Comparison of Markov Chain and Stochastic Differential Equation Population Models Under Higher-Order Moment Closure Approximations

Abstract

Continuous time Markov chain (CTMC) and It stochastic differential equation (SDE) models are derived for a population with births, immigration and deaths (BID model). Differential equations are derived for the moments of the distribution for each stochastic model. Each moment differential equation depends on higher-order moments. Assumptions are made regarding higher-order moments to form a finite, solvable system. Conditions are given under which the CTMC and SDE BID models have the same moment solution or the same stationary solution. The close agreement between the CTMC and SDE models is illustrated in three numerical examples based on normal or log-normal moment closure assumptions.

Keywords:

• Itô stochastic differential equation
• Kolmogorov differential equation
• Markov chain
• Metapopulation
• Moment closure

Mathematics Subject Classification:

• Primary 60J20, 60G35
• Secondary 92D25

Financial support was provided by the Texas Advanced Research Program 003644-0001-2006 and by the National Science Foundation DMS-0718302.