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

Authors: Tufail Malik

Addresses: Department of Mathematics and Applied Sciences, Khalifa University of Science, Technology and Research, Abu Dhabi, UAE; Department of Mathematics, University of Manitoba, Winnipeg, Manitoba, R3T 5N2, Canada

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.

Keywords: stochastic modelling; population dynamics; discrete time Markov chain; dynamical systems; rest phases; cell division; Monte Carlo simulation.

DOI: 10.1504/IJANS.2016.076996

International Journal of Applied Nonlinear Science, 2016 Vol.2 No.3, pp.137 - 152

Received: 15 Sep 2014
Accepted: 22 Feb 2015

Published online: 17 Jun 2016 *

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