How a population's size changes near an absorbing boundary---a threshold, often extinction, from which the population cannot recover---has long been a fixture of ecological theory. Asking whether a small population can invade a resident community, rather than go extinct, is one way to study the ecological mechanisms that maintain biodiversity. Escaping extinction, or any absorbing boundary, is not a black and white matter: some populations go quick, others slow, and all have some probability of being absorbed. The variation in these rates or probabilities may have lasting implications for each population and relate to the nature of the ecological community composed of these populations. I define in a precise way two kinds of variation between species---niche differences, which promote coexistence, and relative fitness differences, which promote competitive exclusion---that depend on the growth rates of small populations invading established communities. In a classic deterministic model of consumer-resource competition, the magnitude of niche and fitness differences are both necessary to understand a relationship between biodiversity and ecosystem-function. In a novel model of a competitive community that includes demographic stochasticity, the variation among species introduced by niche and fitness differences produces dynamics that quickly diverge from predictions of the "neutral theory of biodiversity", while some static properties can remain unaffected. The tools for modeling stochastic population dynamics yield a precise way of calculating the probability that a population fails to escape an absorbing boundary, and this translates into knowledge of the mean time a population persists. In a model of the stochastic infection process for a disease of amphibians, the expected lifetime of an adult host shows little sensitivity to the initial proximity of the pathogen's abundance to its lethal threshold.