---
_id: '63164'
abstract:
- lang: eng
text: Refined investigation of chemotaxis processes has revealed a significant
role of degeneracies in corresponding motilities in a number of application contexts.
A rapidly growing literature concerned with the analysis of resulting mathematical
models has been capable of solving fundamental issues, but various problems have
remained open, or even newly arisen. The goal of the paper consists in a summary
of some developments in this area, and particularly in the discussion of the question
how far the introduction of degeneracies may influence the behavior of solutions
to chemotaxis systems.
author:
- first_name: Michael
full_name: Winkler, Michael
id: '31496'
last_name: Winkler
citation:
ama: Winkler M. Effects of degeneracies in taxis-driven evolution. Mathematical
Models and Methods in Applied Sciences. 2025;35(02):283-343. doi:10.1142/s0218202525400020
apa: Winkler, M. (2025). Effects of degeneracies in taxis-driven evolution. Mathematical
Models and Methods in Applied Sciences, 35(02), 283–343. https://doi.org/10.1142/s0218202525400020
bibtex: '@article{Winkler_2025, title={Effects of degeneracies in taxis-driven evolution},
volume={35}, DOI={10.1142/s0218202525400020},
number={02}, journal={Mathematical Models and Methods in Applied Sciences}, publisher={World
Scientific Pub Co Pte Ltd}, author={Winkler, Michael}, year={2025}, pages={283–343}
}'
chicago: 'Winkler, Michael. “Effects of Degeneracies in Taxis-Driven Evolution.”
Mathematical Models and Methods in Applied Sciences 35, no. 02 (2025):
283–343. https://doi.org/10.1142/s0218202525400020.'
ieee: 'M. Winkler, “Effects of degeneracies in taxis-driven evolution,” Mathematical
Models and Methods in Applied Sciences, vol. 35, no. 02, pp. 283–343, 2025,
doi: 10.1142/s0218202525400020.'
mla: Winkler, Michael. “Effects of Degeneracies in Taxis-Driven Evolution.” Mathematical
Models and Methods in Applied Sciences, vol. 35, no. 02, World Scientific
Pub Co Pte Ltd, 2025, pp. 283–343, doi:10.1142/s0218202525400020.
short: M. Winkler, Mathematical Models and Methods in Applied Sciences 35 (2025)
283–343.
date_created: 2025-12-16T19:23:40Z
date_updated: 2025-12-18T20:16:23Z
doi: 10.1142/s0218202525400020
intvolume: ' 35'
issue: '02'
language:
- iso: eng
page: 283-343
publication: Mathematical Models and Methods in Applied Sciences
publication_identifier:
issn:
- 0218-2025
- 1793-6314
publication_status: published
publisher: World Scientific Pub Co Pte Ltd
status: public
title: Effects of degeneracies in taxis-driven evolution
type: journal_article
user_id: '31496'
volume: 35
year: '2025'
...
---
_id: '54837'
author:
- first_name: Leander
full_name: Claes, Leander
id: '11829'
last_name: Claes
orcid: 0000-0002-4393-268X
- first_name: Johannes
full_name: Lankeit, Johannes
last_name: Lankeit
- first_name: Michael
full_name: Winkler, Michael
id: '31496'
last_name: Winkler
citation:
ama: 'Claes L, Lankeit J, Winkler M. A model for heat generation by acoustic waves
in piezoelectric materials: Global large-data solutions. Mathematical Models
and Methods in Applied Sciences. 2025;35(11):2465-2512. doi:10.1142/s0218202525500447'
apa: 'Claes, L., Lankeit, J., & Winkler, M. (2025). A model for heat generation
by acoustic waves in piezoelectric materials: Global large-data solutions. Mathematical
Models and Methods in Applied Sciences, 35(11), 2465–2512. https://doi.org/10.1142/s0218202525500447'
bibtex: '@article{Claes_Lankeit_Winkler_2025, title={A model for heat generation
by acoustic waves in piezoelectric materials: Global large-data solutions}, volume={35},
DOI={10.1142/s0218202525500447},
number={11}, journal={Mathematical Models and Methods in Applied Sciences}, publisher={World
Scientific Pub Co Pte Ltd}, author={Claes, Leander and Lankeit, Johannes and Winkler,
Michael}, year={2025}, pages={2465–2512} }'
chicago: 'Claes, Leander, Johannes Lankeit, and Michael Winkler. “A Model for Heat
Generation by Acoustic Waves in Piezoelectric Materials: Global Large-Data Solutions.”
