Stochastic Equilibria in Nonhomogeneous Markov Population Replacement Processes

This paper is an exposition on the dynamic stochastic equilibria in multi-type time dependent Markov population processes for different types of environmental behaviour. The joint multivariate cumulant generating function of the various population sizes at any time is derived together with the covariance structure both with and without time lags. An asymptotic analysis shows that there are either one or several equilibria of dynamic character, depending critically on the behaviour of the system parameters as well as on the total population size. The possible stochastic equilibria are characterized as stable or unstable according to whether or not they are independent of initial conditions.