On the reaction-diffusion replicator systems: Spatial patterns and   asymptotic behavior
  The replicator equation is ubiquitous for many areas of mathematical biology. One of major shortcomings of this equation is that it does not allow for an explicit spatial structure. Here we review analytical approaches to include spatial variables to the system. We also provide a concise exposition of the results concerning the appearance of spatial patterns in replicator reaction-diffusion systems. 
Artem S. Novozhilov, Vladimir P. Posvyanskii, and Alexander S. Bratus
q-bio.PE math.AP