Fitness fluctuations in Bak-Sneppen model

We study the one-dimensional Bak-Sneppen model for the evolution of species in an ecosystem. Of particular interest are the temporal fluctuations in the fitness variables. We numerically compute the power spectral density and apply the finite-size scaling method to get data collapse. A clear signature of $1/f^{\alpha}$ noise with $\alpha\approx 1.2$ (long-time correlations) emerges for both local and global (or average) fitness noises. The limiting value 0 or 2 for the spectral exponent corresponds to the no interaction or random neighbors version model, respectively. The local power spectra are spatially uncorrelated and also show an additional scaling $\sim 1/L$ in the frequency regime $L^{-\lambda}\ll f\ll 1/2$, where $L$ is the linear extent of the system.