In 1993, Holt and Lawton introduced a stochastic model of two host species parasitized by a common parasitoid species. We introduce and analyze a generalization of these stochastic difference equations with any number of host species, stochastically varying parasitism rates, stochastically varying host intrinsic fitnesses, and stochastic immigration of parasitoids. Despite the lack of direct, host density-dependence, we show that this system is dissipative i.e. enters a compact set in finite time for all initial conditions. When there is a single host species, stochastic persistence and extinction of the host is characterized using external Lyapunov exponents corresponding to the average per-capita growth rates of the host when rare. When a single host persists, say species $ i $, a explicit expression is derived for the average density, $ P_i^* $, of the parasitoid at the stationary distributions supporting both species. When there are multiple host species, we prove that the host species with the largest $ P_i^* $ value stochastically persists, while the other host species are asymptotically driven to extinction. A review of the main mathematical methods used to prove the results and future challenges are given.

中文翻译：

---

的 \ begin {document} $ P ^ * $ \ end {document} 随机竞争的Holt-Lawton随机模型中的规则

在1993年，Holt和Lawton引入了一个随机模型，该模型随机地将两个寄主物种寄生了一个共同的寄生类物种。我们介绍并分析了这些随机差分方程的泛化，其中包含任意数量的宿主物种，随机变化的寄生率，随机变化的宿主固有适应度以及寄生虫的随机迁移。尽管缺乏直接的，宿主密度依赖性，我们证明了该系统是耗散的，即在所有初始条件下都在有限时间内输入了一个紧凑集合。如果只有一个寄主物种，则使用外部Lyapunov指数来表征寄主的随机持久性和灭绝，而外部Lyapunov指数则对应于稀有主机的平均人均增长率。当单个宿主持续存在时，例如物种$ i $，就得出平均密度$ P_i ^ * $的显式表达式，在支持两个物种的固定分布处的寄生寄生虫的数量。当存在多个寄主物种时，我们证明$ P_i ^ * $值最大的寄主物种将随机存在，而其他寄主物种则被渐进地灭绝。回顾了用于证明结果和未来挑战的主要数学方法。