Limits for density dependent time inhomogeneous Markov processes

• A law of large numbers that is applied to a class of Markov processes is developed.
• Time dependence within patch dynamics is introduced into a metapopulation model.
• The LLN shows that the metapopulation model is approximated by a nonautonomous DE.
• Conditions for extinction and persistence of the deterministic process are derived.

A new functional law of large numbers to approximate a time inhomogeneous Markov process that is only density dependent in the limit as an index parameter goes to infinity is developed. This extends previous results by other authors to a broader class of Markov processes while relaxing some of the conditions required for those results to hold. This result is applied to a stochastic metapopulation model that accounts for spatial structure as well as within patch dynamics with the novel addition of time dependent dynamics. The resulting nonautonomous differential equation is analysed to provide conditions for extinction and persistence for a number of examples. This condition shows that the migration of a species will positively impact the reproduction in less populated areas while negatively impacting densely populated areas.