The spreading behavior of new or invasive species is a central topic in ecology. The modeling of free boundary problems is widely studied to better understand the nature of spreading behaviors of new species. From mathematical modeling point of view, it is a challenge to perform numerical simulations of the free boundary problems, due to the moving boundaries, the topological changes, and the stiffness of the system. \\ \\ Our work is concerned with numerical simulations of the long-term dynamical behavior of invasive species modeled by reaction-diffusion equations with free boundaries. We develop a front-tracking method to track the locations of the moving boundary explicitly in one dimension and higher dimensions with spherical symmetry. In two dimensional cases, we employ the level set method to handle topological bifurcations. For single invasive species, we numerically analyze the spreading-vanishing dichotomy in the diffusive logistic model. Various numerical experiments are presented in the two-dimensional spaces to show that the population range tend to be more and more spherical as time increases no matter what geometrical shape the initial population range has if the invasive species spreads successfully. For two invasive species in a weak-strong competition case, we examine how the long-time dynamics of the model changes as the initial functions are varies. Specifically, we simulate the ``chase-and-run coexistence'' phenomenon by choosing the initial function properly. The spreading behavior under time-periodic perturbation of the environment is also considered in our work.