Bolker and Pacala recently introduced a model of an evolving population in which an individual's fecundity is reduced in proportion to the "local population density." We consider two versions of this model and prove complementary extinction/persistence results, one for each version. Roughly, if individuals in the population disperse sufficiently quickly relative to the range of the interaction induced by the density dependent regulation, then the population has positive chance of survival, whereas, if they do not, then the population will die out.
"Survival and extinction in a locally regulated population." Ann. Appl. Probab. 14 (1) 188 - 214, February 2004. https://doi.org/10.1214/aoap/1075828051