  The goal of this paper is to investigate an instance of the tragedy of the
commons in spatially distributed harvesting games. The model we choose is that
of a fishes' population that is governed by a parabolic bistable equation and
that fishermen harvest. We assume that, when no fisherman is present, the
fishes' population is invading (mathematically, there is an invading travelling
front). Is it possible that fishermen, when acting selfishly, each in his or
her own best interest, might lead to a reversal of the travelling wave and,
consequently, to an extinction of the global population? To answer this
question, we model the behaviour of individual fishermen using a Mean Field
Game approach, and we show that the answer is yes. We then show that, at least
in some cases, if the fishermen coordinated instead of acting selfishly, each
of them could make more benefit, while still guaranteeing the survival of the
population. Our study is illustrated by several numerical simulations.
