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Abstract We consider a collection of Markov chains that model the evolution of multitype biological populations. The state space of the chains is the positive orthant, and the boundary of the orthant is the absorbing state for the Markov chain and represents the extinction states of different population types. We are interested in the long-term behavior of the Markov chain away from extinction, under a small noise scaling. Under this scaling, the trajectory of the Markov process over any compact interval converges in distribution to the solution of an ordinary differential equation (ODE) evolving in the positive orthant. We study the asymptotic behavior of the quasi-stationary distributions (QSD) in this scaling regime. Our main result shows that, under conditions, the limit points of the QSD are supported on the union of interior attractors of the flow determined by the ODE. We also give lower bounds on expected extinction times which scale exponentially with the system size. Results of this type when the deterministic dynamical system obtained under the scaling limit is given by a discrete-time evolution equation and the dynamics are essentially in a compact space (namely, the one-step map is a bounded function) have been studied by Faure and Schreiber (2014). Our results extend these to a setting of an unbounded state space and continuous-time dynamics. The proofs rely on uniform large deviation results for small noise stochastic dynamical systems and methods from the theory of continuous-time dynamical systems. In general, QSD for Markov chains with absorbing states and unbounded state spaces may not exist. We study one basic family of binomial-Poisson models in the positive orthant where one can use Lyapunov function methods to establish existence of QSD and also to argue the tightness of the QSD of the scaled sequence of Markov chains. The results from the first part are then used to characterize the support of limit points of this sequence of QSD.
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When Can We Treat Trajectories As Points?
In the formal verification of dynamical systems, one often looks at a trajectory through a state space as a sample behavior of the system. Thus, metrics on trajectories give important information about the different behavior of the system given different starting states. In the important special case of linear dynamical systems, the set of trajectories forms a finite-dimensional vector space. In this paper, we exploit this vector space structure to define (semi)norms on the trajectories, give an isometric embedding from the trajectory metric into low-dimensional Euclidean space, and bound the Lipschitz constant on the map from start states to trajectories as measured in one of several different metrics. These results show that for an interesting class of trajectories, one can treat the trajectories as points while losing little or no information.
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- Award ID(s):
- 1739936
- PAR ID:
- 10075298
- Date Published:
- Journal Name:
- Canadian Conference on Computational Geometry
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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