TY - JOUR

T1 - Classical and generalized solutions of an alarm-taxis model

AU - Fuest, Mario

AU - Lankeit, Johannes

N1 - Publisher Copyright: © The Author(s) 2024.

PY - 2024/11

Y1 - 2024/11

N2 - 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.

AB - 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.

KW - 35A01

KW - 35D99

KW - 35K20

KW - 35Q92

KW - 92D40

KW - Alarm-taxis

KW - Classical solution

KW - Food-chain

KW - Generalized solution

KW - Predator–prey

KW - Prey-taxis

UR - http://www.scopus.com/inward/record.url?scp=85201386233&partnerID=8YFLogxK

U2 - 10.1007/s00030-024-00989-6

DO - 10.1007/s00030-024-00989-6

M3 - Article

AN - SCOPUS:85201386233

VL - 31

JO - Nonlinear Differential Equations and Applications

JF - Nonlinear Differential Equations and Applications

SN - 1021-9722

IS - 6

M1 - 101

ER -