  Starting from the dynamical system model capturing the
splitting-differentiation process of populations, we extend this notion to show
how the speciation mechanism from a single species leads to the consideration
of several well known evolution cross-diffusion partial differential equations.
  Among the different alternatives for the diffusion terms, we study the model
introduced by Busenberg and Travis, for which we prove the existence of
solutions in the one-dimensional spatial case. Using a direct parabolic
regularization technique, we show that the problem is well posed in the space
of bounded variation functions, and demonstrate with a simple example that this
is the best regularity expected for solutions.
  We numerically compare our approach to other alternative regularizations
previously introduced in the literature, for the particular case of the contact
inhibition problem. Simulation experiments indicate that the numerical scheme
arising from the approximation introduced in this article outperforms those of
the existent models from the stability point of view.
