# American Institute of Mathematical Sciences

December  2018, 15(6): 1345-1385. doi: 10.3934/mbe.2018062

## Early and late stage profiles for a chemotaxis model with density-dependent jump probability

 1 School of Mathematical Sciences, South China Normal University, Guangzhou, Guangdong 510631, China 2 School of Mathematics, South China University of Technology, Guangzhou, Guangdong 510641, China 3 Department of Mathematics, Champlain College Saint-Lambert, Quebec, J4P 3P2, Canada 4 Department of Mathematics and Statistics, McGill University, Montreal, Quebec, H3A 2K6, Canada

* Corresponding author: jism@scut.edu.cn

Received  October 19, 2017 Accepted  July 24, 2018 Published  September 2018

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 $0 < β < \frac{1}{2}$. 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 $c>0$ is also obtained.

Citation: 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 & Engineering, 2018, 15 (6) : 1345-1385. doi: 10.3934/mbe.2018062
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##### References:
(a) Comparison of experimental and simulated cell distribution for MG63 cells. The measured cell density (gray histogram) are fitted using the solution of degenerate nonlinear diffusion model (gray lines). (b) Comparison of experimental and simulated cell distribution for HBMSC cells. The measured cell density (gray histogram) are matched with the solution of linear diffusion model (gray lines). This diagram was redrawn from the one in Ref. [30]
The growth curve of HEPA-1 spheroids. The solid line represents the position of the outer tumour boundary. Dimensional diameters are shown in $\mu m$. This diagram was redrawn from the one in Ref. [27]
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