This research studies the robustness of permanence and the continuous dependence of the stationary distribution on the parameters for a stochastic predator–prey model with Beddington–DeAngelis functional response. We show that if the model is extinct (resp. permanent) for a parameter, it is still extinct (resp. permanent) in a neighbourhood of this parameter. In the case of extinction, the Lyapunov exponent of predator quantity is negative and the prey quantity converges almost to the saturated situation, where the predator is absent at an exponential rate. Under the condition of permanence, the unique stationary distribution converges weakly to the degenerate measure concentrated at the unique limit cycle or at the globally asymptotic equilibrium when the diffusion term tends to 0.

中文翻译：

---

平稳分布对随机捕食者-被捕食者模型参数的连续依赖性

本研究研究了具有贝丁顿-德安吉利斯函数响应的随机捕食者-被捕食者模型的持久性和平稳分布对参数的连续依赖性。我们证明，如果模型对于某个参数来说是灭绝的（或永久的），那么它在该参数的邻域中仍然是灭绝的（或永久的）。在灭绝的情况下，捕食者数量的李亚普诺夫指数为负，猎物数量几乎收敛到饱和状态，此时捕食者以指数速率消失。在永久性条件下，当扩散项趋于0时，唯一平稳分布弱收敛于集中在唯一极限环或全局渐近平衡处的简并测度。