Existence Theory and Qualitative Analysis for a Fully Cross-Diffusive Predator-Prey System

Source abstract

This manuscript considers a Neumann initial-boundary value problem for the predator-prey system $ \{ \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}(\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 is 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 $a_2 \lambda_1$. Adequately coping with this will first yield a result on global existence of weak solutions for arbitrary choices of the parameters in ($\star$) and arbitrarily large positive initial data from $H^1$, and second allow for the conclusion that in both cases $\lambda_2 \star a_2\lambda_1$ and $\lambda_2 \le a_2\lambda_1$, the respectively obtained spatially homogeneous coexistence and prey-extinction states uphold their global asymptotic stability properties well-known to be present in the corresponding ODE setting, provided that both tactic sensitivities $\chi_1$ and $\chi_2$ are suitably small.