The difficulties and hindrances of palaeobiogeography and historical biogeography in its long, sterile search for centres of origin or ancestral areas of species, leading some authors to withdraw the centre-of-origin as a non-scientific concept, are here considered as signals and not artefacts for the recognition of patterns in the biogeographic history of lineages. The time-symmetric model, which assumes gradual origination and extinction processes of a species, is here questioned and an alternative time-asymmetric model is proposed. The origination and expansion processes of a species history would be much faster and more unpredictable than the often gradual, long, predictable extinction process marked by previous signals of geographic area contraction. Monitored biological invasions, plankton blooms and episodes of coordinated migrations illustrate the fast expansion of novel species behaving as dissipative structures. The asymmetric biogeographic model is tested through palaeobiogeographic data on Old World mammal species and by the consistent preference of biostratigraphers for species appearance against species extinction as time-marker bioevents. Time-asymmetry is a common phenomenon of nature, indicating that asymmetry could well be a general property of Time itself.