Share This Article:

Vegetative Rhombic Pattern Formation Driven by Root Suction for an Interaction-Diffusion Plant-Ground Water Model System in an Arid Flat Environment

Abstract Full-Text HTML XML Download Download as PDF (Size:1085KB) PP. 1278-1300
DOI: 10.4236/ajps.2015.68129    2,252 Downloads   2,614 Views   Citations

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.

Conflicts of Interest

The authors declare no conflicts of interest.

Cite this paper

Chaiya, I. , Wollkind, D. , Cangelosi, R. , Kealy-Dichone, B. and Rattanakul, C. (2015) Vegetative Rhombic Pattern Formation Driven by Root Suction for an Interaction-Diffusion Plant-Ground Water Model System in an Arid Flat Environment. American Journal of Plant Sciences, 6, 1278-1300. doi: 10.4236/ajps.2015.68129.

References

[1] von Hardenberg, J., Meron, E., Shachak, M. and Zarmi, Y. (2001) Diversity of Vegetation Patterns and Desertification. Physical Review Letters, 87, Article ID: 198101.
http://dx.doi.org/10.1103/PhysRevLett.87.198101
[2] Rietkerk, M., Dekker, S.C., de Ruiter, P.C. and van de Koppel, J. (2004) Self-Organized Patchiness and Catastrophic Shift In Ecosystems. Science, 305, 1926-1929.
http://dx.doi.org/10.1126/science.1101867
[3] Turing, A.M. (1952) The Chemical Basis of Morphogenesis. Philosophical Transactions of the Royal Society B, 237, 37-72.
http://dx.doi.org/10.1098/rstb.1952.0012
[4] Rietkerk, M., Boerlijst, M.C., van Langevelde, F., HilleRisLambers, R., van de Koppel. J., Kumar, L., Prins, H.H.T. and de Roos, A.M. (2002) Self-Organization of Vegetation in Arid Ecosystems. The American Naturalist, 160, 524-530.
http://dx.doi.org/10.1086/342078
[5] HilleRisLambers, R., Rietkerk, M., van den Bosch, F., Prins, H.T.H. and de Kroon, H. (2001) Vegetation Pattern Formation in Semi-Arid Grazing Systems. Ecology, 82, 50-61.
http://dx.doi.org/10.1890/0012-9658(2001)082[0050:VPFISA]2.0.CO;2
[6] Keller, E.F. and Segel, L.A. (1970) Initiation of Slime Mold Aggregation Viewed as an Instability. Journal of Theoretical Biology, 26, 399-415.
http://dx.doi.org/10.1016/0022-5193(70)90092-5
[7] Stuart, J.T. (1960) On the Nonlinear Mechanics of Wave Disturbances in Stable and Unstable Parallel Flows, Part 1. The Basic Behavior of Plane Poiseuille Flow. Journal of Fluid Mechanics, 9, 353-370.
http://dx.doi.org/10.1017/S002211206000116X
[8] Watson, J. (1960) On the Nonlinear Mechanics of Wave Disturbances in Stable and Unstable Parallel Flows, Part 2. The Development of a Solution for Plane Poiseuille Flow and Plane Couette Flow. Journal of Fluid Mechanics, 9, 371-389.
http://dx.doi.org/10.1017/S0022112060001171
[9] Wollkind, D.J., Manoranjan, V.S. and Zhang, L. (1994) Weakly Nonlinear Stability Analyses of Reaction-Diffusion Model Equations. SIAM Review, 36, 176-214.
http://dx.doi.org/10.1137/1036052
[10] Wollkind, D.J. and Stephenson, L.E. (2000) Chemical Turing Pattern Formation Analyses: Comparison of Theory with Experiment. SIAM Journal of Applied Mathematics, 61, 387-431.
http://dx.doi.org/10.1137/S0036139997326211
[11] Geddes, J.B., Indik, R.A., Moloney, J.V. and Firth, W.J. (1994) Hexagons and Squares in a Passive Nonlinear Optical System. Physical Review A, 50, 3471-3485.
http://dx.doi.org/10.1103/PhysRevA.50.3471
[12] Cross, M.C. and Hohenberg, P.C. (1993) Pattern Formation outside of Equilibrium. Reviews of Modern Physics, 65, 851-1112.
http://dx.doi.org/10.1103/RevModPhys.65.851
[13] Boonkorkuea, N., Lenbury, Y., Alvarado, F.J. and Wollkind, D.J. (2010) Nonlinear Stability Analyses of Vegetative Pattern Formation in an Arid Environment. Journal of Biological Dynamics, 4, 346-380.
http://dx.doi.org/10.1080/17513750903301954
[14] Cangelosi, R.A., Wollkind, D.J., Kealy-Dichone, B.J. and Chaiya, I. (2014) Nonlinear Turing Patterns for a Mussel-Algae Model. Journal of Mathematical Biology, 70, 1249-1294.
http://dx.doi.org/10.1007/s00285-014-0794-7
[15] Segel, L.A. (1965) The Nonlinear Interaction of a Finite Number of Disturbances to a Fluid Layer Heated from Below. Journal of Fluid Mechanics, 21, 359-384.
http://dx.doi.org/10.1017/S002211206500023X
[16] Segel, L.A. and Stuart, J.T. (1962) On the Question of the Preferred Mode in Cellular Thermal Convection. Journal of Fluid Mechanics, 13, 289-306.
http://dx.doi.org/10.1017/S0022112062000683
