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 planfor...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.展开更多
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 ...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.展开更多
文摘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.
文摘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.
