Competition between populations: preventing domination of resistant population using optimal control

Abstract

Emergence of pesticide-resistant strains of pests, antibiotic-resistant bacteria or chemotherapy-resistant tumor cells is a major obstacle standing in the way of effective external control of undesirable growth of these populations. Once the resistant subpopulation becomes dominant the means to control the population are effectively lost. Prevention of resistance may be therefore considered one of the optimal control goals to ensure long-term controllability. In this paper we formulate nonlinear optimal control problem motivated by the question of controlling the growth of two coexisting subpopulations representing two phenotypes of the same species. However, the model we consider is a standard competition model with populations following logistic growth, and therefore it is more general and can be used also when two competing populations are taken into account. We consider a situation when the control (e.g. pesticide or drug) has effect only on the specimen belonging to one (sensitive) population while having no effect on the other (resistant) population. We use a non-standard, nonlinear objective functional in which, aside from penalizing the overall size of both populations, we include an additional, nonlinear term to ensure that the control-sensitive population remains the dominant one. It is shown how the resistance penalty gives rise to locally-optimal singular controls which are key to achieving the balance between resistant and sensitive populations whenever the subpopulations are similar in terms of the model parameters. Analytic properties of the singular arcs and the structure of optimal control are investigated. Both theoretical and numerical results strongly suggest that the optimal control contains a singular interval.