  Population genetics struggles to model extinction; standard models track the
relative rather than absolute fitness of genotypes, while the exceptions
describe only the short-term transition from imminent doom to evolutionary
rescue. But extinction can result from failure to adapt not only to
catastrophes, but also to a backlog of environmental challenges. We model
long-term evolution to long series of small challenges, where fitter
populations reach higher population sizes. The population's long-term fitness
dynamic is well approximated by a simple stochastic Markov chain model.
Long-term persistence occurs when the rate of adaptation exceeds the rate of
environmental deterioration for some genotypes. Long-term persistence times are
consistent with typical fossil species persistence times of several million
years. Immediately preceding extinction, fitness declines rapidly, appearing as
though a catastrophe disrupted a stably established population, even though
gradual evolutionary processes are responsible. New populations go through an
establishment phase where, despite being demographically viable, their
extinction risk is elevated. Should the population survive long enough,
extinction risk later becomes constant over time.
