Widening Disparity and its Suppression in a Stochastic Replicator Model

Winner-take-all phenomena are observed in various competitive systems. We find similar phenomena in replicator models with randomly fluctuating growth rates. The disparity between winners and losers increases indefinitely, even if all elements are statistically equivalent. A lognormal distribution describes well the nonstationary time evolution. If a nonlinear load corresponding to progressive taxation is introduced, a stationary distribution is obtained and disparity widening is suppressed.