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In the context of conservation under uncertainty, the problem of designing and analysing adaptive switching feedback control schemes for systems of positive difference equations is considered. The aim is to ensure persistence of the population, corresponding to the zero equilibrium of the closed-loop control system being unstable in a certain sense. A robust control approach is adopted, where multiple discrete control actions are available, corresponding to different management strategies or policies. However, the exact effect of each strategy is assumed to be uncertain. Based on principles from both positive dynamical systems and simple adaptive feedback control, a suite of control schemes is proposed from which a destabilising (persistent) strategy is selected based on a switching process, should such a strategy exist. We demonstrate that the switching rules can be augmented with several variations, altering the transient behaviour and thus tailoring them to the particular requirements of the user. The current work substantially builds upon and enhances earlier results of the authors, by establishing key and quite general hypotheses of the underlying model and control scheme to ensure persistence, so that the results are applicable to a wide range of model types. The proposed control schemes are illustrated with examples.
Smith et al. (Mon,) studied this question.