  We study a dynamical model of a population of cooperators and defectors whose
actions have long-term consequences on environmental "commons" - what we term
the "resource". Cooperators contribute to restoring the resource whereas
defectors degrade it. The population dynamics evolve according to a replicator
equation coupled with an environmental state. Our goal is to identify methods
of influencing the population with the objective to maximize accumulation of
the resource. In particular, we consider strategies that modify
individual-level incentives. We then extend the model to incorporate a public
opinion state that imperfectly tracks the true environmental state, and study
strategies that influence opinion. We formulate optimal control problems and
solve them using numerical techniques to characterize locally optimal control
policies for three problem formulations: 1) control of incentives, and control
of opinions through 2) propaganda-like strategies and 3) awareness campaigns.
We show numerically that the resulting controllers in all formulations achieve
the objective, albeit with an unintended consequence. The resulting dynamics
include cycles between low and high resource states - a dynamical regime termed
an "oscillating tragedy of the commons". This outcome may have desirable
average properties, but includes risks to resource depletion. Our findings
suggest the need for new approaches to controlling coupled
population-environment dynamics.