Mathematical Models and Methods in Applied Sciences 35, no. 11 (2025):
2465–2512. https://doi.org/10.1142/s0218202525500447.'
ieee: 'L. Claes, J. Lankeit, and M. Winkler, “A model for heat generation by acoustic
waves in piezoelectric materials: Global large-data solutions,” Mathematical
Models and Methods in Applied Sciences, vol. 35, no. 11, pp. 2465–2512, 2025,
doi: 10.1142/s0218202525500447.'
mla: 'Claes, Leander, et al. “A Model for Heat Generation by Acoustic Waves in Piezoelectric
Materials: Global Large-Data Solutions.” Mathematical Models and Methods in
Applied Sciences, vol. 35, no. 11, World Scientific Pub Co Pte Ltd, 2025,
pp. 2465–512, doi:10.1142/s0218202525500447.'
short: L. Claes, J. Lankeit, M. Winkler, Mathematical Models and Methods in Applied
Sciences 35 (2025) 2465–2512.
date_created: 2024-06-20T13:43:42Z
date_updated: 2026-01-05T07:59:41Z
department:
- _id: '90'
- _id: '49'
doi: 10.1142/s0218202525500447
external_id:
arxiv:
- '2411.14900'
intvolume: ' 35'
issue: '11'
language:
- iso: eng
main_file_link:
- open_access: '1'
url: https://arxiv.org/pdf/2411.14900
oa: '1'
page: 2465-2512
project:
- _id: '245'
name: 'FOR 5208: Modellbasierte Bestimmung nichtlinearer Eigenschaften von Piezokeramiken
für Leistungsschallanwendungen (NEPTUN)'
publication: Mathematical Models and Methods in Applied Sciences
publication_identifier:
issn:
- 1793-6314
publisher: World Scientific Pub Co Pte Ltd
status: public
title: 'A model for heat generation by acoustic waves in piezoelectric materials:
Global large-data solutions'
type: journal_article
user_id: '11829'
volume: 35
year: '2025'
...
---
_id: '53328'
abstract:
- lang: eng
text: As a simplified version of a three-component taxis cascade model
accounting for different migration strategies of two population groups in search
of food, a two-component nonlocal nutrient taxis system is considered in a two-dimensional
bounded convex domain with smooth boundary. For any given conveniently regular
and biologically meaningful initial data, smallness conditions on the prescribed
resource growth and on the initial nutrient signal concentration are identified
which ensure the global existence of a global classical solution to the corresponding
no-flux initial-boundary value problem. Moreover, under additional assumptions
on the food production source these solutions are shown to be bounded, and to
stabilize toward semi-trivial equilibria in the large time limit, respectively.
author:
- first_name: Youshan
full_name: Tao, Youshan
last_name: Tao
- first_name: Michael
full_name: Winkler, Michael
last_name: Winkler
citation:
ama: Tao Y, Winkler M. Small-signal solutions to a nonlocal cross-diffusion model
for interaction of scroungers with rapidly diffusing foragers. Mathematical
Models and Methods in Applied Sciences. 2023;33(01):103-138. doi:10.1142/s0218202523500045
apa: Tao, Y., & Winkler, M. (2023). Small-signal solutions to a nonlocal cross-diffusion
model for interaction of scroungers with rapidly diffusing foragers. Mathematical
Models and Methods in Applied Sciences, 33(01), 103–138. https://doi.org/10.1142/s0218202523500045
bibtex: '@article{Tao_Winkler_2023, title={Small-signal solutions to a nonlocal
cross-diffusion model for interaction of scroungers with rapidly diffusing foragers},
volume={33}, DOI={10.1142/s0218202523500045},
number={01}, journal={Mathematical Models and Methods in Applied Sciences}, publisher={World
Scientific Pub Co Pte Ltd}, author={Tao, Youshan and Winkler, Michael}, year={2023},
pages={103–138} }'
chicago: 'Tao, Youshan, and Michael Winkler. “Small-Signal Solutions to a Nonlocal
Cross-Diffusion Model for Interaction of Scroungers with Rapidly Diffusing Foragers.”
Mathematical Models and Methods in Applied Sciences 33, no. 01 (2023):
103–38. https://doi.org/10.1142/s0218202523500045.'
ieee: 'Y. Tao and M. Winkler, “Small-signal solutions to a nonlocal cross-diffusion
model for interaction of scroungers with rapidly diffusing foragers,” Mathematical
Models and Methods in Applied Sciences, vol. 33, no. 01, pp. 103–138, 2023,
doi: 10.1142/s0218202523500045.'
mla: Tao, Youshan, and Michael Winkler. “Small-Signal Solutions to a Nonlocal Cross-Diffusion
Model for Interaction of Scroungers with Rapidly Diffusing Foragers.” Mathematical
Models and Methods in Applied Sciences, vol. 33, no. 01, World Scientific
Pub Co Pte Ltd, 2023, pp. 103–38, doi:10.1142/s0218202523500045.
short: Y. Tao, M. Winkler, Mathematical Models and Methods in Applied Sciences 33
(2023) 103–138.
date_created: 2024-04-07T12:43:13Z
date_updated: 2024-04-07T12:43:17Z
doi: 10.1142/s0218202523500045
intvolume: ' 33'
issue: '01'
keyword:
- Applied Mathematics
- Modeling and Simulation
language:
- iso: eng
page: 103-138
publication: Mathematical Models and Methods in Applied Sciences
publication_identifier:
issn:
- 0218-2025
- 1793-6314
publication_status: published
publisher: World Scientific Pub Co Pte Ltd
status: public
title: Small-signal solutions to a nonlocal cross-diffusion model for interaction
of scroungers with rapidly diffusing foragers
type: journal_article
user_id: '31496'
volume: 33
year: '2023'
...
