Multiple travelling waves in evolutionary game dynamics
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Spatial models for the spread of favourable alleles have a distinguished place in the history of mathematical genetics. The realisation that reaction-diffusion equations often have travelling-wave solutions has been influential in the analysis of many practical problems as well as posing interesting theoretical problems. More recently, these same ideas have been successfully applied to evolutionary game dynamics, both in biological and economic contexts. Classically, if there are two alleles (or strategies) then the problem 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 there are two equations) then there may be many types of waves even for the classical, replicator dynamic. Thus the initial conditions are crucial in determining the outcome of contests. The existence of the waves is established by a bifurcation technique based upon a result from Conley index theory. Extensive numerical calculations are also reported.
Key wordsevolutionary games global bifurcation Conley index travelling waves
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