  The conditions that can lead to the exploitative depletion of a shared
resource, i.e, the tragedy of the commons, can be reformulated as a game of
prisoner's dilemma: while preserving the common resource is in the best
interest of the group, over-consumption is in the interest of each particular
individual at any given point in time. One way to try and prevent the tragedy
of the commons is through infliction of punishment for over-consumption and/or
encouraging under-consumption, thus selecting against over-consumers. Here, the
effectiveness of various punishment functions in an evolving consumer-resource
system is evaluated within a framework of a parametrically heterogeneous system
of ordinary differential equations (ODEs). Conditions leading to the
possibility of sustainable coexistence with the common resource for a subset of
cases are identified analytically using adaptive dynamics; the effects of
punishment on heterogeneous populations with different initial composition are
evaluated using the Reduction theorem for replicator equations. Obtained
results suggest that one cannot prevent the tragedy of the commons through
rewarding of under-consumers alone - there must also be an implementation of
some degree of punishment that increases in a non-linear fashion with respect
to over-consumption and which may vary depending on the initial distribution of
clones in the population.
