Travelling wave solutions to a microtube-driven glioma invasion model

In this article, we establish the existence of travelling wave solutions for a non-cooperative reaction-diffusion model representing glioma cell invasion. The model describes the microtube-driven migration of glioma consisting of an ODE equation describing the dynamics of the tumour bulk and a reaction-diffusion equation for the tumour microtubes. We derive an explicit formula for the minimum wave speed $\overline{c}$ based on system parameters such that travelling waves exist for speeds $c\ge \overline{c}$ while no travelling wave solution exists for $c<\overline{c}$. We prove the existence of travelling wave solutions by constructing upper and lower solutions and employing Schauder's fixed point theorem. We obtain non-existence for small speeds by use of the negative one-sided Laplace transform. Our result is one of the few complete results on travelling waves of a non-cooperative partially degenerate reaction-diffusion systems. The findings have implications for understanding glioma spread dynamics and potential modelling applications in predicting tumour progression based on cellular migration speeds.