Numerical Methods for a Two-Species Competition-Diffusion Model with Free Boundaries

The systems of reaction-diffusion equations coupled with moving boundaries defined by Stefan condition have been widely used to describe the dynamics of spreading population and with competition of two species. To solve these systems numerically, new numerical challenges arise from the competition of two species due to the interaction of their free boundaries. On the one hand, extremely small time steps are usually needed due to the stiffness of the system. On the other hand, it is always difficult to efficiently and accurately handle the moving boundaries especially with competition of two species. To overcome these numerical difficulties, we introduce a front tracking method coupled with an implicit solver for the 1D model. For the general 2D model, we use a level set approach to handle the moving boundaries to efficiently treat complicated topological changes. Several numerical examples are examined to illustrate the efficiency, accuracy and consistency for different approaches.