  Two mathematical models of macroevolution are studied. These models have
population dynamics at the species level, and mutations and extinction of
species are also included. The population dynamics are updated by difference
equations with stochastic noise terms that characterize population
fluctuations. The effects of the stochastic population fluctuations on
diversity and total population sizes on evolutionary time scales are studied.
In one model, species can make either predator-prey, mutualistic, or
competitive interactions, while the other model allows only predator-prey
interactions. When the noise in the population dynamics is strong enough, both
models show intermittent behavior and their power spectral densities show
approximate $1/f$ fluctuations. In the noiseless limit, the two models have
different power spectral densities. For the predator-prey model, $1/f^2$
fluctuations appears, indicating random-walk like behavior, while the other
model still shows $1/f$ noise. These results indicate that stochastic
population fluctuations may significantly affect long-time evolutionary
dynamics.
