Linear stability for a free boundary tumor model with a periodic supply of external nutrients

Huang, Yaodan; Zhang, Zhengce; Hu, Bei

doi:10.1002/mma.5412

Cited by 23 publications

(12 citation statements)

References 65 publications

(119 reference statements)

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“…By our method, one can easily improve results of [1] and [12] to obtain that the corresponding nonnecrotic tumor model has a unique periodic solution if and only if 1 ω ω 0 φ(t)dt >σ, and it is also asymptotically stable under radial perturbations and linearly stable under small non-radial perturbations.…”

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confidence: 96%

“…In the limiting casê σ = 0 of problem (1) with (A4) and linear functions f , g given in (2), i.e., for a similar nonnecrotic tumor model, Bai and Xu [1] proved that if min 0≤t≤ω φ(t) >σ, then there exists a unique periodic nonnecrotic solution, and it is asymptotically stable under radial perturbations. Recently, Huang, Zhang and Hu [12] further proved the periodic nonnecrotic solution is linearly stable under small non-radial perturbations.…”

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confidence: 99%

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Asymptotic behavior of a nonlinear necrotic tumor model with a periodic external nutrient supply

Wu¹,

Xu²

2020

Discrete &Amp; Continuous Dynamical Systems - B

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In this paper we study a nonlinear free boundary problem for the growth of radially symmetric tumor with a necrotic core. The proliferation of tumor cells depends on the concentration of nutrient which satisfies a diffusion equation within tumor and is periodically supplied by external tissues. The tumor outer surface and the inner interface of the necrotic core are both free boundaries. We give a sufficient and necessary condition for the existence and uniqueness of positive periodic solution, and show it is globally asymptotically stable under radial perturbations. Our analysis implies that tumor growth may finally synchronize the periodic external nutrient supply.

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confidence: 96%

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confidence: 99%

Asymptotic behavior of a nonlinear necrotic tumor model with a periodic external nutrient supply

Wu¹,

Xu²

2020

Discrete &Amp; Continuous Dynamical Systems - B

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“…Proof of Theorem 1.2. Using Propositions 1, 2, 3 and (141), (142), (149), we can derive (15) by an argument quite similar to that in the proof of Theorem 1.1 of [21]. On the other hand, the linear instability in the case µ > µ * follows by taking n = 2 in (97).…”

Section: Proposition 4 Rmentioning

confidence: 87%

The impact of time delay and angiogenesis in a tumor model

Wang¹,

Zhou²,

Song³

2022

DCDS-B

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<p style='text-indent:20px;'>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 <inline-formula><tex-math id="M1">\begin{document}$ \alpha $\end{document}</tex-math></inline-formula>, and a parameter <inline-formula><tex-math id="M2">\begin{document}$ \mu $\end{document}</tex-math></inline-formula> is proportional to the 'aggressiveness' of the tumor. In this paper, we first prove that there exists a unique radially symmetric stationary solution <inline-formula><tex-math id="M3">\begin{document}$ \left(\sigma_{*}, p_{*}, R_{*}\right) $\end{document}</tex-math></inline-formula> for all positive <inline-formula><tex-math id="M4">\begin{document}$ \alpha $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M5">\begin{document}$ \mu $\end{document}</tex-math></inline-formula>. Then a threshold value <inline-formula><tex-math id="M6">\begin{document}$ \mu_\ast $\end{document}</tex-math></inline-formula> is found such that the radially symmetric stationary solution is linearly stable if <inline-formula><tex-math id="M7">\begin{document}$ \mu<\mu_\ast $\end{document}</tex-math></inline-formula> and linearly unstable if <inline-formula><tex-math id="M8">\begin{document}$ \mu>\mu_\ast $\end{document}</tex-math></inline-formula>. Our results indicate that the increase of the angiogenesis parameter <inline-formula><tex-math id="M9">\begin{document}$ \alpha $\end{document}</tex-math></inline-formula> would result in the reduction of the threshold value <inline-formula><tex-math id="M10">\begin{document}$ \mu_\ast $\end{document}</tex-math></inline-formula>, adding the time delay would not alter the threshold value <inline-formula><tex-math id="M11">\begin{document}$ \mu_\ast $\end{document}</tex-math></inline-formula>, but would result in a larger stationary tumor, and the larger the tumor aggressiveness parameter <inline-formula><tex-math id="M12">\begin{document}$ \mu $\end{document}</tex-math></inline-formula> is, the greater impact of time delay would have on the size of the stationary tumor.</p>

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“…Over the past decades, extensive studies have been done on free boundary problems modeling the growth of tumors (see, e.g., [2,4,3,6,15,16,20,21]). In this paper we consider a spherically symmetric non-necrotic tumor in R 3 and study the concentration of a certain type of nutrient within the tumor.…”

Section: Introductionmentioning

confidence: 99%

“…As far as we know, some relevant mathematical model has been investigated when the nutrition supply α(t) on the tumor surface is periodic (see, e.g. [6,16,21]). When a Gibbs-Thmson relation is taken into account, Wu [20] established the existence and uniqueness of solutions of the tumor model for the linear case.…”

Section: Introductionmentioning

confidence: 99%