Multiple travelling waves in evolutionary game dynamics

Spatial models for the spread of favourable alleles have a distinguished pl ace in the history of mathematical genetics. The realisation that reaction- diffusion equations often have travelling-wave solutions has been influenti al in the analysis of many practical problems as well as posing interesting theoretical problems. More recently, these same ideas have been successful ly applied to evolutionary game dynamics, both in biological and economic c ontexts. Classically, if there are two alleles (or strategies) then the pro blem reduces to just a single equation (the frequency of an allele or of a particular strategy) in which case there is only one possible wave. Here it is shown that, if the number of alleles (strategies) is three (so that the re are two equations) then there may be many types of waves even for the cl assical, replicator dynamic. Thus the initial conditions are crucial in det ermining the outcome of contests. The existence of the waves is established by a bifurcation technique based upon a result from Conley index theory. E xtensive numerical calculations are also reported.