The spreading behavior of new or invasive species is a central topic in ecology. The modelings of free boundary problems are widely studied to better understand the nature of spreading behavior of new species. From mathematical modeling point of view, it is a challenge to perform numerical simulations of free boundary problems, due to the moving boundary, the stiffness of the system and topological changes.
In this work, we design numerical methods to investigate the spreading behavior of new species for a diffusive logistic model with a free boundary and a diffusive competition system with free boundaries. We develop a front-tracking method, which explicitly tracks the location of the moving boundary, in one dimension and higher dimensions with spherical symmetry. In higher dimensional cases, we introduce level set method to handle topological bifurcations. Various numerical simulations in one and two dimensional spaces are presented to validate the accuracy, and stability of the proposed numerical methods. To efficiently solve stiff reaction-diffusion equations, we also develop implicit integration factor (IIF) method combining Krylov subspace to solve the diffusive logistic model with a free boundary in one dimension. Compared with different numerical schemes, it can be observed that Krylov IIF is advantageous to other approaches in terms of stability and efficiency by direct comparison through numerical examples