---
_id: '63271'
abstract:
- lang: eng
text: As a simplified version of a three-component taxis cascade model
accounting for different migration strategies of two population groups in search
of food, a two-component nonlocal nutrient taxis system is considered in a two-dimensional
bounded convex domain with smooth boundary. For any given conveniently regular
and biologically meaningful initial data, smallness conditions on the prescribed
resource growth and on the initial nutrient signal concentration are identified
which ensure the global existence of a global classical solution to the corresponding
no-flux initial-boundary value problem. Moreover, under additional assumptions
on the food production source these solutions are shown to be bounded, and to
stabilize toward semi-trivial equilibria in the large time limit, respectively.
author:
- first_name: Youshan
full_name: Tao, Youshan
last_name: Tao
- first_name: Michael
full_name: Winkler, Michael
id: '31496'
last_name: Winkler
citation:
ama: Tao Y, Winkler M. Small-signal solutions to a nonlocal cross-diffusion model
for interaction of scroungers with rapidly diffusing foragers. Mathematical
Models and Methods in Applied Sciences. 2023;33(01):103-138. doi:10.1142/s0218202523500045
apa: Tao, Y., & Winkler, M. (2023). Small-signal solutions to a nonlocal cross-diffusion
model for interaction of scroungers with rapidly diffusing foragers. Mathematical
Models and Methods in Applied Sciences, 33(01), 103–138. https://doi.org/10.1142/s0218202523500045
bibtex: '@article{Tao_Winkler_2023, title={Small-signal solutions to a nonlocal
cross-diffusion model for interaction of scroungers with rapidly diffusing foragers},
volume={33}, DOI={10.1142/s0218202523500045},
number={01}, journal={Mathematical Models and Methods in Applied Sciences}, publisher={World
Scientific Pub Co Pte Ltd}, author={Tao, Youshan and Winkler, Michael}, year={2023},
pages={103–138} }'
chicago: 'Tao, Youshan, and Michael Winkler. “Small-Signal Solutions to a Nonlocal
Cross-Diffusion Model for Interaction of Scroungers with Rapidly Diffusing Foragers.”
Mathematical Models and Methods in Applied Sciences 33, no. 01 (2023):
103–38. https://doi.org/10.1142/s0218202523500045.'
ieee: 'Y. Tao and M. Winkler, “Small-signal solutions to a nonlocal cross-diffusion
model for interaction of scroungers with rapidly diffusing foragers,” Mathematical
Models and Methods in Applied Sciences, vol. 33, no. 01, pp. 103–138, 2023,
doi: 10.1142/s0218202523500045.'
mla: Tao, Youshan, and Michael Winkler. “Small-Signal Solutions to a Nonlocal Cross-Diffusion
Model for Interaction of Scroungers with Rapidly Diffusing Foragers.” Mathematical
Models and Methods in Applied Sciences, vol. 33, no. 01, World Scientific
Pub Co Pte Ltd, 2023, pp. 103–38, doi:10.1142/s0218202523500045.
short: Y. Tao, M. Winkler, Mathematical Models and Methods in Applied Sciences 33
(2023) 103–138.
date_created: 2025-12-18T19:12:35Z
date_updated: 2025-12-18T20:10:55Z
doi: 10.1142/s0218202523500045
intvolume: ' 33'
issue: '01'
language:
- iso: eng
page: 103-138
publication: Mathematical Models and Methods in Applied Sciences
publication_identifier:
issn:
- 0218-2025
- 1793-6314
publication_status: published
publisher: World Scientific Pub Co Pte Ltd
status: public
title: Small-signal solutions to a nonlocal cross-diffusion model for interaction
of scroungers with rapidly diffusing foragers
type: journal_article
user_id: '31496'
volume: 33
year: '2023'
...
---
_id: '45970'
abstract:
- lang: eng
text: ' We introduce a new phase field model for tumor growth where viscoelastic
effects are taken into account. The model is derived from basic thermodynamical
principles and consists of a convected Cahn–Hilliard equation with source terms
for the tumor cells and a convected reaction–diffusion equation with boundary
supply for the nutrient. Chemotactic terms, which are essential for the invasive
behavior of tumors, are taken into account. The model is completed by a viscoelastic
system consisting of the Navier–Stokes equation for the hydrodynamic quantities,
and a general constitutive equation with stress relaxation for the left Cauchy–Green
tensor associated with the elastic part of the total mechanical response of the
viscoelastic material. For a specific choice of the elastic energy density and
with an additional dissipative term accounting for stress diffusion, we prove
existence of global-in-time weak solutions of the viscoelastic model for tumor
growth in two space dimensions [Formula: see text] by the passage to the limit
in a fully-discrete finite element scheme where a CFL condition, i.e. [Formula:
see text], is required. Moreover, in arbitrary dimensions [Formula:
see text], we show stability and existence of solutions for the fully-discrete
finite element scheme, where positive definiteness of the discrete Cauchy–Green
tensor is proved with a regularization technique that was first introduced by
Barrett and Boyaval [Existence and approximation of a (regularized) Oldroyd-B
model, Math. Models Methods Appl. Sci. 21 (2011) 1783–1837]. After that, we improve
the regularity results in arbitrary dimensions [Formula: see text] and in two
dimensions [Formula: see text], where a CFL condition is required. Then, in two
dimensions [Formula: see text], we pass to the limit in the discretization parameters
and show that subsequences of discrete solutions converge to a global-in-time
weak solution. Finally, we present numerical results in two dimensions [Formula:
see text]. '
author:
- first_name: Harald
full_name: Garcke, Harald
last_name: Garcke
- first_name: Balázs
full_name: Kovács, Balázs
id: '100441'
last_name: Kovács
orcid: 0000-0001-9872-3474
- first_name: Dennis
full_name: Trautwein, Dennis
last_name: Trautwein
citation:
ama: Garcke H, Kovács B, Trautwein D. Viscoelastic Cahn–Hilliard models for tumor
growth. Mathematical Models and Methods in Applied Sciences. 2022;32(13):2673-2758.
