---
_id: '63355'
abstract:
- lang: eng
  text: "<jats:title>Abstract</jats:title><jats:p>This work studies the two‐species
    Shigesada–Kawasaki–Teramoto model with cross‐diffusion for one species, as given by\r\n<jats:disp-formula>\r\n</jats:disp-formula>with
    positive parameters  and , and nonnegative constants  and . Beyond some statements
    on global existence, the literature apparently provides only few results on qualitative
    behavior of solutions; in particular, questions related to boundedness as well
    as to large time asymptotics in <jats:ext-link xmlns:xlink=\"http://www.w3.org/1999/xlink\"
    xlink:href=\"#plms12276-disp-0001\" /> seem unsolved so far.</jats:p><jats:p>In
    the present paper it is <jats:italic>inter alia</jats:italic> shown that if  and
    \ is a bounded convex domain with smooth boundary, then whenever  and  are nonnegative,
    the associated Neumann initial‐boundary value problem for <jats:ext-link xmlns:xlink=\"http://www.w3.org/1999/xlink\"
    xlink:href=\"#plms12276-disp-0001\" /> possesses a global classical solution which
    in fact is bounded in the sense that\r\n<jats:disp-formula>\r\n</jats:disp-formula>Moreover,
    the asymptotic behavior of arbitrary nonnegative solutions enjoying the boundedness
    property is studied in the general situation when  is arbitrary and  no longer
    necessarily convex. If , then in both cases  and , an explicit smallness condition
    on  is identified as sufficient for stabilization of any nontrivial solutions
    toward a corresponding unique nontrivial spatially homogeneous steady state. If
    \ and , then without any further assumption all nonzero solutions are seen to
    approach the equilibrium (0,1). As a by‐product, this particularly improves previous
    knowledge on nonexistence of nonconstant equilibria of <jats:ext-link xmlns:xlink=\"http://www.w3.org/1999/xlink\"
    xlink:href=\"#plms12276-disp-0001\" />.</jats:p>"
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. Boundedness and stabilization in a population model with
    cross‐diffusion for one species. <i>Proceedings of the London Mathematical Society</i>.
    2019;119(6):1598-1632. doi:<a href="https://doi.org/10.1112/plms.12276">10.1112/plms.12276</a>
  apa: Tao, Y., &#38; Winkler, M. (2019). Boundedness and stabilization in a population
    model with cross‐diffusion for one species. <i>Proceedings of the London Mathematical
    Society</i>, <i>119</i>(6), 1598–1632. <a href="https://doi.org/10.1112/plms.12276">https://doi.org/10.1112/plms.12276</a>
  bibtex: '@article{Tao_Winkler_2019, title={Boundedness and stabilization in a population
    model with cross‐diffusion for one species}, volume={119}, DOI={<a href="https://doi.org/10.1112/plms.12276">10.1112/plms.12276</a>},
    number={6}, journal={Proceedings of the London Mathematical Society}, publisher={Wiley},
    author={Tao, Youshan and Winkler, Michael}, year={2019}, pages={1598–1632} }'
  chicago: 'Tao, Youshan, and Michael Winkler. “Boundedness and Stabilization in a
    Population Model with Cross‐diffusion for One Species.” <i>Proceedings of the
    London Mathematical Society</i> 119, no. 6 (2019): 1598–1632. <a href="https://doi.org/10.1112/plms.12276">https://doi.org/10.1112/plms.12276</a>.'
  ieee: 'Y. Tao and M. Winkler, “Boundedness and stabilization in a population model
    with cross‐diffusion for one species,” <i>Proceedings of the London Mathematical
    Society</i>, vol. 119, no. 6, pp. 1598–1632, 2019, doi: <a href="https://doi.org/10.1112/plms.12276">10.1112/plms.12276</a>.'
  mla: Tao, Youshan, and Michael Winkler. “Boundedness and Stabilization in a Population
    Model with Cross‐diffusion for One Species.” <i>Proceedings of the London Mathematical
    Society</i>, vol. 119, no. 6, Wiley, 2019, pp. 1598–632, doi:<a href="https://doi.org/10.1112/plms.12276">10.1112/plms.12276</a>.
  short: Y. Tao, M. Winkler, Proceedings of the London Mathematical Society 119 (2019)
    1598–1632.
date_created: 2025-12-19T10:54:01Z
date_updated: 2025-12-19T10:54:09Z
doi: 10.1112/plms.12276
intvolume: '       119'
issue: '6'
language:
- iso: eng
page: 1598-1632
publication: Proceedings of the London Mathematical Society
publication_identifier:
  issn:
  - 0024-6115
  - 1460-244X
publication_status: published
publisher: Wiley
status: public
title: Boundedness and stabilization in a population model with cross‐diffusion for
  one species
type: journal_article
user_id: '31496'
volume: 119
year: '2019'
...