Wavespeed selection and interstitial gap formation in an acid-mediated cancer invasion model

Abstract

We consider a two-component reaction-diffusion system that has previously been developed to model invasion of cells into a resident cell population. The system is an idealised version of models of tumour growth in which tumour cells degrade the surrounding tissue by increasing the acidity of the local environment. By numerically computing families of travelling wave solutions to this problem, we observe that a general initial condition with either compact support, or sufficiently large exponential decay in the far field, tends to the travelling wave solution that has the largest possible decay at its front. Initial conditions with sufficiently slow exponential decay tend to those travelling wave solutions that have the same exponential decay as their initial conditions. We also show that in the limit that the (nondimensional) degradation rate of resident cells is large, the system has similar asymptotic structure as previously observed in perturbed Fisher--KPP models. The asymptotic analysis in this limit explains the formation of an interstitial gap (a region between the invading and receding fronts, in which both cell populations are small), the width of which is logarithmically large in the limit of large degradation rate. These results show that the general mechanism behind the formation of the interstitial gap in reaction-diffusion tumour models is connected to perturbations of the Fisher-KPP system. Biologically, this implies that order of magnitude difference in degradation rate is required to produce appreciably different gap sizes, while the velocity of the invading front is largely determined by the Fisher-KPP velocity, and only very weakly affected by the presence of the interstitial gap.