Large-time asymptotics of a public goods game model with diffusion

TL;DR

This paper analyzes the long-term behavior of a spatially inhomogeneous public goods game model with diffusion, demonstrating convergence to simpler systems and periodic solutions using advanced mathematical techniques.

Contribution

It introduces a novel analysis of the large-time asymptotics of a diffusive public goods game using a generalized Hamiltonian structure.

Findings

PDE solutions converge to ODE system asymptotically

Periodic behavior of solutions in large time

Convergence of PDE to shadow system in fast-diffusion limit

Abstract

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.