We analyze ecological systems that are influenced by random environmental fluctuations. We first provide general conditions which ensure that the species coexist and the system converges to a unique invariant probability measure (stationary distribution). Since it is usually impossible to characterize this invariant probability measure analytically, we develop a powerful method for numerically approximating invariant probability measures. This allows us to shed light upon how the various parameters of the ecosystem impact the stationary distribution. We analyze different types of environmental fluctuations. At first we study ecosystems modeled by stochastic differential equations. In the second setting we look at piecewise deterministic Markov processes. These are processes where one follows a system of differential equations for a random time, after which the environmental state changes, and one follows a different set of differential equations—this procedure then gets repeated indefinitely. Finally, we look at stochastic differential equations with switching, which take into account both the white noise fluctuations and the random environmental switches. As applications of our theoretical and numerical analysis, we look at competitive Lotka–Volterra, Beddington–DeAngelis predator–prey, and rock–paper–scissors dynamics. We highlight new biological insights by analyzing the stationary distributions of the ecosystems and by seeing how various types of environmental fluctuations influence the long term fate of populations.

我们分析受随机环境波动影响的生态系统。我们首先提供确保物种共存并且系统收敛到唯一不变概率度量（平稳分布）的一般条件。由于通常不可能通过分析来表征这种不变概率度量，因此我们开发了一种强大的方法来数值逼近不变概率度量。这使我们能够阐明生态系统的各种参数如何影响平稳分布。我们分析不同类型的环境波动。首先，我们研究由随机微分方程建模的生态系统。在第二个设置中，我们查看分段确定性马尔可夫过程。在这些过程中，人们在随机时间内遵循微分方程组，之后环境状态发生变化，并且遵循一组不同的微分方程——然后这个过程无限重复。最后，我们看一下带有切换的随机微分方程，它同时考虑了白噪声波动和随机环境切换。作为我们的理论和数值分析的应用，我们研究了竞争性 Lotka-Volterra、Beddington-DeAngelis 捕食者-猎物和岩石-纸-剪刀动力学。我们通过分析生态系统的平稳分布以及了解各种类型的环境波动如何影响种群的长期命运来突出新的生物学见解。我们看一下带有切换的随机微分方程，它同时考虑了白噪声波动和随机环境切换。作为我们的理论和数值分析的应用，我们研究了竞争性 Lotka-Volterra、Beddington-DeAngelis 捕食者-猎物和岩石-纸-剪刀动力学。我们通过分析生态系统的平稳分布以及了解各种类型的环境波动如何影响种群的长期命运来突出新的生物学见解。我们看一下带有切换的随机微分方程，它同时考虑了白噪声波动和随机环境切换。作为我们的理论和数值分析的应用，我们研究了竞争性 Lotka-Volterra、Beddington-DeAngelis 捕食者-猎物和岩石-纸-剪刀动力学。我们通过分析生态系统的平稳分布以及了解各种类型的环境波动如何影响种群的长期命运来突出新的生物学见解。