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Global generalized solutions to a multi-dimensional doubly tactic resource consumption model accounting for social interactions

Michael Winkler

Institut für Mathematik, Universität Paderborn, Warburger Straße 100, 33098 Paderborn, Germany

E-mail Address: michael.winkler@math.upb.de

https://doi.org/10.1142/S021820251950012XCited by:44 (Source: Crossref)

Abstract

This work is concerned with a prototypical model for the spatio-temporal evolution of a forager–exploiter system, consisting of two species which simultaneously consume a common nutrient, and which interact through a taxis-type mechanism according to which individuals from the exploiter subpopulation move upward density gradients of the forager subgroup. Specifically, the model

for the population densities

$u$ and

$v$ of foragers and exploiters, as well as the nutrient concentration

$w$ , is considered in smoothly bounded domains

$Ω \subset ℝ^{n}$ ,

$n \geq 1$ . It is first shown that under an explicit condition linking the sizes of the resource production rate

$r$ and of the initial nutrient concentration, an associated Neumann-type initial-boundary value problem admits a global solution within an appropriate generalized concept. The second of the main results asserts stabilization of these solutions toward spatially homogeneous equilibria in the large time limit, provided that

$r$ satisfies a mild assumption on temporal decay. To the best of our knowledge, these are the first rigorous analytical results addressing taxis-type cross-diffusion mechanisms coupled in a cascade-like manner as in (⋆).

Communicated by N. Bellomo

Keywords:

AMSC: 35Q91, 35Q92, 35B40, 35K55, 35D30

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