  We study an initial boundary value problem for a cross-diffusion system in
population dynamics. The mathematical challenge is due to the fact that the
determinant of the coefficient matrix of the system changes signs. As a result,
the system is only partially parabolic. We design an approximation scheme. The
sequence of approximate solutions generated by our scheme converges and its
limit satisfies the original system in the parabolic region. It remains open if
one can construct a vector-valued function that satisfies the system in both
the parabolic region and the anti-parabolic one.
