Far-from-Equilibrium Traveling Pulses in Sloped Semiarid Environments Driven by Autotoxicity Effects

In this work, we study the influence of autotoxicity on vegetation patterns, by studying the existence and properties of traveling vegetation pulses. To that end, we consider an extension of the one-dimensional Klausmeier model that accounts for the toxicity compounds. Numerical simulations are first conducted to capture the qualitative behaviors of the pulse--type solutions and, then, geometric singular perturbation theory is used to prove the existence of such traveling pulses by constructing the corresponding homoclinic orbits in the associated four-dimensional system. A scaling analysis on the investigated model is performed to identify the asymptotic scaling regime in which traveling pulses can be constructed. Interestingly, due to the autotoxicity extension, the use of geometric blowup techniques can be avoided, in contrast with previous work [P. Carter and A. Doelman, SIAM J. Appl. Math., 78 (2023), pp. 3213--3237]. Some biological observations are extracted from the analytical results and the role of autotoxicity in traveling patterns is emphasized. Finally, the analytically constructed solutions are compared with the numerical ones, leading to a good agreement that confirms the validity of the conducted analysis. Numerical investigations are also carried out in order to gain additional information on vegetation dynamics.