Rise and Fall of Periodic Patterns for a Generalized Klausmeier–Gray–Scott Model

@article{Stelt2013RiseAF,
  title={Rise and Fall of Periodic Patterns for a Generalized Klausmeier–Gray–Scott Model},
  author={Sjors van der Stelt and Arjen Doelman and Geertje Hek and Jens D. M. Rademacher},
  journal={Journal of Nonlinear Science},
  year={2013},
  volume={23},
  pages={39-95}
}
In this paper we introduce a conceptual model for vegetation patterns that generalizes the Klausmeier model for semi-arid ecosystems on a sloped terrain (Klausmeier in Science 284:1826–1828, 1999). Our model not only incorporates downhill flow, but also linear or nonlinear diffusion for the water component.To relate the model to observations and simulations in ecology, we first consider the onset of pattern formation through a Turing or a Turing–Hopf bifurcation. We perform a Ginzburg–Landau… 
Localised pattern formation in a model for dryland vegetation
TLDR
The von Hardenberg model displays a sequence of (often hysteretic) transitions from a non-vegetated state, to localised patches of vegetation that exist with uniform low-level vegetation, to periodic patterns, to higher-level uniform vegetation as the precipitation parameter increases.
Spatially Periodic Multipulse Patterns in a Generalized Klausmeier-Gray-Scott Model
TLDR
A reaction-advection-diffusion model is used that describes the interaction of vegetation and water supply on gentle slopes and establishes the existence of long wavelength patterns in this model, which are typically observed on sloped terrains.
Spatial patterns in a vegetation model with internal competition and feedback regulation
The vegetation patterns are a characteristic particularity of semiarid zones, which can be the future of modern ecology for its importance. This paper aims to study a diffusive vegetation model for
Transitions between patterned states in vegetation models for semiarid ecosystems.
TLDR
The robustness of this "standard" sequence of qualitatively different patterned states, "gaps → labyrinth → spots," is explored in the generic setting of a bifurcation problem on a hexagonal lattice, as well as in a particular reaction-diffusion model for vegetation pattern formation.
Striped pattern selection by advective reaction-diffusion systems: resilience of banded vegetation on slopes.
TLDR
It is numerically show that the resilience of the vegetation bands is larger on steeper slopes by computing the stability regions (Busse balloons) of striped patterns with respect to 1D and transverse 2D perturbations, and proves a "Squire theorem" for a class of two-component reaction-advection-diffusion systems that includes this model.
Analysis of a model for banded vegetation patterns in semi-arid environments with nonlocal dispersal
TLDR
This paper replaces the diffusion term describing plant dispersal by a more realistic nonlocal convolution integral to account for the possibility of long-range dispersal of seeds, and numerically extends the results to other dispersal kernels, showing that the tendency to form patterns depends on the type of decay of the kernel.
Dynamics of patchy vegetation patterns in the two-dimensional generalized Klausmeier model
  • Tony Wong, M. Ward
  • Mathematics
    Discrete & Continuous Dynamical Systems - S
  • 2022
We study the dynamical and steady-state behavior of self-organized spatially localized patches or "spots" of vegetation for the Klausmeier reaction-diffusion (RD) system of spatial ecology that
Stable planar vegetation stripe patterns on sloped terrain in dryland ecosystems
In water-limited regions, competition for water resources results in the formation of vegetation patterns; on sloped terrain, one finds that the vegetation typically aligns in stripes or arcs. We
...
1
2
3
4
5
...

References

SHOWING 1-10 OF 158 REFERENCES
Pattern solutions of the Klausmeier Model for banded vegetation in semi-arid environments I
In many semi-arid environments, vegetation cover is sparse, and is self-organized into large-scale spatial patterns. In particular, banded vegetation is typical on hillsides. Mathematical modelling
A Nonlinear Stability Analysis of Vegetative Turing Pattern Formation for an Interaction–Diffusion Plant-Surface Water Model System in an Arid Flat Environment
TLDR
A weakly nonlinear diffusive instability analysis is applied to the appropriate model system for the development of spontaneous stationary vegetative patterns in an arid flat environment and theoretical predictions are compared with relevant observational evidence and existing numerical simulations of similar model systems.
Hopf dances near the tips of Busse balloons
TLDR
A novel generic destabilization mechanism for (reversible) spatially periodic patterns in reaction-diffusion equations in one spatial dimension that occurs for long wavelength patterns near the homoclinic tip of the associated Busse balloon.
Stationary periodic patterns in the 1D Gray–Scott model
In this work, we study the existence and stability of a family of stationary periodic patterns in the ID Gray-Scott model. First, it is shown that these periodic solutions are born at a critical
A Putative Mechanism for Bog Patterning
TLDR
This work reports on regular “maze patterns” on flat ground, consisting of bands densely vegetated by vascular plants in a more sparsely vegetated matrix of nonvascular plant communities, and proposes that the patterns are self‐organized and originate from a nutrient accumulation mechanism.
Mathematical models of vegetation pattern formation in ecohydrology
Highly organized vegetation patterns can be found in a number of landscapes around the world. In recent years, several authors have investigated the processes underlying vegetation pattern formation.
...
1
2
3
4
5
...