Numerical Algorithms for a Variational Problem of the Spatial Segregation of Reaction-Diffusion Systems
This work provides a numerical method for spatial segregation problems in reaction-diffusion systems, but the results are incremental and limited to specific cases.
The authors develop a numerical approximation scheme for stationary states of reaction-diffusion systems with disjoint supports, proving convergence in particular cases and demonstrating simulations for Lotka-Volterra models with high competition.
In this paper, we study a numerical approximation for a class of stationary states for reaction-diffusion system with m densities having disjoint support, which are governed by a minimization problem. We use quantitative properties of both solutions and free boundaries to derive our scheme. Furthermore, the proof of convergence of the numerical method is given in some particular cases. We also apply our numerical simulations for the spatial segregation limit of diffusive Lotka-Volterra models in presence of high competition and inhomogeneous Dirichlet boundary conditions. We discuss numerical implementations of the resulting approach and present computational tests.