doi:10.1142/s0218202522500634
apa: Garcke, H., Kovács, B., & Trautwein, D. (2022). Viscoelastic Cahn–Hilliard
models for tumor growth. Mathematical Models and Methods in Applied Sciences,
32(13), 2673–2758. https://doi.org/10.1142/s0218202522500634
bibtex: '@article{Garcke_Kovács_Trautwein_2022, title={Viscoelastic Cahn–Hilliard
models for tumor growth}, volume={32}, DOI={10.1142/s0218202522500634},
number={13}, journal={Mathematical Models and Methods in Applied Sciences}, publisher={World
Scientific Pub Co Pte Ltd}, author={Garcke, Harald and Kovács, Balázs and Trautwein,
Dennis}, year={2022}, pages={2673–2758} }'
chicago: 'Garcke, Harald, Balázs Kovács, and Dennis Trautwein. “Viscoelastic Cahn–Hilliard
Models for Tumor Growth.” Mathematical Models and Methods in Applied Sciences
32, no. 13 (2022): 2673–2758. https://doi.org/10.1142/s0218202522500634.'
ieee: 'H. Garcke, B. Kovács, and D. Trautwein, “Viscoelastic Cahn–Hilliard models
for tumor growth,” Mathematical Models and Methods in Applied Sciences,
vol. 32, no. 13, pp. 2673–2758, 2022, doi: 10.1142/s0218202522500634.'
mla: Garcke, Harald, et al. “Viscoelastic Cahn–Hilliard Models for Tumor Growth.”
Mathematical Models and Methods in Applied Sciences, vol. 32, no. 13, World
Scientific Pub Co Pte Ltd, 2022, pp. 2673–758, doi:10.1142/s0218202522500634.
short: H. Garcke, B. Kovács, D. Trautwein, Mathematical Models and Methods in Applied
Sciences 32 (2022) 2673–2758.
date_created: 2023-07-10T11:47:27Z
date_updated: 2024-04-03T09:15:35Z
department:
- _id: '841'
doi: 10.1142/s0218202522500634
intvolume: ' 32'
issue: '13'
keyword:
- Applied Mathematics
- Modeling and Simulation
language:
- iso: eng
page: 2673-2758
publication: Mathematical Models and Methods in Applied Sciences
publication_identifier:
issn:
- 0218-2025
- 1793-6314
publication_status: published
publisher: World Scientific Pub Co Pte Ltd
status: public
title: Viscoelastic Cahn–Hilliard models for tumor growth
type: journal_article
user_id: '100441'
volume: 32
year: '2022'
...
---
_id: '63290'
abstract:
- lang: eng
text: This paper proposes a review focused on exotic chemotaxis and cross-diffusion
models in complex environments. The term exotic is used to denote the dynamics
of models interacting with a time-evolving external system and, specifically,
models derived with the aim of describing the dynamics of living systems. The
presentation first, considers the derivation of phenomenological models of chemotaxis
and cross-diffusion models with particular attention on nonlinear characteristics.
Then, a variety of exotic models is presented with some hints toward the derivation
of new models, by accounting for a critical analysis looking ahead to perspectives.
The second part of the paper is devoted to a survey of analytical problems concerning
the application of models to the study of real world dynamics. Finally, the focus
shifts to research perspectives within the framework of a multiscale vision, where
different paths are examined to move from the dynamics at the microscopic scale
to collective behaviors at the macroscopic scale.
author:
- first_name: N.
full_name: Bellomo, N.
last_name: Bellomo
- first_name: N.
full_name: Outada, N.
last_name: Outada
- first_name: J.
full_name: Soler, J.
last_name: Soler
- first_name: Y.
full_name: Tao, Y.
last_name: Tao
- first_name: Michael
full_name: Winkler, Michael
id: '31496'
last_name: Winkler
citation:
ama: 'Bellomo N, Outada N, Soler J, Tao Y, Winkler M. Chemotaxis and cross-diffusion
models in complex environments: Models and analytic problems toward a multiscale
vision. Mathematical Models and Methods in Applied Sciences. 2022;32(04):713-792.
