This work focuses on multi-species Lotka–Volterra models with regime switching modulated by a continuous-time Markov chain involving a small parameter. The small parameter is used to reflect different rates of the switching among a large number of states representing the discrete events. Using perturbed Lyapunov function methods and the structure of the limit system as a bridge, stochastic permanence and extinction are obtained. Sufficient conditions under which the measures of the original system converge to the invariant measure of that of the limit system are provided. A couple of examples and numerical simulations are given to illustrate our results.

中文翻译：

---

具有弱相互作用和强相互作用的区域切换的多种种群Lotka-Volterra模型的渐近性质

这项工作着重于多物种Lotka–Volterra模型，其模式切换由涉及小参数的连续时间马尔可夫链调制。小参数用于反映代表离散事件的大量状态之间切换的不同速率。利用扰动的Lyapunov函数方法和极限系统的结构作为桥梁，获得了随机的持久性和灭绝性。提供了原始系统的度量收敛到极限系统的不变度量的充分条件。给出了一些例子和数值模拟来说明我们的结果。