Trélat E., Zhu J., Zuazua, E. Allee optimal control of a system in ecology , Math. Models Methods Appl. Sci., Vol. 28, No. 09 (2018), pp. 1665-1697. DOI: 10.1142/S021820251840002X
Abstract: 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 objectives 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 individual behaviours are driven by stochastic laws. Numerical simulations of the stochastic process show that population propagations are governed by wave-like solutions corresponding to traveling solutions of the macroscopic reaction-diffusion system. An optimal control problem for the macroscopic model is then introduced with the objective of steering the system to a target traveling wave. The relevance of this problem is motivated by the fact that traveling wave solutions model the fact that bounded space domains reach asymptotically an equilibrium configuration. Using well known analytical results and stability properties of traveling 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 efficiency 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 article with some open problems and directions for future research.