[{"intvolume":"       119","page":"1598-1632","citation":{"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>.","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>","short":"Y. Tao, M. Winkler, Proceedings of the London Mathematical Society 119 (2019) 1598–1632.","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} }","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>."},"year":"2019","issue":"6","publication_identifier":{"issn":["0024-6115","1460-244X"]},"publication_status":"published","doi":"10.1112/plms.12276","title":"Boundedness and stabilization in a population model with cross‐diffusion for one species","volume":119,"author":[{"full_name":"Tao, Youshan","last_name":"Tao","first_name":"Youshan"},{"first_name":"Michael","last_name":"Winkler","full_name":"Winkler, Michael","id":"31496"}],"date_created":"2025-12-19T10:54:01Z","date_updated":"2025-12-19T10:54:09Z","publisher":"Wiley","status":"public","abstract":[{"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>","lang":"eng"}],"publication":"Proceedings of the London Mathematical Society","type":"journal_article","language":[{"iso":"eng"}],"user_id":"31496","_id":"63355"}]