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We consider a collection of Markov chains that model the evolution ofmultitype biological populations. The state space of the chains is the positiveorthant, and the boundary of the orthant is absorbing representing theextinction states of different population types. We are interested in thelong-term behavior of the Markov chain away from extinction, under a smallnoise scaling. Under this scaling, the trajectory of the Markov process overany compact interval converges in distribution to the solution of an ordinarydifferential equation (ODE) evolving in the positive orthant. We study theasymptotic behavior of the quasi-stationary distributions (QSD) in this scalingregime. Our main result shows that the limit points of the QSD are supported onthe union of interior attractors of the flow determined by the ODE. We alsogive lower bounds on expected extinction times which scale exponentially withthe system size. Results of this type when the deterministic dynamical systemobtained under the scaling limit is given by a discrete time evolution equationand the dynamics are essentially in a compact space (namely, the one step mapis a bounded function) have been studied by Faure and Schreiber (2014). Ourresults extend these to a setting of an unbounded state space and continuoustime dynamics. The proofs rely on uniform large deviation results for smallnoise stochastic dynamical systems and methods from the theory of dynamicalsystems. In general QSD for Markov chains with absorbing states and unboundedstate spaces may not exist. We study one basic family of Binomial-Poissonmodels in the positive orthant where one can use Lyapunov function methods toestablish existence of QSD and also to argue the tightness of the QSD of thescaled sequence of Markov chains. The results from the first part are then usedto characterize the support of limit points of this sequence of QSD.

Asymptotics of Quasi-Stationary Distributions of Small Noise Stochastic Dynamical Systems in Unbounded Domains