Here is a recent paper by Alexander Bratus, Vladimir Posvyanskii, and yself, that we uploaded to arXiv: Replicator equations and space
We chose quite an ad hoc method to model the spatial evolution of the frequencies of interacting species, but the results, especially numerical, are very interesting. Here is the abstract
A reaction–diffusion replicator equation is studied. A novel method to apply the principle of global regulation is used to write down the model with explicit spatial structure. Properties of stationary solutions together with their stability are analyzed analytically, and relationships between stability of the rest points of the non-distributed replicator equation and distributed system are shown. A numerical example is given to show that the spatial variable in this particular model promotes the system’s permanence.
Mathematical Biology 