Mathematical Biosciences and Engineering, 2016, 13(1): 193-207. doi: 10.3934/mbe.2016.13.193

Primary: 35B35, 35B40, 92C50; Secondary: 35A01, 35K55, 35K57.

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Nonlinear stability of a heterogeneous state in a PDE-ODE model for acid-mediated tumor invasion

1. Department of Applied Mathematics, Dong Hua University, Shanghai 200051
2. Departamento de Matemática Aplicada, E.T.S.I. Sistemas Informáticos, Universidad Politécnica de Madrid, 28031 Madrid

This work studies a general reaction-diffusion model foracid-mediated tumor invasion, where tumor cells produce excess acidthat primarily kills healthy cells, and thereby invade the microenvironment. The acid diffuses and could be cleared byvasculature, and the healthy and tumor cells are viewed as twospecies following logistic growth with mutual competition. A keyfeature of this model is the density-limited diffusion for tumorcells, reflecting that a healthy tissue will spatially constrain atumor unless shrunk. Under appropriate assumptions on modelparameters and on initial data, it is shown that the uniqueheterogeneous state is nonlinearly stable, which implies a long-term coexistence of the healthy and tumor cells in certainparameter space. Our theoretical result suggests that acidity mayplay a significant role in heterogeneous tumor progression.
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