  This manuscript considers a Neumann initial-boundary value problem for the
predator-prey system $$
  \left\{ \begin{array}{l}
  u_t = D_1 u_{xx} - \chi_1 (uv_x)_x + u(\lambda_1-u+a_1 v), \\[1mm]
  v_t = D_2 v_{xx} + \chi_2 (vu_x)_x + v(\lambda_2-v-a_2 u),
  \end{array} \right.
  \qquad \qquad (\star) $$ in an open bounded interval $\Omega$ as the spatial
domain, where for $i\in\{1,2\}$ the parameters $D_i, a_i, \lambda_i$ and
$\chi_i$ are positive.
  Due to the simultaneous appearance of two mutually interacting taxis-type
cross-diffusive mechanisms, one of which even being attractive, it seems
unclear how far a solution theory can be built upon classical results on
parabolic evolution problems. In order to nevertheless create an analytical
setup capable of providing global existence results as well as detailed
information on qualitative behavior, this work pursues a strategy via parabolic
regularization, in the course of which ($\star$) is approximated by means of
certain fourth-order problems involving degenerate diffusion operators of thin
film type.
  During the design thereof, a major challenge is related to the ambition to
retain consistency with some fundamental entropy-like structures formally
associated with ($\star$); in particular, this will motivate the construction
of an approximation scheme including two free parameters which will finally be
fixed in different ways, depending on the size of $\lambda_2$ relative to $a_2
\lambda_1$.
