# Pulse solutions for an extended Klausmeier model with spatially varying coefficients

## TL;DR

This paper analyzes an extended Klausmeier ecological model with spatially varying coefficients, establishing the existence and stability of pulse solutions, and revealing how spatial heterogeneity influences their stability and bifurcations.

## Contribution

It provides a rigorous mathematical framework for existence and spectral stability of pulse solutions in a spatially heterogeneous Klausmeier model, including bifurcation analysis.

## Key findings

- Spatial heterogeneity can stabilize or destabilize pulse solutions.
- Existence of stationary multi-pulse solutions due to bifurcations.
- Identification of a pitchfork bifurcation affecting pulse stability.

## Abstract

Motivated by its application in ecology, we consider an extended Klausmeier model, a singularly perturbed reaction-advection-diffusion equation with spatially varying coefficients. We rigorously establish existence of stationary pulse solutions by blending techniques from geometric singular perturbation theory with bounds derived from the theory of exponential dichotomies. Moreover, the spectral stability of these solutions is determined, using similar methods. It is found that, due to the break-down of translation invariance, the presence of spatially varying terms can stabilize or destabilize a pulse solution. In particular, this leads to the discovery of a pitchfork bifurcation and existence of stationary multi-pulse solutions.