Stability analysis of traveling wave fronts in a model for tumor growth

Abstract

In this paper, we study the orbital stability of traveling wave solutions to the Gallay and Mascia (GM) reduction of the Gatenby–Gawlinski model. The heteroclinic solutions provided by Gallay and Mascia represent the propagation of a tumor front into healthy tissue. Orbital stability is crucial to investigating models as it determines which solutions are likely to be observed in practice. Through constructing the unstable manifold to connect fixed states of the GM model and applying a shooting argument, we constructed front solutions. After numerically generating front solutions, we studied stability by constructing the spectrum for various parameters of the GM model. We see no evidence of point eigenvalues in the right half-plane, leaving the essential spectrum as the only possible source of instability. These findings show that Gallay and Mascia’s derived heteroclinic solutions are likely to be observed physically in biological systems and are stable for various tumor growth speeds.