  Biodiversity and extinction are central issues in evolution. Dynamical
balance among different species in ecosystems is often described by
deterministic replicator equations with moderate success. However, fluctuations
are inevitable, either caused by external environment or intrinsic random
competitions in finite populations, and the evolutionary dynamics is stochastic
in nature. Here we show that, after appropriate coarse-graining, random
fluctuations generate dissipation towards extinction because the evolution
trajectories in the phase space of all competing species possess positive
curvature. As a demonstrating example, we compare the fluctuation-induced
dissipative dynamics in Lotka-Volterra model with numerical simulations and
find impressive agreement. Our finding is closely related to the
fluctuation-dissipation theorem in statistical mechanics but the marked
difference is the non-equilibrium essence of the generic evolutionary dynamics.
As the evolving ecosystems are far from equilibrium, the relation between
fluctuations and dissipations is often complicated and dependent on microscopic
details. It is thus remarkable that the generic positivity of the trajectory
curvature warrants dissipation arisen from the seemingly harmless fluctuations.
The unexpected dissipative dynamics is beyond the reach of conventional
replicator equations and plays a crucial role in investigating the biodiversity
in ecosystems.
