Details
| Original language | English |
|---|---|
| Article number | 101 |
| Number of pages | 25 |
| Journal | Nonlinear Differential Equations and Applications |
| Volume | 31 |
| Issue number | 6 |
| Early online date | 16 Aug 2024 |
| Publication status | Published - Nov 2024 |
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.
Keywords
- 35A01, 35D99, 35K20, 35Q92, 92D40, Alarm-taxis, Classical solution, Food-chain, Generalized solution, Predator–prey, Prey-taxis
ASJC Scopus subject areas
- Mathematics(all)
- Analysis
- Mathematics(all)
- Applied Mathematics
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In: Nonlinear Differential Equations and Applications, Vol. 31, No. 6, 101, 11.2024.
Research output: Contribution to journal › Article › Research › peer review
}
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 -