On The Drift Paradox in a Regime-Switching Model

This note is motivated by the article by F. Lutscher, E. Pachepsky, and M. Lewis (2005), The Effect of Dispersal Patterns on Stream Populations SIAM Rev. Vol. 47 No. 4 pp. 749-772 on the drift paradox. We consider the case of a regime switching probabilistic model for a population of organisms living in a one dimensional environment with drift towards an absorbing boundary of the type introduced by Lutscher et al. (2005). In particular, the two regimes consist of birth/death style demographics governing the evolution of the immobile regime, and some form of advection-dispersion governing the evolution of the mobile regime, together with a regime-switching mechanism linking the two. In the present note it is shown for the regime-switching model, and in contrast to the results in the afore cited work, for any finite advection speed, no matter how large, there is a finite critical length of the domain above which the population can persist.