doi:10.1142/s0218202522500166'
apa: 'Bellomo, N., Outada, N., Soler, J., Tao, Y., & Winkler, M. (2022). Chemotaxis
and cross-diffusion models in complex environments: Models and analytic problems
toward a multiscale vision. Mathematical Models and Methods in Applied Sciences,
32(04), 713–792. https://doi.org/10.1142/s0218202522500166'
bibtex: '@article{Bellomo_Outada_Soler_Tao_Winkler_2022, title={Chemotaxis and cross-diffusion
models in complex environments: Models and analytic problems toward a multiscale
vision}, volume={32}, DOI={10.1142/s0218202522500166},
number={04}, journal={Mathematical Models and Methods in Applied Sciences}, publisher={World
Scientific Pub Co Pte Ltd}, author={Bellomo, N. and Outada, N. and Soler, J. and
Tao, Y. and Winkler, Michael}, year={2022}, pages={713–792} }'
chicago: 'Bellomo, N., N. Outada, J. Soler, Y. Tao, and Michael Winkler. “Chemotaxis
and Cross-Diffusion Models in Complex Environments: Models and Analytic Problems
toward a Multiscale Vision.” Mathematical Models and Methods in Applied Sciences
32, no. 04 (2022): 713–92. https://doi.org/10.1142/s0218202522500166.'
ieee: 'N. Bellomo, N. Outada, J. Soler, Y. Tao, and M. Winkler, “Chemotaxis and
cross-diffusion models in complex environments: Models and analytic problems toward
a multiscale vision,” Mathematical Models and Methods in Applied Sciences,
vol. 32, no. 04, pp. 713–792, 2022, doi: 10.1142/s0218202522500166.'
mla: 'Bellomo, N., et al. “Chemotaxis and Cross-Diffusion Models in Complex Environments:
Models and Analytic Problems toward a Multiscale Vision.” Mathematical Models
and Methods in Applied Sciences, vol. 32, no. 04, World Scientific Pub Co
Pte Ltd, 2022, pp. 713–92, doi:10.1142/s0218202522500166.'
short: N. Bellomo, N. Outada, J. Soler, Y. Tao, M. Winkler, Mathematical Models
and Methods in Applied Sciences 32 (2022) 713–792.
date_created: 2025-12-18T19:20:25Z
date_updated: 2025-12-18T20:07:51Z
doi: 10.1142/s0218202522500166
intvolume: ' 32'
issue: '04'
language:
- iso: eng
page: 713-792
publication: Mathematical Models and Methods in Applied Sciences
publication_identifier:
issn:
- 0218-2025
- 1793-6314
publication_status: published
publisher: World Scientific Pub Co Pte Ltd
status: public
title: 'Chemotaxis and cross-diffusion models in complex environments: Models and
analytic problems toward a multiscale vision'
type: journal_article
user_id: '31496'
volume: 32
year: '2022'
...
---
_id: '63291'
abstract:
- lang: eng
text: An initial-boundary value problem for a coupled chemotaxis-Navier–Stokes
model with porous medium type diffusion is considered. Previous related literature
has provided profound knowledge in cases when the system is augmented with no-flux/no-flux/no-slip
boundary conditions for the density of cells, the chemical concentration and the
fluid velocity field, respectively; in particular, available qualitative results
strongly indicate that only trivial solution behavior can be expected on large
time scales. In line with refined modeling approaches to oxygen evolution near
fluid-air interfaces, this study now focuses on situations involving a fixed chemoattractant
concentration on the boundary. Despite an apparent loss of mathematically favorable
energy structures thereby induced, by means of an alternative variational approach
a basic theory of global existence is developed in a natural framework of weak
solvability. Beyond this, some additional qualitative information on the large
time behavior of these solutions is derived by identifying a certain global relaxation
property. Specifically, a second result asserts, within a suitable topological
setting, the existence of a bounded set which eventually absorbs each individual
of the obtained trajectories, and the diameter of which is bounded only by the
physically relevant quantities of total population size and prescribed boundary
concentration of the chemical signal.
author:
- first_name: Tobias
full_name: Black, Tobias
id: '23686'
last_name: Black
orcid: 0000-0001-9963-0800
- first_name: Michael
full_name: Winkler, Michael
id: '31496'
last_name: Winkler
citation:
ama: Black T, Winkler M. Global weak solutions and absorbing sets in a chemotaxis-Navier–Stokes
system with prescribed signal concentration on the boundary. Mathematical Models
and Methods in Applied Sciences. 2021;32(01):137-173. doi:10.1142/s021820252250004x
apa: Black, T., & Winkler, M. (2021). Global weak solutions and absorbing sets
in a chemotaxis-Navier–Stokes system with prescribed signal concentration on the
boundary. Mathematical Models and Methods in Applied Sciences, 32(01),
137–173. https://doi.org/10.1142/s021820252250004x
bibtex: '@article{Black_Winkler_2021, title={Global weak solutions and absorbing
sets in a chemotaxis-Navier–Stokes system with prescribed signal concentration
on the boundary}, volume={32}, DOI={10.1142/s021820252250004x},
number={01}, journal={Mathematical Models and Methods in Applied Sciences}, publisher={World
Scientific Pub Co Pte Ltd}, author={Black, Tobias and Winkler, Michael}, year={2021},
pages={137–173} }'
chicago: 'Black, Tobias, and Michael Winkler. “Global Weak Solutions and Absorbing
Sets in a Chemotaxis-Navier–Stokes System with Prescribed Signal Concentration
on the Boundary.” Mathematical Models and Methods in Applied Sciences 32,
no. 01 (2021): 137–73. https://doi.org/10.1142/s021820252250004x.'
