# American Institute of Mathematical Sciences

doi: 10.3934/dcdsb.2021219
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## The impact of time delay and angiogenesis in a tumor model

 School of Mathematics and Statistics, Jiangxi Normal University, Nanchang 330022, China

*Corresponding author: Huijuan Song

Received  March 2021 Revised  June 2021 Early access September 2021

We consider a free boundary tumor model under the presence of angiogenesis and time delays in the process of proliferation, in which the cell location is incorporated. It is assumed that the tumor attracts blood vessels at a rate proportional to $\alpha$, and a parameter $\mu$ is proportional to the 'aggressiveness' of the tumor. In this paper, we first prove that there exists a unique radially symmetric stationary solution $\left(\sigma_{*}, p_{*}, R_{*}\right)$ for all positive $\alpha$, $\mu$. Then a threshold value $\mu_\ast$ is found such that the radially symmetric stationary solution is linearly stable if $\mu<\mu_\ast$ and linearly unstable if $\mu>\mu_\ast$. Our results indicate that the increase of the angiogenesis parameter $\alpha$ would result in the reduction of the threshold value $\mu_\ast$, adding the time delay would not alter the threshold value $\mu_\ast$, but would result in a larger stationary tumor, and the larger the tumor aggressiveness parameter $\mu$ is, the greater impact of time delay would have on the size of the stationary tumor.

Citation: Zejia Wang, Haihua Zhou, Huijuan Song. The impact of time delay and angiogenesis in a tumor model. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021219
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