Stochastic functional Kolmogorov equations II: Extinction

This work, Part II, together with its companion-Part I develops a new framework for stochastic functional Kolmogorov equations, which are nonlinear differential equations depending on the current as well past states. Because of complexity problems, it is natural to divide our contributions into two parts answer long-standing question in biology and ecology. What minimal conditions long-term persistence extinction population? work provides characterization persistence, whereas this part, main focus. The techniques used paper combination newly developed Itô formula dynamic system approach. Compared study systems without delays, difficulty that infinite dimensional have be treated. characterized after investigating random occupation measures examining behavior around boundaries. Our characterizations reduce delay when there no dependence. A number applications also examined.