Allee optimal control of a system in ecology

The Allee threshold of an ecological system distinguishes the sign of population growth either towards extinction or to carrying capacity. In practice, human interventions can tune the Allee threshold for instance thanks to the sterile male technique and the mating disruption. In this paper, we address various control problems for a system described by a diffusion-reaction equation regulating the Allee threshold, viewed as a real parameter determining the unstable equilibrium of the bistable nonlinear reaction term. We prove that this system is the mean field limit of an interacting system of particles in which the individual behaviour is driven by stochastic laws. Numerical simulations of the stochastic process show that the propagation of population is governed by travelling wave solutions of the macroscopic reaction diffusion system, which model the fact that solutions, in bounded space domains, reach asymptotically an equilibrium configuration. An optimal control problem for the macroscopic model is then introduced with the objective of steering the system to a target travelling wave. Using well-known analytical results and stability properties of travelling waves, we show that well-chosen piecewise constant controls allow to reach the target approximately in sufficiently long time. We then develop a direct. computational method and show its efficiency for computing such controls in various numerical simulations. Finally, we show the effectiveness of the obtained macroscopic optimal controls in the microscopic system of interacting particles and we discuss their advantage when addressing situations that are out of reach for the analytical methods. We conclude the paper with some open problems and directions for future research.