Optimal Dirac controls for time-periodic bistable ODEs, application to population replacement

Abstract: This work addresses an optimal control problem on a dynamics governed by a nonlinear differential equation with a bistable time-periodic nonlinearity. This problem, relevant in population dynamics, models the strategy of replacing a population of A-type individuals by a population of B-type individuals in a time-varying environment, focusing on the evolution of the proportion of B-type individuals among the whole population. The control term accounts for the instant release of B-type individuals. Our main goal, after noting some interesting properties on the differential equation, is to determine the optimal time at which this release should be operated to ensure population replacement while minimizing the release effort. The results establish that the optimal release time appears to be the minimizer of a function involving the carrying capacity of the environment and the threshold periodic solution of the dynamics; they also describe the convergence of the whole optimal release strategy. An application to the biocontrol of mosquito populations using Wolbachia-infected individuals illustrates the relevance of the theoretical results. Wolbachia is a bacterium that helps preventing the transmission of some viruses from mosquitoes to humans, making the optimization of Wolbachia propagation in a mosquito population a crucial issue.