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Linear stability for a free boundary tumor model with a periodic supply of external nutrients
Author(s) -
Huang Yaodan,
Zhang Zhengce,
Hu Bei
Publication year - 2019
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.5412
Subject(s) - mathematics , periodic boundary conditions , boundary (topology) , function (biology) , mathematical analysis , stability (learning theory) , boundary value problem , combinatorics , evolutionary biology , machine learning , computer science , biology
In this paper, we consider a free boundary tumor model with a periodic supply of external nutrients, so that the nutrient concentration σ satisfies σ = ϕ ( t ) on the boundary, where ϕ ( t ) is a positive periodic function with period T . A parameter μ in the model is proportional to the “aggressiveness” of the tumor. If 0 < σ ˜ < min 0 ≤ t ≤ T ϕ ( t ) , where σ ˜ is 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 ( σ ∗ ( r , t ), p ∗ ( r , t ), R ∗ ( t )), which is stable for any μ > 0 with respect to all radially symmetric perturbations.[17][Bai M, 2013] We prove that under nonradially symmetric perturbations, there exists a number μ ∗ such that if 0 < μ < μ ∗ , then the T ‐periodic solution is linearly stable, whereas if μ > μ ∗ , then the T ‐periodic solution is linearly unstable.