  We consider a spatially inhomogeneous public goods game model with diffusion.
By utilising a generalised Hamiltonian structure of the model we study the
existence of global classical solutions as well as the large time behaviour:
First, the asymptotic convergence of the PDE to the corresponding ODE system is
proven. This result entails also the periodic behaviour of PDE solutions in the
large time limit. Secondly, a shadow system approximation is considered and the
convergence of the PDE to the shadow system in the associated fast-diffusion
limit is shown. Finally, the asymptotic convergence of the shadow to the ODE
system is proven.
