Bifurcation for a free-boundary tumor model with angiogenesis

Huang, Yaodan; Zhang, Zhengce; Hu, Bei

doi:10.1016/j.nonrwa.2016.12.003

Cited by 46 publications

(44 citation statements)

References 45 publications

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“…Combining with the condition (21), there is a unique * > 0 such that (25) holds. This establishes Theorem 1.…”

Section: Preliminariesmentioning

confidence: 96%

“…We shall investigate the relation of parameter χ and the stationary solution ( , M, p, * ), which partly reflects the effect of haptotaxis on tumor growth. By (25), it is obvious that the thickness of stationary tumor * is only determined bỹ, which does not depend on χ. Furthermore, Figure 1 shows that and M are independent of χ and p is monotonically increasing in χ, respectively.…”

Section: Preliminariesmentioning

confidence: 99%

“…These models, which consider the tumor tissue as a density of proliferating cells, are based on mass conservation laws and on reaction‐diffusion processes. The influences of different factors on tumor growth are investigated, such as the effect of inhibitor, cell cycle, angiogenesis, and so on. Lowengrub et al provided a systematic survey of tumor model studies.…”

Section: Introducionmentioning

confidence: 99%

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Bifurcation for a free boundary problem modeling the growth of multilayer tumors with ECM and MDE interactions

2020

Math Methods in App Sciences

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We study a free boundary problem modeling the growth of multilayer tumors. This model describes the invasion of tumors: the tumor cells produce matrix degrading enzymes (MDEs) to degrade the extracellular matrix (ECM) which provides structural support of the surrounding tissue. As in Pan and Xing, the influence of ECM and MDE interactions is considered in this paper. The model equations include two diffusion equations for the nutrient concentration and MDE concentration and an ordinary differential equation for ECM concentration. The tumor cell proliferation is at the rate normalλ, which characterizes the aggressiveness of the tumor. In contrast to Pan and Xing, we consider “flat stationary solution” in this paper. We first show that there exists a unique flat stationary solution for any normalλ>0. Then, we prove that there are infinite branches of bifurcation solutions from the flat stationary solutions at the bifurcation points normalλ=normalλkfalse(ρ*false) ( k≥1).

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“…Combining with the condition (21), there is a unique * > 0 such that (25) holds. This establishes Theorem 1.…”

Section: Preliminariesmentioning

confidence: 96%

Section: Preliminariesmentioning

confidence: 99%

Section: Introducionmentioning

confidence: 99%

See 1 more Smart Citation

Bifurcation for a free boundary problem modeling the growth of multilayer tumors with ECM and MDE interactions

2020

Math Methods in App Sciences

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“…Over the past 40 years, an increasing number of mathematical models describing tumor growth have been developed and studied, and many theoretical and numerical results have been established; see the review papers [1][2][3][4][5][6], the recent papers [7][8][9][10][11][12][13][14][15], and the references therein.…”

Section: Introductionmentioning

confidence: 99%

“…The asymptotic stability of stationary solutions was studied in [22][23][24]. Later, for problem (1.1)-(1.7) in the absence of inhibitors (β = 0), Friedman and Lam [8] proved that, for any α > 0, 0 <σ <σ , there exists a unique radially symmetric solution σ * (r) with radius R * ; furthermore, Hu et al [10] showed that for each μ m (m ≥ 2), there exist symmetry-breaking solutions bifurcating from the radially symmetric solution. While, if the inhibitor is present (β = 0) and the boundary conditions (1.4) and (1.5) are replaced by the boundary conditions σ =σ , β =β (which is formally the case α = ∞, τ = ∞), the existence and asymptotic stability of radially symmetric solutions were studied in [17,25,26].…”

Section: Introductionmentioning

confidence: 99%

Bifurcation analysis for a free-boundary tumor model with angiogenesis and inhibitor

Wang

Song

2018

Bound Value Probl

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This paper is concerned with the bifurcation phenomenon of a free-boundary problem modeling the tumor growth under the action of angiogenesis and inhibitor.Taking the surface tension coefficient γ as a bifurcation parameter, we prove that there exist a positive integer m * * and a sequence of γ m such that, for every γ m (m > m * * ), symmetry-breaking stationary solutions bifurcate from the radially symmetric stationary solutions. MSC: Primary 35B35; 35K35; secondary 35Q80

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

Huang

Zhang

2018

Math Methods in App Sciences

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In this paper, we consider a free boundary tumor model with a periodic supply of external nutrients, so that the nutrient concentration satisfies = (t) on the boundary, where (t) is a positive periodic function with period T. A parameter in the model is proportional to the "aggressiveness" of the tumor.If 0 <̃< min 0≤t≤T (t), wherẽis a threshold concentration for proliferation, Bai and Xu [Pac J Appl Math. 2013;5;217-223] proved that there exists a unique radially symmetric T-periodic positive solution ( * (r, t), p * (r, t), R * (t)), which is stable for any > 0 with respect to all radially symmetric perturbations. 17 We prove that under nonradially symmetric perturbations, there exists a number * such that if 0 < < * , then the T-periodic solution is linearly stable, whereas if > * , then the T-periodic solution is linearly unstable.

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Bifurcation for a free-boundary tumor model with angiogenesis

Cited by 46 publications

References 45 publications

Bifurcation for a free boundary problem modeling the growth of multilayer tumors with ECM and MDE interactions

Bifurcation for a free boundary problem modeling the growth of multilayer tumors with ECM and MDE interactions

Bifurcation analysis for a free-boundary tumor model with angiogenesis and inhibitor

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

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