Narrow Results

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.

In this paper, the diffusion-driven instability of the Leslie–Gower competition model with the periodic boundary conditions is investigated. By using the linearization method and the inner product techniques, the instability conditions of this model at the coexistence fixed point and the competitive exclusion fixed points are obtained, respectively. As an example, the diffusion-driven instability conditions of a symmetric Leslie–Gower competition model at the coexistence fixed point is obtained when the diffusion coefficients are equal. Under these instability conditions, various patterns, including spirals, traveling waves and disorders, are observed in the numerical simulations. On the other hand, we also numerically investigate the effects of diffusion coefficient and the strength of the interspecific competition on the wave patterns.

We study travelling wave solutions describing non-isothermal subsonic phase transitions in a van der Waals fluid. We apply the center manifold method and establish the existence of viscosity-capillarity profiles for small viscosity, large heat conduction, and large heat capacity.

This paper presents the simulation of general and travelling dielectrophoretic forces, as well as the movement of the particles in a sandwich structure micro-device. The electrode geometry of the micro device used for simulation is an interdigitated bar electrode. The simulation method used to solve the equations is based on the least square finite difference method (LSFD). The simulation first calculates all forces acting at any place in the chamber, with these forces the trajectory of a particle can now be proposed. All of the particles parameters like radius, voltage, initial height, etc can easily be changed and the simulation can be redone. With this continuous trial we receive different behavior of the particles and examine the relevancy of the different changes made. This detailed information about the influences of the parameters on the procedure in the micro-device can be used for the development of further micro-devices.