Global stability of a PDE-ODE model for acid-mediated tumor invasion

Abstract

In this paper, we study the global dynamics of a general reaction-diffusion model based on acid-mediated invasion hypothesis, which is a candidate explanation for the Warburg effect. Our theoretical results characterize the effects of acid resistance and mutual competition of healthy cells and tumor cells on tumor progression in the long term, i.e., whether the healthy cells and tumor cells coexist or the tumor cells prevail after tumor invasion. A key feature of this model is the density-limited tumor diffusion term for tumor cells, which might give rise to the degeneracy of the parabolic equation. To overcome this difficulty, we combine the construction of suitable Lyapunov functionals and upper/lower solutions. This paper continues and improves the work begun in [15].