On the basic reproduction number in a random environment

The concept of basic reproduction number R0 in population dynamics is studied in the case of random environments. For simplicity the dependence between successive environments is supposed to follow a Markov chain. R0 is the spectral radius of a next-generation operator. Its position with respect to 1 always determines population growth or decay in simulations, unlike another parameter suggested in a recent article (Hernandez-Suarez et al., Theor Popul Biol, doi: 10.1016/j.tpb.2012.05.004 , 2012). The position of the latter with respect to 1 determines growth or decay of the population's expectation. R0 is easily computed in the case of scalar population models without any structure. The main emphasis is on discrete-time models but continuous-time models are also considered.