ieee: 'T. Black and M. Winkler, “Global weak solutions and absorbing sets in a chemotaxis-Navier–Stokes
system with prescribed signal concentration on the boundary,” Mathematical
Models and Methods in Applied Sciences, vol. 32, no. 01, pp. 137–173, 2021,
doi: 10.1142/s021820252250004x.'
mla: Black, Tobias, and Michael Winkler. “Global Weak Solutions and Absorbing Sets
in a Chemotaxis-Navier–Stokes System with Prescribed Signal Concentration on the
Boundary.” Mathematical Models and Methods in Applied Sciences, vol. 32,
no. 01, World Scientific Pub Co Pte Ltd, 2021, pp. 137–73, doi:10.1142/s021820252250004x.
short: T. Black, M. Winkler, Mathematical Models and Methods in Applied Sciences
32 (2021) 137–173.
date_created: 2025-12-18T19:20:48Z
date_updated: 2025-12-18T20:08:01Z
doi: 10.1142/s021820252250004x
intvolume: ' 32'
issue: '01'
language:
- iso: eng
page: 137-173
publication: Mathematical Models and Methods in Applied Sciences
publication_identifier:
issn:
- 0218-2025
- 1793-6314
publication_status: published
publisher: World Scientific Pub Co Pte Ltd
status: public
title: Global weak solutions and absorbing sets in a chemotaxis-Navier–Stokes system
with prescribed signal concentration on the boundary
type: journal_article
user_id: '31496'
volume: 32
year: '2021'
...
---
_id: '34670'
author:
- first_name: Tobias
full_name: Black, Tobias
id: '23686'
last_name: Black
orcid: 0000-0001-9963-0800
citation:
ama: Black T. Global generalized solutions to a forager–exploiter model with superlinear
degradation and their eventual regularity properties. Mathematical Models and
Methods in Applied Sciences. 2020;30(06):1075-1117. doi:10.1142/s0218202520400072
apa: Black, T. (2020). Global generalized solutions to a forager–exploiter model
with superlinear degradation and their eventual regularity properties. Mathematical
Models and Methods in Applied Sciences, 30(06), 1075–1117. https://doi.org/10.1142/s0218202520400072
bibtex: '@article{Black_2020, title={Global generalized solutions to a forager–exploiter
model with superlinear degradation and their eventual regularity properties},
volume={30}, DOI={10.1142/s0218202520400072},
number={06}, journal={Mathematical Models and Methods in Applied Sciences}, publisher={World
Scientific Pub Co Pte Lt}, author={Black, Tobias}, year={2020}, pages={1075–1117}
}'
chicago: 'Black, Tobias. “Global Generalized Solutions to a Forager–Exploiter Model
with Superlinear Degradation and Their Eventual Regularity Properties.” Mathematical
Models and Methods in Applied Sciences 30, no. 06 (2020): 1075–1117. https://doi.org/10.1142/s0218202520400072.'
ieee: 'T. Black, “Global generalized solutions to a forager–exploiter model with
superlinear degradation and their eventual regularity properties,” Mathematical
Models and Methods in Applied Sciences, vol. 30, no. 06, pp. 1075–1117, 2020,
doi: 10.1142/s0218202520400072.'
mla: Black, Tobias. “Global Generalized Solutions to a Forager–Exploiter Model with
Superlinear Degradation and Their Eventual Regularity Properties.” Mathematical
Models and Methods in Applied Sciences, vol. 30, no. 06, World Scientific
Pub Co Pte Lt, 2020, pp. 1075–117, doi:10.1142/s0218202520400072.
short: T. Black, Mathematical Models and Methods in Applied Sciences 30 (2020) 1075–1117.
date_created: 2022-12-21T09:48:11Z
date_updated: 2023-07-10T11:39:59Z
department:
- _id: '34'
- _id: '10'
- _id: '90'
doi: 10.1142/s0218202520400072
intvolume: ' 30'
issue: '06'
keyword:
- Applied Mathematics
- Modeling and Simulation
language:
- iso: eng
page: 1075-1117
publication: Mathematical Models and Methods in Applied Sciences
publication_identifier:
issn:
- 0218-2025
- 1793-6314
publication_status: published
publisher: World Scientific Pub Co Pte Lt
status: public
title: Global generalized solutions to a forager–exploiter model with superlinear
degradation and their eventual regularity properties
type: journal_article
user_id: '23686'
volume: 30
year: '2020'
...
---
_id: '63331'
abstract:
- lang: eng
text: We consider a class of macroscopic models for the spatio-temporal
evolution of urban crime, as originally going back to Ref. 29 [M. B. Short, M.
