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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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This content will become publicly available on July 2, 2026
What is the graph of a dynamical system?
Some of the basic properties of any dynamical system can be summarized by a graph. The dynamical systems in our theory run from maps like the logistic map to ordinary differential equations to dissipative partial differential equations. Our goal has been to define a meaningful concept of graph of any dynamical system. As a result, we base our definition of “chain graph” on “epsilon-chains”, defining both nodes and edges of the graph in terms of chains. In particular, nodes are often maximal limit sets and there is an edge between two nodes if there is a trajectory whose forward limit set is in one node and its backward limit set is in the other. Our initial goal was to prove that every “chain graph” of a dynamical system is, in some sense, connected, and we prove connectedness under mild hypotheses.
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- Award ID(s):
- 2308225
- PAR ID:
- 10612866
- Publisher / Repository:
- Springer
- Date Published:
- Journal Name:
- Nonlinear Dynamics
- ISSN:
- 0924-090X
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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