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arXiv · 1310.7210

Transitions between patterned states in vegetation models for semi-arid ecosystems

Karna Gowda·Hermann Riecke·Mary Silber
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Abstract

A feature common to many models of vegetation pattern formation in semi-arid ecosystems is a sequence of qualitatively different patterned states, "gaps -> labyrinth -> spots", that occurs as a parameter representing precipitation decreases. We explore the robustness of this "standard" sequence 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. Specifically, we consider a degeneracy of the bifurcation equations that creates a small bubble in parameter space in which stable small-amplitude patterned states may exist near two Turing bifurcations. Pattern transitions between these bifurcation points can then be analyzed in a weakly nonlinear framework. We find that a number of transition scenarios besides the standard sequence are generically possible, which calls into question the reliability of any particular pattern or sequence as a precursor to vegetation collapse. Additionally, we find that clues to the robustness of the standard sequence lie in the nonlinear details of a particular model.

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Karna Gowda, Hermann Riecke, Mary Silber. 2014-01-16. Transitions between patterned states in vegetation models for semi-arid ecosystems. https://doi.org/10.1103/physreve.89.022701

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