EXISTENCE THEORY AND QUALITATIVE ANALYSIS FOR A FULLY CROSS-DIFFUSIVE PREDATOR-PREY SYSTEM

This manuscript considers a Neumann initial-boundary value problem for the predator-prey system {(vt = D2vxx + chi 2(vux)x + v(lambda 2 - v - a2u),) (ut = D1uxx - chi 1(uvx)x + u(lambda 1 - u + a1v),) ((*)) in an open bounded interval Omega as the spatial domain, where for i is an element of {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 ((*)) 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 ((*)); 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 ((*)) and arbitrarily large positive initial data from H-1, and second allow for the conclusion that in both cases lambda(2) (*) a(2)lambda(1) and lambda(2) <= 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.

Automated summary

This manuscript investigates a Neumann initial-boundary value problem for a predator-prey system. By employing a strategy of parabolic regularization, the study provides global existence results and detailed information on qualitative behavior. The research involves two mutually interacting degenerate diffusion mechanisms and fundamental entropy-like structures.