[{"doi":"10.1112/plms.12276","language":[{"iso":"eng"}],"date_updated":"2025-12-19T10:54:09Z","publication_status":"published","intvolume":" 119","title":"Boundedness and stabilization in a population model with cross‐diffusion for one species","year":"2019","author":[{"full_name":"Tao, Youshan","first_name":"Youshan","last_name":"Tao"},{"id":"31496","last_name":"Winkler","first_name":"Michael","full_name":"Winkler, Michael"}],"publication_identifier":{"issn":["0024-6115","1460-244X"]},"type":"journal_article","date_created":"2025-12-19T10:54:01Z","abstract":[{"text":"AbstractThis work studies the two‐species Shigesada–Kawasaki–Teramoto model with cross‐diffusion for one species, as given by\r\n\r\nwith 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 seem unsolved so far.In the present paper it is inter alia 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 possesses a global classical solution which in fact is bounded in the sense that\r\n\r\nMoreover, 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 .","lang":"eng"}],"publication":"Proceedings of the London Mathematical Society","issue":"6","user_id":"31496","volume":119,"page":"1598-1632","_id":"63355","publisher":"Wiley","status":"public","citation":{"bibtex":"@article{Tao_Winkler_2019, title={Boundedness and stabilization in a population model with cross‐diffusion for one species}, volume={119}, DOI={10.1112/plms.12276}, number={6}, journal={Proceedings of the London Mathematical Society}, publisher={Wiley}, author={Tao, Youshan and Winkler, Michael}, year={2019}, pages={1598–1632} }","ama":"Tao Y, Winkler M. Boundedness and stabilization in a population model with cross‐diffusion for one species. Proceedings of the London Mathematical Society. 2019;119(6):1598-1632. doi:10.1112/plms.12276","mla":"Tao, Youshan, and Michael Winkler. “Boundedness and Stabilization in a Population Model with Cross‐diffusion for One Species.” Proceedings of the London Mathematical Society, vol. 119, no. 6, Wiley, 2019, pp. 1598–632, doi:10.1112/plms.12276.","chicago":"Tao, Youshan, and Michael Winkler. “Boundedness and Stabilization in a Population Model with Cross‐diffusion for One Species.” Proceedings of the London Mathematical Society 119, no. 6 (2019): 1598–1632. https://doi.org/10.1112/plms.12276.","short":"Y. Tao, M. Winkler, Proceedings of the London Mathematical Society 119 (2019) 1598–1632.","ieee":"Y. Tao and M. Winkler, “Boundedness and stabilization in a population model with cross‐diffusion for one species,” Proceedings of the London Mathematical Society, vol. 119, no. 6, pp. 1598–1632, 2019, doi: 10.1112/plms.12276.","apa":"Tao, Y., & Winkler, M. (2019). Boundedness and stabilization in a population model with cross‐diffusion for one species. Proceedings of the London Mathematical Society, 119(6), 1598–1632. https://doi.org/10.1112/plms.12276"}},{"citation":{"ieee":"M. Ulirsch, “Functorial tropicalization of logarithmic schemes: the case of constant coefficients,” Proceedings of the London Mathematical Society, vol. 114, no. 6, pp. 1081–1113, 2017, doi: 10.1112/plms.12031.","apa":"Ulirsch, M. (2017). Functorial tropicalization of logarithmic schemes: the case of constant coefficients. Proceedings of the London Mathematical Society, 114(6), 1081–1113. https://doi.org/10.1112/plms.12031","short":"M. Ulirsch, Proceedings of the London Mathematical Society 114 (2017) 1081–1113.","chicago":"Ulirsch, Martin. “Functorial Tropicalization of Logarithmic Schemes: The Case of Constant Coefficients.” Proceedings of the London Mathematical Society 114, no. 6 (2017): 1081–1113. https://doi.org/10.1112/plms.12031.","mla":"Ulirsch, Martin. “Functorial Tropicalization of Logarithmic Schemes: The Case of Constant Coefficients.” Proceedings of the London Mathematical Society, vol. 114, no. 6, Wiley, 2017, pp. 1081–113, doi:10.1112/plms.12031.","bibtex":"@article{Ulirsch_2017, title={Functorial tropicalization of logarithmic schemes: the case of constant coefficients}, volume={114}, DOI={10.1112/plms.12031}, number={6}, journal={Proceedings of the London Mathematical Society}, publisher={Wiley}, author={Ulirsch, Martin}, year={2017}, pages={1081–1113} }","ama":"Ulirsch M. Functorial tropicalization of logarithmic schemes: the case of constant coefficients. Proceedings of the London Mathematical Society. 2017;114(6):1081-1113. doi:10.1112/plms.12031"},"publication":"Proceedings of the London Mathematical Society","issue":"6","date_created":"2026-07-08T07:24:50Z","type":"journal_article","publication_identifier":{"issn":["0024-6115","1460-244X"]},"author":[{"first_name":"Martin","last_name":"Ulirsch","full_name":"Ulirsch, Martin","id":"114697"}],"title":"Functorial tropicalization of logarithmic schemes: the case of constant coefficients","year":"2017","status":"public","intvolume":" 114","publication_status":"published","date_updated":"2026-07-09T07:31:48Z","language":[{"iso":"eng"}],"_id":"66354","publisher":"Wiley","page":"1081-1113","volume":114,"user_id":"82981","doi":"10.1112/plms.12031"}]