Markov population replacement processes

The paper considers a migration process whose singleton process is a time-dependent Markov replacement process. For the singleton process, which may be treated as either open or closed, it studies the limiting distribution, the distribution of the time to replacement and related quantities. For a replacement process in equilibrium the paper obtains a version of LIttle’s law and it provides conditions for reversibility. For the resulting linear population process the paper characterizes exponential ergodicity for two types of environmental behaviour, i.e. either convergent or cyclic, and finally for large population sizes a diffusion approximation analysis is provided.