EARLY AND LATE STAGE PROFILES FOR A CHEMOTAXIS MODEL WITH DENSITY-DEPENDENT JUMP PROBABILITY
In this paper, we derive a chemotaxis model with degenerate diffusion and density-dependent chemotactic sensitivity, and we provide a more realistic description of cell migration process for its early and late stages. Different from the existing studies focusing on the case of non-degenerate diffusi...
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| Format: | Article |
| Language: | English |
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AIMS Press
2018-11-01
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| Series: | Mathematical Biosciences and Engineering |
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| Online Access: | https://www.aimspress.com/article/doi/10.3934/mbe.2018062 |
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| author | Tianyuan Xu Shanming Ji Chunhua Jin Ming Mei Jingxue Yin |
| author_facet | Tianyuan Xu Shanming Ji Chunhua Jin Ming Mei Jingxue Yin |
| author_sort | Tianyuan Xu |
| collection | DOAJ |
| description | In this paper, we derive a chemotaxis model with degenerate diffusion and density-dependent chemotactic sensitivity, and we provide a more realistic description of cell migration process for its early and late stages. Different from the existing studies focusing on the case of non-degenerate diffusion, this model with degenerate diffusion causes us some essential difficulty on the boundedness estimates and the propagation behavior of its compact support. In the presence of logistic damping, for the early stage before tumour cells spread to the whole domain, we first estimate the expanding speed of tumour region as $O(t^{β})$ for $ 0 \lt β \lt \frac{1}{2}$ . Then, for the late stage of cell migration, we further prove that the asymptotic profile of the original system is just its corresponding steady state. The global convergence of the original weak solution to the steady state with exponential rate $O(e^{-ct})$ for some $c \gt 0$ is also obtained. |
| format | Article |
| id | doaj-art-77d02f42077e49d5b500461361fbc803 |
| institution | Kabale University |
| issn | 1551-0018 |
| language | English |
| publishDate | 2018-11-01 |
| publisher | AIMS Press |
| record_format | Article |
| series | Mathematical Biosciences and Engineering |
| spelling | doaj-art-77d02f42077e49d5b500461361fbc8032025-01-24T02:41:08ZengAIMS PressMathematical Biosciences and Engineering1551-00182018-11-011561345138510.3934/mbe.2018062EARLY AND LATE STAGE PROFILES FOR A CHEMOTAXIS MODEL WITH DENSITY-DEPENDENT JUMP PROBABILITYTianyuan Xu0Shanming Ji1Chunhua Jin2Ming Mei3Jingxue Yin4School of Mathematical Sciences, South China Normal University, Guangzhou, Guangdong 510631, ChinaSchool of Mathematics, South China University of Technology, Guangzhou, Guangdong 510641, ChinaSchool of Mathematical Sciences, South China Normal University, Guangzhou, Guangdong 510631, ChinaDepartment of Mathematics, Champlain College Saint-Lambert, Quebec, J4P 3P2, CanadaSchool of Mathematical Sciences, South China Normal University, Guangzhou, Guangdong 510631, ChinaIn this paper, we derive a chemotaxis model with degenerate diffusion and density-dependent chemotactic sensitivity, and we provide a more realistic description of cell migration process for its early and late stages. Different from the existing studies focusing on the case of non-degenerate diffusion, this model with degenerate diffusion causes us some essential difficulty on the boundedness estimates and the propagation behavior of its compact support. In the presence of logistic damping, for the early stage before tumour cells spread to the whole domain, we first estimate the expanding speed of tumour region as $O(t^{β})$ for $ 0 \lt β \lt \frac{1}{2}$ . Then, for the late stage of cell migration, we further prove that the asymptotic profile of the original system is just its corresponding steady state. The global convergence of the original weak solution to the steady state with exponential rate $O(e^{-ct})$ for some $c \gt 0$ is also obtained.https://www.aimspress.com/article/doi/10.3934/mbe.2018062chemotaxis modeldegenerate diffusiondensity-dependent jump probabilityfinite speed propagationtumour invasions modelsporous media diffusion. |
| spellingShingle | Tianyuan Xu Shanming Ji Chunhua Jin Ming Mei Jingxue Yin EARLY AND LATE STAGE PROFILES FOR A CHEMOTAXIS MODEL WITH DENSITY-DEPENDENT JUMP PROBABILITY Mathematical Biosciences and Engineering chemotaxis model degenerate diffusion density-dependent jump probability finite speed propagation tumour invasions models porous media diffusion. |
| title | EARLY AND LATE STAGE PROFILES FOR A CHEMOTAXIS MODEL WITH DENSITY-DEPENDENT JUMP PROBABILITY |
| title_full | EARLY AND LATE STAGE PROFILES FOR A CHEMOTAXIS MODEL WITH DENSITY-DEPENDENT JUMP PROBABILITY |
| title_fullStr | EARLY AND LATE STAGE PROFILES FOR A CHEMOTAXIS MODEL WITH DENSITY-DEPENDENT JUMP PROBABILITY |
| title_full_unstemmed | EARLY AND LATE STAGE PROFILES FOR A CHEMOTAXIS MODEL WITH DENSITY-DEPENDENT JUMP PROBABILITY |
| title_short | EARLY AND LATE STAGE PROFILES FOR A CHEMOTAXIS MODEL WITH DENSITY-DEPENDENT JUMP PROBABILITY |
| title_sort | early and late stage profiles for a chemotaxis model with density dependent jump probability |
| topic | chemotaxis model degenerate diffusion density-dependent jump probability finite speed propagation tumour invasions models porous media diffusion. |
| url | https://www.aimspress.com/article/doi/10.3934/mbe.2018062 |
| work_keys_str_mv | AT tianyuanxu earlyandlatestageprofilesforachemotaxismodelwithdensitydependentjumpprobability AT shanmingji earlyandlatestageprofilesforachemotaxismodelwithdensitydependentjumpprobability AT chunhuajin earlyandlatestageprofilesforachemotaxismodelwithdensitydependentjumpprobability AT mingmei earlyandlatestageprofilesforachemotaxismodelwithdensitydependentjumpprobability AT jingxueyin earlyandlatestageprofilesforachemotaxismodelwithdensitydependentjumpprobability |