Shock-fronted traveling waves in a tumor-invasion model with generalized weak Allee effect

Advection-reaction-diffusion models resulting in singular perturbation problems have been proposed to describe the invasion of tumor cells influenced by hapto- or chemotaxis. The strong Allee effect, where tumor cells have negative growth rates in low densities, was considered an appropriate growth mechanism in the tumor models, since it leads to traveling wave solutions with well-defined edges. Motivated by experimental observations, it is expected that the tumor models can generate biologically relevant invasion fronts even if the growth rates of tumor cells are non-negative. In this work, we consider a specific type of reaction term for tumor growth, named the generalized weak Allee effect. Based on geometric singular perturbation theory (GSPT), canard theory and the blow-up technique, we investigate the influence of the higher-order nonlinearity, that is, the parameter m (m≥2) on the existence and types of traveling waves. It turns out that the generalized weak Allee effect model also admits traveling wave solutions with well-defined edges when m is small, while large m leads to shock-fronted traveling waves with irregular profiles, suggesting that a higher growth threshold may forbid tumor cells to form stable invasion fronts.