Considering the complexity of random variations in ecosystem, a n-species Gilpin-Ayala competition system with nonlinear noises is studied in this paper. Using our developed $ϵ$-stochastic criterion method in eliminating nonlinear perturbations, we construct several suitable Lyapunov functions to obtain the threshold for the existence and uniqueness of an ergodic stationary distribution, which reflects population coexistence in a long term. Moreover, we establish the sufficient conditions for population extinction. Finally, several numerical simulations are performed and our analytical results are discussed by comparison with the existing papers.

中文翻译：

---

具有非线性随机摄动的n种群Gilpin–Ayala竞争系统的遍历平稳分布和灭绝

考虑到生态系统随机变化的复杂性，本文研究了具有非线性噪声的n种Gilpin-Ayala竞争系统。使用我们开发的$ϵ$随机准则消除非线性扰动，我们构造了几个合适的Lyapunov函数来获得遍历平稳分布的存在性和唯一性的阈值，该阈值长期反映了种群的共存性。此外，我们为人口灭绝建立了充分的条件。最后，进行了一些数值模拟，并与现有论文进行了比较，讨论了我们的分析结果。