Spatially Periodic Multipulse Patterns in a Generalized Klausmeier--Gray--Scott Model

Abstract

Semiarid ecosystems form the stage for a plethora of vegetation patterns, a feature that has been captured in terms of mathematical models since the beginning of this millennium. To study these patterns, we use a reaction-advection-diffusion model that describes the interaction of vegetation and water supply on gentle slopes. As water diffuses much faster than vegetation, this model operates on multiple timescales. While many types of patters are observed in the field, our focus is on two-dimensional stripe patterns. These patterns are typically observed on sloped terrains. In the present idealized setting they correspond to solutions of the model that are spatially periodic in one direction---the $x$-direction---and are extended trivially in the perpendicular $y$-direction. The existence of long wavelength patterns in our model is established analytically using methods from geometric singular perturbation theory, in which a correct parameter scaling is crucial. Subsequently, an Evans function approach yields statements about their stability. Previous work has shown numerically that in order for stripe patterns to be stable, a sufficient slope is needed. The instability of the two-dimensional patterns in parameter regimes corresponding to gentle slopes is confirmed analytically in this article, and we show that the ecological resilience of stripe patterns increases with increasing slope. A full stability proof for steep slopes is, however, beyond the scope of this asymptotic analysis. Since the main destabilization mechanism for the constructed two-dimensional stripes is via perturbations in the transverse direction, we provide a detailed overview of stability results of one-dimensional, spatially periodic patterns and show that they can be stable.