A numerical method is presented to solve single species population dynamics problems, with focus on the spread if invading organisms. This problem is mathematically modeled by a non-linear advection-diffusion-reaction transport equation, which is numerically solved by a finite element methodology. Some examples are conducted, showing the stability and accuracy of the proposed methodology. The role of the weak and strong Alee effects is analyzed on the spread of a single species invading organism in a one-dimensional domain. In this case the population dynamics describes propagation of traveling population fronts that represent either single-species invasion or single-species retreat. Some scenarios are investigated so as to evaluate the interaction of traveling population fronts under different parameter values, allowing to derive numerically whether they can reverse or block the species invasion.