Global generalized solutions to a forager–exploiter model with superlinear degradation and their eventual regularity properties

Abstract

In this paper, we consider a cascaded taxis model for two proliferating and degrading species which thrive on the same nutrient but orient their movement according to different schemes. In particular, we assume the first group, the foragers, to orient their movement directly along an increasing gradient of the food density, while the second group, the exploiters, instead track higher densities of the forager group. Specifically, we will investigate an initial boundary-value problem for a prototypical forager–exploiter model of the form [Formula: see text] in a smoothly bounded domain [Formula: see text], where [Formula: see text], [Formula: see text] is nonnegative and the functions [Formula: see text] are assumed to satisfy [Formula: see text], [Formula: see text] as well as [Formula: see text] respectively, with constants [Formula: see text], [Formula: see text] and [Formula: see text] and [Formula: see text]. Assuming that [Formula: see text], [Formula: see text] and that [Formula: see text] satisfies certain structural conditions, we establish the global solvability of this system with respect to a suitable generalized solution concept and then, for the more restrictive case of [Formula: see text] and [Formula: see text], investigate an eventual regularity effect driven by the decay of the nutrient density [Formula: see text].