We study a Markov birth-and-death process on a space of locally finite configurations, which describes an ecological model with a density-dependent fecundity regulation mechanism. We establish existence and uniqueness of this process and analyze its properties. In particular, we show global time-space boundedness of the population density and, using a constructed Foster–Lyapunov-type function, we study return times to certain level sets of tempered configurations. We also find sufficient conditions that the degenerate invariant distribution is unique for the considered process.

中文翻译：

---

空间生死过程中的生育力调控

我们研究了局部有限配置空间上的马尔可夫生死过程，该过程描述了具有密度依赖性繁殖力调节机制的生态模型。我们确定了这个过程的存在性和唯一性，并分析了它的性质。特别是，我们展示了人口密度的全局时空有界性，并且使用构造的 Foster-Lyapunov 型函数，我们研究了返回时间到特定水平的回火配置集。我们还发现了退化不变分布对于所考虑的过程是唯一的充分条件。