  The goal of this paper is to study a Mean Field Game (MFG) system stemming
from the harvesting of resources. Modelling the latter through a
reaction-diffusion equation and the harvesters as competing rational agents, we
are led to a non-local (in time and space) MFG system that consists of three
equations, the study of which is quite delicate. The main focus of this paper
is on the derivation of analytical results (e.g existence, uniqueness) and of
long time behaviour (here, convergence to the ergodic system). We provide some
explicit solutions to this ergodic system.
