  In this work we study the permanence and extinction of a regime-switching
predator-prey model with Beddington-DeAngelis functional response. The
switching process is used to describe the random changing of corresponding
parameters such as birth and death rates of a species in different
environments. Our criteria can justify whether a prey die out or not when it
will die out in some environments and will not in others. Our criteria are
rather sharp, and they cover the known on-off type results on permanence of
predator-prey models without switching. Our method relies on the recent study
of ergodicity of regime-switching diffusion processes.
