  We study a prototypical non-polynomial decision-making model for which agents
in a population potentially alternate between two consumption strategies, one
related to the exploitation of an unlimited but considerably expensive resource
and the other a comparably cheaper but restricted and slowly renewable source.
In particular, we study a model following a Boltzmann-like exploration policy,
enhancing the accuracy at which the exchange rates are captured with respect to
classical polynomial approaches by considering sigmoidal functions to represent
the cost-profit relation in both exploit strategies. Additionally, given the
intrinsic timescale separation between the decision-making process and recovery
rates of the renewable resource, we use geometric singular perturbation theory
to analyze the model. We further use numerical analysis to determine parameter
ranges for which the model undergoes bifurcations. These bifurcations, being
related to critical states of the system, are relevant to the fast transitions
between strategies. Hence, we design controllers to regulate such rapid
transitions by taking advantage of the system's criticality.
