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It is widely recognized that unregulated harvesting and hunting of biological resources can be harmful and endanger ecosystems. Therefore, various measures to prevent the biological resources from destruction and to protect the ecological environment have been taken. An effective resolution is to designate protection zones where harvesting and hunting are prohibited. Assuming that migration can occur between protected and unprotected areas, a fundamental question is, how large should a protection zone be so that the species in both the protection subregion and natural environment are able to survive. Devoted to answering the question, this paper aims at studying ecosystems that are subject to random noise represented by Brownian motion. Sufficient conditions for permanence and extinction are obtained, which are sharp and close to necessary conditions. Moreover, ergodicity, convergence of probability measures to that of the invariant measure under total variation norm, and rates of convergence are obtained. [ABSTRACT FROM AUTHOR]

Aiming at reduction of complexity, this work is concerned with two-time-scale Markov chains and applications to quasi-birth-death queues. Asymptotic expansions of probability vectors are constructed and justified. Lumping all states of the Markov chain in each subspace into a single state, an aggregated process is shown to converge to a continuous-time Markov chain whose generator is an average with respect to the stationary measures. Then a suitably scaled sequence is shown to converge to a switching diffusion process. Extensions of the results are presented together with examples of quasi-birth-death queues. [ABSTRACT FROM AUTHOR]