Travelling waves in dryland ecology: continuous and discontinuous connections in a hyperbolic vegetation model

This manuscript tackles the study of the emergence of travelling wave solutions in a hyperbolic extension of the well-known Klausmeier model for flat arid environments. In particular, the one-dimensional two-compartment reaction-transport system is taken into account to describe the evolution of biomass and water densities by emphasizing the role of the internal competition among plants, the water uptake ability of plant roots and the inertial effects. To this aim, focus is given to the formation of continuous and discontinuous heteroclinic connections as consequences of the occurrence of gradual or abrupt phenomena that are not directly included in the model under investigation, such as floods, drought periods or huge grazing localized effects. Results conduct to the emergence of both desertification and colonization processes, mainly characterized by the direction of the excited migration speed. Numerical investigations are also addressed to validate the theoretical results and to extract additional information on the environmental response to a change in the ecosystem conditions.