Fisher equation with turbulence in one dimension

As an example of life at high Reynolds number, we investigate the dynamics of the Fisher equation for the spreading of micro-organisms in one dimension subject to both turbulent convection and diffusion. We show that for strong enough turbulence, bacteria, for example, track, in a quasilocalized fashion (with remarkably long persistence times), sinks in the turbulent field. An important consequence is a large reduction in the carrying capacity of the fluid medium. We analytically determine the regimes where this quasi-localized behavior occurs and test our predictions by numerical simulations.