Classical and generalized solutions of an alarm-taxis model

authored by

Mario Fuest, Johannes Lankeit

Abstract

In bounded, spatially two-dimensional domains, the system (Formula presented.) complemented with initial and homogeneous Neumann boundary conditions, models the interaction between prey (with density u), predator (with density v) and superpredator (with density w), which preys on both other populations. Apart from random motion and prey-tactical behavior of the primary predator, the key aspect of this system is that the secondary predator reacts to alarm calls of the prey, issued by the latter whenever attacked by the primary predator. We first show in the pure alarm-taxis model, i.e. if ξ=0, that global classical solutions exist. For the full model (with ξ>0), the taxis terms and the presence of the term -a₂uw in the first equation apparently hinder certain bootstrap procedures, meaning that the available regularity information is rather limited. Nonetheless, we are able to obtain global generalized solutions. An important technical challenge is to guarantee strong convergence of (weighted) gradients of the first two solution components in order to conclude that approximate solutions converge to a generalized solution of the limit problem.

Organisation(s)

Institute of Applied Mathematics

Type

Article

Journal

Nonlinear Differential Equations and Applications

Volume

31

No. of pages

25

ISSN

1021-9722

Publication date

11.2024

Publication status

Published

Peer reviewed

Yes

ASJC Scopus subject areas

Analysis, Applied Mathematics

Electronic version(s)

https://doi.org/10.1007/s00030-024-00989-6 (Access: Open)