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Abstract

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 has a distinct number of fold points of its critical manifold. These points, 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.