R. D’Orsogna, V. B. Pasour, G. E. Tita, P. J. Brantingham, A. L. Bertozzi and
L. B. Chayes, A statistical model of criminal behavior, Math. Models Methods Appl.
Sci. 18 (2008) 1249–1267]. The focus here is on the question of how far a certain
porous medium enhancement in the random diffusion of criminal agents may exert
visible relaxation effects. It is shown that sufficient regularity of the non-negative
source terms in the system and a sufficiently strong nonlinear enhancement ensure
that a corresponding Neumann-type initial–boundary value problem, posed in a smoothly
bounded planar convex domain, admits locally bounded solutions for a wide class
of arbitrary initial data. Furthermore, this solution is globally bounded under
mild additional conditions on the source terms. These results are supplemented
by numerical evidence which illustrates smoothing effects in solutions with sharply
structured initial data in the presence of such porous medium-type diffusion and
support the existence of singular structures in the linear diffusion case, which
is the type of diffusion proposed in Ref. 29.
author:
- first_name: Nancy
full_name: Rodríguez, Nancy
last_name: Rodríguez
- first_name: Michael
full_name: Winkler, Michael
id: '31496'
last_name: Winkler
citation:
ama: Rodríguez N, Winkler M. Relaxation by nonlinear diffusion enhancement in a
two-dimensional cross-diffusion model for urban crime propagation. Mathematical
Models and Methods in Applied Sciences. 2020;30(11):2105-2137. doi:10.1142/s0218202520500396
apa: Rodríguez, N., & Winkler, M. (2020). Relaxation by nonlinear diffusion
enhancement in a two-dimensional cross-diffusion model for urban crime propagation.
Mathematical Models and Methods in Applied Sciences, 30(11), 2105–2137.
https://doi.org/10.1142/s0218202520500396
bibtex: '@article{Rodríguez_Winkler_2020, title={Relaxation by nonlinear diffusion
enhancement in a two-dimensional cross-diffusion model for urban crime propagation},
volume={30}, DOI={10.1142/s0218202520500396},
number={11}, journal={Mathematical Models and Methods in Applied Sciences}, publisher={World
Scientific Pub Co Pte Ltd}, author={Rodríguez, Nancy and Winkler, Michael}, year={2020},
pages={2105–2137} }'
chicago: 'Rodríguez, Nancy, and Michael Winkler. “Relaxation by Nonlinear Diffusion
Enhancement in a Two-Dimensional Cross-Diffusion Model for Urban Crime Propagation.”
Mathematical Models and Methods in Applied Sciences 30, no. 11 (2020):
2105–37. https://doi.org/10.1142/s0218202520500396.'
ieee: 'N. Rodríguez and M. Winkler, “Relaxation by nonlinear diffusion enhancement
in a two-dimensional cross-diffusion model for urban crime propagation,” Mathematical
Models and Methods in Applied Sciences, vol. 30, no. 11, pp. 2105–2137, 2020,
doi: 10.1142/s0218202520500396.'
mla: Rodríguez, Nancy, and Michael Winkler. “Relaxation by Nonlinear Diffusion Enhancement
in a Two-Dimensional Cross-Diffusion Model for Urban Crime Propagation.” Mathematical
Models and Methods in Applied Sciences, vol. 30, no. 11, World Scientific
Pub Co Pte Ltd, 2020, pp. 2105–37, doi:10.1142/s0218202520500396.
short: N. Rodríguez, M. Winkler, Mathematical Models and Methods in Applied Sciences
30 (2020) 2105–2137.
date_created: 2025-12-18T19:38:42Z
date_updated: 2025-12-18T20:00:53Z
doi: 10.1142/s0218202520500396
intvolume: ' 30'
issue: '11'
language:
- iso: eng
page: 2105-2137
publication: Mathematical Models and Methods in Applied Sciences
publication_identifier:
issn:
- 0218-2025
- 1793-6314
publication_status: published
publisher: World Scientific Pub Co Pte Ltd
status: public
title: Relaxation by nonlinear diffusion enhancement in a two-dimensional cross-diffusion
model for urban crime propagation
type: journal_article
user_id: '31496'
volume: 30
year: '2020'
...
---
_id: '63356'
abstract:
- lang: eng
text: This work deals with a taxis cascade model for food consumption in
two populations, namely foragers directly orienting their movement upward the
gradients of food concentration and exploiters taking a parasitic strategy in
search of food via tracking higher forager densities. As a consequence, the dynamics
of both populations are adapted to the space distribution of food which is dynamically
modified in time and space by the two populations. This model extends the classical
one-species chemotaxis-consumption systems by additionally accounting for a second
taxis mechanism coupled to the first in a consecutive manner. It is rigorously
proved that for all suitably regular initial data, an associated Neumann-type
initial-boundary value problem for the spatially one-dimensional version of this
model possesses a globally defined bounded classical solution. Moreover, it is
asserted that the considered two populations will approach spatially homogeneous
distributions in the large time limit, provided that either the total population
number of foragers or that of exploiters is appropriately small.
author:
- first_name: Youshan
full_name: Tao, Youshan
last_name: Tao
- first_name: Michael
full_name: Winkler, Michael
id: '31496'
last_name: Winkler
citation:
ama: Tao Y, Winkler M. Large time behavior in a forager–exploiter model with different
taxis strategies for two groups in search of food. Mathematical Models and
Methods in Applied Sciences. 2019;29(11):2151-2182. doi:10.1142/s021820251950043x
apa: Tao, Y., & Winkler, M. (2019). Large time behavior in a forager–exploiter
model with different taxis strategies for two groups in search of food. Mathematical
Models and Methods in Applied Sciences, 29(11), 2151–2182. https://doi.org/10.1142/s021820251950043x
bibtex: '@article{Tao_Winkler_2019, title={Large time behavior in a forager–exploiter
model with different taxis strategies for two groups in search of food}, volume={29},
DOI={10.1142/s021820251950043x},
number={11}, journal={Mathematical Models and Methods in Applied Sciences}, publisher={World
Scientific Pub Co Pte Ltd}, author={Tao, Youshan and Winkler, Michael}, year={2019},
pages={2151–2182} }'
chicago: 'Tao, Youshan, and Michael Winkler. “Large Time Behavior in a Forager–Exploiter
Model with Different Taxis Strategies for Two Groups in Search of Food.” Mathematical
Models and Methods in Applied Sciences 29, no. 11 (2019): 2151–82. https://doi.org/10.1142/s021820251950043x.'
