Travelling pulses on three spatial scales in a Klausmeier-type vegetation-autotoxicity model

TL;DR

This paper investigates complex traveling pulse solutions in an advanced vegetation-water-autotoxicity reaction-diffusion model, revealing how different small-scale parameters influence pattern formation and stability.

Contribution

It introduces a new model with biomass carrying capacity, analyzes three types of traveling pulses across different parameter regimes, and combines analytical and numerical methods for validation.

Findings

Existence of three distinct homoclinic traveling pulses.

Influence of small parameters on phase space and pulse structure.

Numerical confirmation of pulse stability and existence.

Abstract

Reaction-diffusion models describing interactions between vegetation and water reveal the emergence of several types of patterns and travelling wave solutions corresponding to structures observed in real-life. Increasing their accuracy by also considering the ecological factor known as autotoxicity has lead to more involved models supporting the existence of complex dynamic patterns. In this work, we include an additional carrying capacity for the biomass in a Klausmeier-type vegetation-water-autotoxicity model, which induces the presence of two asymptotically small parameters: $\varepsilon$, representing the usual scale separation in vegetation-water models, and $\delta$, directly linked to autotoxicity. We construct three separate types of homoclinic travelling pulse solutions based on two different scaling regimes involving $\varepsilon$ and $\delta$, with and without a so-called superslow plateau. The relative ordering of the small parameters significantly influences the phase space geometry underlying the construction of the pulse solutions. We complement the analysis by numerical continuation of the constructed pulse solutions, and demonstrate their existence (and stability) by direct numerical simulation of the full PDE model.