Discrete time Markov chain of a dynamical system with a rest phase. (2016)

• Record Type: Journal Article
• Authors: Malik, Tufail
• Is Part Of: International journal of applied nonlinear science. Volume 2:Number 3(2016)
• Page Start: 137
• Page End: 152
• Publication Date: 2016
• Subjects: stochastic modelling -- population dynamics -- discrete time Markov chain -- dynamical systems -- rest phases -- cell division -- Monte Carlo simulation; Science -- Mathematical models -- Periodicals; Engineering -- Mathematical models -- Periodicals; Nonlinear theories -- Periodicals; 501.5118
• Languages: English
• ISSNs: 1752-2862

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.