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This work is concerned with optimal harvesting problems in random environments. In contrast to the existing literature, the Markov chain is hidden and can only be observed in a Gaussian white noise in our work. We first use the Wonham filter to estimate the Markov chain from the observable evolution of the given process so as to convert the original problem to a completely observable one. Then we treat the resulting optimal control problem. Because the problem is virtually impossible to solve in closed form, our main effort is devoted to developing numerical approximation algorithms. To approximate the value function and optimal strategies, Markov chain approximation methods are used to construct a discrete-time controlled Markov chain. Convergence of the algorithm is proved by weak convergence method and suitable scaling. A numerical example is provided to demonstrate the results. [ABSTRACT FROM AUTHOR]

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]