We have studied a free boundary problem for a certain class of reaction-diffusion equations. Such a problem models the invasion or migration of a biological species which moves toward a new habitat.  The problem has two unknown functions: one is the population density of the species and the other is (a part of ) the boundary of its habitat. The population density is governed by a reaction-diffusion equation and the moving boundary is controlled by Stefan condition.  When a reaction term has two stable and positive equilibrium states, some numerical simulations exhibit different large-time behaviors from known ones.  We have succeeded in getting various theoretical results such as the classification of asymptotic behaviors of solutions into four patterns and the derivation of speeds of spreading free boundaries as time goes to infinity. These result help us to understand the invasion model from the mathematical view-point.