Abstract
A rhombic planform nonlinear cross-diffusive instability analysis is applied to a particular interaction-diffusion plant-ground water model system in an arid flat environment. This model contains a plant root suction effect as a cross-diffusion term in the ground water equation. In addition a threshold-dependent paradigm that differs from the usually employed implicit zero-threshold methodology is introduced to interpret stable rhombic patterns. These patterns are driven by root suction since the plant equation does not yield the required positive feedback necessary for the generation of standard Turing-type self-diffusive instabilities. The results of that analysis can be represented by plots in a root suction coefficient versus rainfall rate dimensionless parameter space. From those plots regions corresponding to bare ground and vegetative patterns consisting of isolated patches, rhombic arrays of pseudo spots or gaps separated by an intermediate rectangular state, and homogeneous distributions from low to high density may be identified in this parameter space. Then, a morphological sequence of stable vegetative states is produced upon traversing an experimentally-determined root suction characteristic curve as a function of rainfall through these regions. Finally, that predicted sequence along a rainfall gradient is compared with observational evidence relevant to the occurrence of leopard bush, pearled bush, or labyrinthine tiger bush vegetative patterns, used to motivate an aridity classification scheme, and placed in the context of some recent biological nonlinear pattern formation studies.
Highlights
Introduction vonHardenberg et al [1] devised a plant-ground water interaction-diffusion system to model self-organized vegetative pattern formation in arid environments
That system basically was formed from two existing models by coupling a simplified version of the interaction terms of one of those systems with the exact diffusion terms of the other. These terms included a crossdiffusion effect in the ground water equation due to plant root suction and a nonautocatalytic effect in the plant equation that precluded the occurrence of Turing self-diffusive instabilities
Our main result could be plotted in a coefficient of root suction versus a rate of rainfall parameter space
Summary
Since quadratics of the form of (2.22) have roots with negative real parts provided their coefficients are positive, we can conclude that the community equilibrium point is stable to linear homogeneous perturbations for which q2 = 0. ( ) For fixed α , γ , ∆ , and μ , the= curve β β0 q2 ;α ,γ , ∆, μ in the first quadrant of the q2 − β plane is ( ) marginal since it serves as a boundary between the linearly stable region where 0 ≤ β < β0 q2 ;α ,γ , ∆, μ and ( ) the unstable region of (2.23) This marginal stability curve has a minimum point at qc , βc given by ( ) qc2 = ∆ μ α , βc = 2 μ ∆ 1 α + (γμ ∆)(1 α ) + μ. (2 3) mm d < R ≤ 3 m= m d , DN 0.1 m= 2 d , DW 10 m2 d
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