Summary: We investigate the time evolution of a prototypical population biology reaction which involves reproduction, self-regulation and competitive annihilation of two distinct species. In one dimension, we use a quasistatic analysis to argue that for a system with equal initial densities of two strongly competing species, an alternating pattern of domains forms whose lengths grow logarithmically with time. A scaling analysis of the underlying master equation, as well as numerical integration of the reaction-diffusion equations support this result. For unequal initial densities, the concentration of the minority species undergoes a power-law decay with a non-universal exponent.
We generalize the model by allowing for a nonlinear self-regulation term in the rate equations. As a function of the exponent of this nonlinearity, the typical domain size may grow either as a power law with time or saturate at a finite value. Our general approach also suggests that a coarsening domain mosaic occurs in arbitrary spatial dimensions. In two dimensions, numerical integration of the reaction-diffusion equations indicates that the average domain area grows approximately as \(t^{0.84}\).