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This manuscript considers a Neumann initial-boundary value problem for thepredator-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 spatialdomain, 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-typecross-diffusive mechanisms, one of which even being attractive, it seemsunclear how far a solution theory can be built upon classical results onparabolic evolution problems. In order to nevertheless create an analyticalsetup capable of providing global existence results as well as detailedinformation on qualitative behavior, this work pursues a strategy via parabolicregularization, in the course of which ($\star$) is approximated by means ofcertain fourth-order problems involving degenerate diffusion operators of thinfilm type.  During the design thereof, a major challenge is related to the ambition toretain consistency with some fundamental entropy-like structures formallyassociated with ($\star$); in particular, this will motivate the constructionof an approximation scheme including two free parameters which will finally befixed in different ways, depending on the size of $\lambda_2$ relative to $a_2\lambda_1$.

Existence theory and qualitative analysis for a fully cross-diffusive predator-prey system