We consider the species-area problem in ecology. The aim is to estimate thetotal number of species in a large area by extrapolating a species-area curve.Starting with a stationary random field as the background process that isdecisive of species occurrence, we argue that asymptotically, the number ofunseen species and the species number increment will decrease hyperbolicallywith increasing area. In particular, stationarity of the background processleads to a fast hyperbolic rate. Some vascular plant speciesrichness datafrom Switzerland are used to motivate discussions.