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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]

Focusing on asymptotic behavior of a stochastic SIR epidemic model represented by a system of stochastic differential equations with a degenerate diffusion, this paper provides suficient conditions that are very close to the necessary ones for the permanence. In addition, this paper develops ergodicity of the underlying system. It is proved that the transition probabilities converge in total variation norm to the invariant measure. Our result gives a precise characterization of the support of the invariant measure. Rates of convergence are also ascertained. It is shown that the rate is not too far from exponential in that the convergence speed is of the form of a polynomial of any degree. [ABSTRACT FROM AUTHOR]