以模拟环境波动、种群结构、生态环境反馈或其他内部或外部因素。我们能够将 Benaim 和 Schreiber（J Math Biol 79:393-431, 2019）最近的离散时间结果显著推广到非紧状态空间，并提供更强的持久性和灭绝结果。Hening 和 Nguyen（Ann Appl Probab 28(3):1893-1942, 2018a）的连续时间结果得到了强化，包括退化噪声和辅助变量。我们使用一般理论解决了几个例子。在离散时间中，我们对一个或两个物种的动力学进行分类，并研究了 Ricker 模型、对数正态分布后代模型、彩票模型、离散 Lotka-Volterra 模型以及多年生和一年生生物模型。对于连续时间设置，我们探索了带有资源变量的模型、随机复制模型和三维 Lotka-Volterra 模型。