[17] Kuske, R. and Matkowsky, B.J. (1994) On Roll, Square, and Hexagonal Cellular Flames. European Journal of Applied Mathematics, 5, 65-93.
http://dx.doi.org/10.1017/S0956792500001303
[18] Wollkind, D.J. (2001) Rhombic and Hexagonal Weakly Nonlinear Stability Analyses: Theory and Applications. In: Debnath, L., Ed., Nonlinear Stability Analysis, Volume II, WIT Press, Southampton, 221-272.
[19] Lejeune, O., Tildi, M. and Couteron, P. (2002) Localized Vegetation Patches: A Self-Organized Response to Resource Scarcity. Physical Review E, 66, Article ID: 010901.
http://dx.doi.org/10.1103/PhysRevE.66.010901
[20] Chen, W. and Ward, M.J. (2011) The Stability and Dynamics of Localized Spot Patterns in the Two-Dimensional Gray-Scott Model. SIAM Journal of Dynamical Systems, 10, 586-666.
http://dx.doi.org/10.1137/09077357x
[21] Wang, R.-H., Liu, Q.-X., Sun, G.-Q., Zhen, J. and van de Koppel, J. (2009) Nonlinear Dynamic and Pattern Bifurcation in a Model for Spatial Patterns in Young Mussel Beds. Journal of the Royal Society Interface, 6, 705-718.
http://dx.doi.org/10.1098/rsif.2008.0439
[22] Liu, Q.-X., Doelman, A., Rottschäfer, V., de Jager, M., Herman, P.M.J., Rietkerk, M. and van de Koppel, J. (2013) Phase Separation Explains a New Class of Self-Organized Spatial Patterns in Ecological Systems. Proceedings of the National Academy of Sciences of the United States of America, 110, 11905-11910.
http://dx.doi.org/10.1073/pnas.1222339110
[23] Oulton, D.B. and Wollkind, D.J. (1982) A Three-Dimensional Nonlinear Stability Analysis of the Solidification of a Dilute Binary Alloy. Old Dominion University Research Foundation, Norfolk.
[24] Kealy, B.J. and Wollkind, D.J. (2012) A Nonlinear Stability Analysis of Vegetative Turing Pattern Formation for an Interaction-Diffusion Plant-Surface Water Model System in an Arid Flat Environment. Bulletin of Mathematical Biology, 74, 803-833.
http://dx.doi.org/10.1007/s11538-011-9688-7
[25] Roose, T. and Fowler, A.C. (2004) A Model for Water Uptake by Plant Roots. Journal of Theoretical Biology, 228, 155-171.
http://dx.doi.org/10.1016/j.jtbi.2003.12.012
[26] Lejeune, O., Tildi, M. and Lefever, R. (2004) Vegetation Spots and Stripes in Arid Landscapes. International Journal of Quantum Chemistry, 98, 261-271. Couteron, P., Mahamane, A., Ouedraogo, P. and Seghieri, J. (2000) Differences between Banded Thickets (Tiger Bush) in Two Sites in West Africa. Journal of Vegetation Sciences, 11, 321-328.
[27] Gowda, K., Riecke, H. and Silber, M. (2014) Transitions between Patterned States in Vegetation Models for Semiarid Ecosystems. Physical Review E, 89, Article ID: 022701(1-8).
http://dx.doi.org/10.1103/PhysRevE.89.022701
[28] Couteron, P., Mahamane,A., Ouedraogo, P. and Seghieri, J. (2000) Differences between Banded Thickets (Tiger Bush) in Two Sites in West Africa. Journal of Vegetation Sciences, 11, 321-328.
[29] Meron, E., Gilad, E., von Hardenberg, J., Shachuk, M. and Zarmi, Y. (2004) Vegetation Patterns along a Rainfall Gradient. Chaos, Solitons, and Fractals, 19, 367-376.
http://dx.doi.org/10.1016/S0960-0779(03)00049-3
[30] Stancevic, O., Angstmann, C.N., Murray, J.M. and Henry, B.I. (2013) Turing Patterns from Dynamics of Early HIV Infection. Bulletin of Mathematical Biology, 75, 774-795.
http://dx.doi.org/10.1007/s11538-013-9834-5
[31] Wang, W.-M., Wang, W.-J., Lin, Y.-Z. and Tan, Y.-J. (2011) Pattern Selection in a Predation Model with Self and Cross Diffusion. Chinese Physics B, 20, Article ID: 034702(1-8).
http://dx.doi.org/10.1088/1674-1056/20/3/034702
[32] Graham, M.D., Kevrekidis, J.G., Asakura, K., Lauterbach, J., Krishner, K., Rotermund, H.-H. and Ertl, G. (1994) Effects of Boundaries on Pattern Formation: Catalytic Oxidation of CO on Platinum. Science, 264, 80-82.
http://dx.doi.org/10.1126/science.264.5155.80
[33] Golovin, A.A., Matkowsky, B.J. and Volpert, V.A. (2008) Turing Pattern Formation in the Brusselator Model with Superdiffusion. SIAM Journal of Applied Mathematics, 69, 251-272.
http://dx.doi.org/10.1137/070703454

  
comments powered by Disqus

Copyright © 2018 by authors and Scientific Research Publishing Inc.

Creative Commons License

This work and the related PDF file are licensed under a Creative Commons Attribution 4.0 International License.