  The recent discovery of potentially habitable exoplanets around Proxima
Centauri and TRAPPIST-1 has attracted much attention due to their potential for
hosting life. We delineate a simple model that accurately describes the
evolution of biological diversity on Earth. Combining this model with
constraints on atmospheric erosion and the maximal evolutionary timescale
arising from the star's lifetime, we arrive at two striking conclusions: (i)
Earth-analogs orbiting low-mass M-dwarfs are unlikely to be inhabited, and (ii)
K-dwarfs and some G-type stars are potentially capable of hosting more complex
biospheres than the Earth. Hence, future searches for biosignatures may have
higher chances of success when targeting planets around K-type stars.