ieee: 'Y. Tao and M. Winkler, “Large time behavior in a forager–exploiter model
with different taxis strategies for two groups in search of food,” Mathematical
Models and Methods in Applied Sciences, vol. 29, no. 11, pp. 2151–2182, 2019,
doi: 10.1142/s021820251950043x.'
mla: Tao, Youshan, and Michael Winkler. “Large Time Behavior in a Forager–Exploiter
Model with Different Taxis Strategies for Two Groups in Search of Food.” Mathematical
Models and Methods in Applied Sciences, vol. 29, no. 11, World Scientific
Pub Co Pte Ltd, 2019, pp. 2151–82, doi:10.1142/s021820251950043x.
short: Y. Tao, M. Winkler, Mathematical Models and Methods in Applied Sciences 29
(2019) 2151–2182.
date_created: 2025-12-19T10:54:36Z
date_updated: 2025-12-19T10:54:44Z
doi: 10.1142/s021820251950043x
intvolume: ' 29'
issue: '11'
language:
- iso: eng
page: 2151-2182
publication: Mathematical Models and Methods in Applied Sciences
publication_identifier:
issn:
- 0218-2025
- 1793-6314
publication_status: published
publisher: World Scientific Pub Co Pte Ltd
status: public
title: Large time behavior in a forager–exploiter model with different taxis strategies
for two groups in search of food
type: journal_article
user_id: '31496'
volume: 29
year: '2019'
...
---
_id: '63363'
abstract:
- lang: eng
text: ' 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 [Formula: see text] for the population
densities [Formula: see text] and [Formula: see text] of foragers and exploiters,
as well as the nutrient concentration [Formula: see text], is considered in smoothly
bounded domains [Formula: see text], [Formula: see text]. It is first shown that
under an explicit condition linking the sizes of the resource production rate
[Formula: see text] 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 [Formula: see text] 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 (⋆).
'
author:
- first_name: Michael
full_name: Winkler, Michael
id: '31496'
last_name: Winkler
citation:
ama: Winkler M. Global generalized solutions to a multi-dimensional doubly tactic
resource consumption model accounting for social interactions. Mathematical
Models and Methods in Applied Sciences. 2019;29(03):373-418. doi:10.1142/s021820251950012x
apa: Winkler, M. (2019). Global generalized solutions to a multi-dimensional doubly
tactic resource consumption model accounting for social interactions. Mathematical
Models and Methods in Applied Sciences, 29(03), 373–418. https://doi.org/10.1142/s021820251950012x
bibtex: '@article{Winkler_2019, title={Global generalized solutions to a multi-dimensional
doubly tactic resource consumption model accounting for social interactions},
volume={29}, DOI={10.1142/s021820251950012x},
number={03}, journal={Mathematical Models and Methods in Applied Sciences}, publisher={World
Scientific Pub Co Pte Ltd}, author={Winkler, Michael}, year={2019}, pages={373–418}
}'
chicago: 'Winkler, Michael. “Global Generalized Solutions to a Multi-Dimensional
Doubly Tactic Resource Consumption Model Accounting for Social Interactions.”
Mathematical Models and Methods in Applied Sciences 29, no. 03 (2019):
373–418. https://doi.org/10.1142/s021820251950012x.'
ieee: 'M. Winkler, “Global generalized solutions to a multi-dimensional doubly tactic
resource consumption model accounting for social interactions,” Mathematical
Models and Methods in Applied Sciences, vol. 29, no. 03, pp. 373–418, 2019,
doi: 10.1142/s021820251950012x.'
mla: Winkler, Michael. “Global Generalized Solutions to a Multi-Dimensional Doubly
Tactic Resource Consumption Model Accounting for Social Interactions.” Mathematical
Models and Methods in Applied Sciences, vol. 29, no. 03, World Scientific
Pub Co Pte Ltd, 2019, pp. 373–418, doi:10.1142/s021820251950012x.
short: M. Winkler, Mathematical Models and Methods in Applied Sciences 29 (2019)
373–418.
date_created: 2025-12-19T10:59:03Z
date_updated: 2025-12-19T10:59:10Z
doi: 10.1142/s021820251950012x
intvolume: ' 29'
issue: '03'
language:
- iso: eng
page: 373-418
publication: Mathematical Models and Methods in Applied Sciences
publication_identifier:
issn:
- 0218-2025
- 1793-6314
publication_status: published
publisher: World Scientific Pub Co Pte Ltd
status: public
title: Global generalized solutions to a multi-dimensional doubly tactic resource
consumption model accounting for social interactions
type: journal_article
user_id: '31496'
volume: 29
year: '2019'
...