Discrete time Markov chain of a dynamical system with a rest phase

Abstract

A stochastic model, in the form of a discrete time Markov chain, is constructed to describe the dynamics of a population that grows through cell-division, and which has a rest state (no growth or death). Transition probabilities are described and the asymptotic dynamics are compared with those of a deterministic discrete dynamical system. In contrast with the deterministic model, the stochastic model predicts extinction even with positive net growth. With zero net growth, the deterministic model has a locally-asymptotically stable steady state, but the stochastic model asserts no absorbing states (except for certain limiting cases). The existence of a unique stationary probability distribution is established in this case, and a transition probability matrix is constructed to plot such distributions. Expected extinction times of populations, with and without a rest phase, are compared numerically, using Monte Carlo simulations.