We consider a dynamical model for evolutionary games, and enquire how
the introduction of diffusion may lead to the formation of stationary
spatially inhomogeneous solutions, that is patterns. For the model equ
ations being used it is already known that if there is an evolutionari
ly stable strategy (ESS), then it is stable. Equilibrium solutions whi
ch are not ESS's and which are stable with respect to spatially consta
nt perturbations may be unstable for certain choices of the dispersal
rates. We prove by a bifurcation technique that under appropriate cond
itions the instability leads to patterns. Computations using a curve-f
ollowing technique show that the bifurcations exhibit a rich structure
with loops joined by symmetry-breaking branches.