Rhombic Analysis Extension of a Plant-Surface Water Interaction-Diffusion Model for Hexagonal Pattern Formation in an Arid Flat Environment

Bonni J. Kealy-Dichone¹, David J. Wollkind², Richard A. Cangelosi¹

¹Department of Mathematics, Gonzaga University, Spokane, USA.
²Department of Mathematics, Washington State University, Pullman, USA.

DOI: 10.4236/ajps.2015.68128                

Abstract

An existing weakly nonlinear diffusive instability hexagonal planform analysis for an interaction-diffusion plant-surface water model system in an arid flat environment [11] is extended by performing a rhombic planform analysis as well. In addition a threshold-dependent paradigm that differs from the usually employed implicit zero-threshold methodology is introduced to interpret stable rhombic patterns. The results of that analysis are synthesized with those of the existing hexagonal planform analysis. In particular these synthesized results can be represented by closed-form plots in the rate of precipitation versus the specific rate of plant density loss parameter space. From those plots, regions corresponding to bare ground and vegetative Turing patterns consisting of tiger bush (parallel stripes and labyrinthine mazes), pearled bush (hexagonal gaps and rhombic pseudo-gaps), and homogeneous distributions of vegetation, respectively, may be identified in this parameter space. Then that predicted sequence of stable states along a rainfall gradient is both compared with observational evidence and used to motivate an aridity classification scheme. Finally this system is shown to be isomorphic to the chemical reaction-diffusion Gray-Scott model and that isomorphism is employed to draw some conclusions about sideband instabilities as applied to vegetative patterning.

Keywords

Tiger Bush, Pearled Bush, Nonlinear Stability, Threshold-Dependent Patterns

Conflicts of Interest

The authors declare no conflicts of interest.

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