Striped pattern selection by advective reaction-diffusion systems: resilience of banded vegetation on slopes.

For water-limited arid ecosystems, where water distribution and infiltration play a vital role, various models have been set up to explain vegetation patterning. On sloped terrains, vegetation aligned in bands has been observed ubiquitously. In this paper, we consider the appearance, stability, and bifurcations of 2D striped or banded patterns in an arid ecosystem model. We numerically show that the resilience of the vegetation bands is larger on steeper slopes by computing the stability regions (Busse balloons) of striped patterns with respect to 1D and transverse 2D perturbations. This is corroborated by numerical simulations with a slowly decreasing water input parameter. Here, long wavelength striped patterns are unstable against transverse perturbations, which we also rigorously prove on flat ground through an Evans function approach. In addition, we prove a "Squire theorem" for a class of two-component reaction-advection-diffusion systems that includes our model, showing that the onset of pattern formation in 2D is due to 1D instabilities in the direction of advection, which naturally leads to striped patterns.

[1]  E. Meron,et al.  Reversing desertification as a spatial resonance problem. , 2015, Physical review. E, Statistical, nonlinear, and soft matter physics.

[2]  Jonathan A. Sherratt,et al.  An Analysis of Vegetation Stripe Formation in Semi-Arid Landscapes , 2005, Journal of mathematical biology.

[3]  Juncheng Wei,et al.  The Stability of a Stripe for the Gierer-Meinhardt Model and the Effect of Saturation , 2006, SIAM J. Appl. Dyn. Syst..

[4]  H. Squire.  On the Stability for Three-Dimensional Disturbances of Viscous Fluid Flow between Parallel Walls , 1933 .

[5]  D. G. Míguez,et al.  Transverse instabilities in chemical Turing patterns of stripes. , 2003, Physical review. E, Statistical, nonlinear, and soft matter physics.

[6]  M. Menzinger,et al.  Self-organization induced by the differential flow of activator and inhibitor. , 1993, Physical review letters.

[7]  W. Chen,et al.  The Stability and Dynamics of Localized Spot Patterns in the Two-Dimensional Gray-Scott Model , 2010, SIAM J. Appl. Dyn. Syst..

[8]  L. Gunderson.  Ecological Resilience—In Theory and Application , 2000 .

[9]  Wentao Sun,et al.  The Slow Dynamics of Two-Spike Solutions for the Gray-Scott and Gierer-Meinhardt Systems: Competition and Oscillatory Instabilities , 2005, SIAM J. Appl. Dyn. Syst..

[10]  B. Sandstede,et al.  Chapter 18 - Stability of Travelling Waves , 2002 .

[11]  Rovinsky,et al.  Chemical instability induced by a differential flow. , 1992, Physical review letters.

[12]  Arjen Doelman,et al.  Pattern formation in the one-dimensional Gray - Scott model , 1997 .

[13]  Philip K. Maini,et al.  Phase differences in reaction-diffusion-advection systems and applications to morphogenesis , 1995 .

[14]  Geertje Hek,et al.  Rise and Fall of Periodic Patterns for a Generalized Klausmeier–Gray–Scott Model , 2013, J. Nonlinear Sci..

[15]  Luca Ridolfi,et al.  Mathematical models of vegetation pattern formation in ecohydrology , 2009 .

[16]  Arjen Doelman,et al.  Semistrong Pulse Interactions in a Class of Coupled Reaction-Diffusion Equations , 2003, SIAM J. Appl. Dyn. Syst..

[17]  Arjen Doelman,et al.  Hopf dances near the tips of Busse balloons , 2011 .

[18]  E. Meron,et al.  Diversity of vegetation patterns and desertification. , 2001, Physical review letters.

[19]  Robert Gardner,et al.  A stability index analysis of 1-D patterns of the Gray-Scott model , 2002 .

[20]  J. Chomaz,et al.  Extended Squire’s transformation and its consequences for transient growth in a confined shear flow , 2014, Journal of Fluid Mechanics.

[21]  L. Kumar,et al.  Self‐Organization of Vegetation in Arid Ecosystems , 2002, The American Naturalist.

[22]  S. Scott,et al.  The development of spatial structure in an ionic chemical system induced by applied electric fields , 2000 .

[23]  F. Busse,et al.  Non-linear properties of thermal convection , 1978 .

[24]  Arjen Doelman,et al.  Stability of spatially periodic pulse patterns in a class of singularly perturbed reaction-diffusion equations. , 2005 .

[25]  M. Silber,et al.  Transitions between patterned states in vegetation models for semiarid ecosystems. , 2013, Physical review. E, Statistical, nonlinear, and soft matter physics.

[26]  A. P. Muñuzuri,et al.  Effect of electric field on Turing patterns in a microemulsion , 2012 .

[27]  Martin Golubitsky,et al.  Symmetries and pattern selection in Rayleigh-Bénard convection , 1984 .

[28]  Jonathan A. Sherratt,et al.  Pattern Solutions of the Klausmeier Model for Banded Vegetation in Semiarid Environments V: The Transition from Patterns to Desert , 2013, SIAM J. Appl. Math..

[29]  C. S. Holling.  Resilience and Stability of Ecological Systems , 1973 .

[30]  Zhen Jin,et al.  Nonlinear dynamic and pattern bifurcations in a model for spatial patterns in young mussel beds , 2008, Journal of The Royal Society Interface.

[31]  Jens D. M. Rademacher,et al.  pde 2 path – a Matlab package for continuation and bifurcation in 2 D elliptic systems , 2013 .

[32]  Jean Poesen,et al.  Soil and water components of banded vegetation patterns , 1999 .

[33]  Su Zhao,et al.  Operator splitting implicit integration factor methods for stiff reaction-diffusion-advection systems , 2011, J. Comput. Phys..

[34]  Arjen Doelman,et al.  Slowly Modulated Two-Pulse Solutions in the Gray--Scott Model II: Geometric Theory, Bifurcations, and Splitting Dynamics , 2001, SIAM J. Appl. Math..

[35]  Hezi Yizhaq,et al.  Banded vegetation: biological productivity and resilience , 2005 .

[36]  J. Bogaert,et al.  Determinants and dynamics of banded vegetation pattern migration in arid climates , 2012 .

[37]  Nicolas Barbier,et al.  Environmental modulation of self‐organized periodic vegetation patterns in Sudan , 2011 .

[38]  Lateral instabilities of cubic autocatalytic reaction fronts in a constant electric field , 1999 .

[39]  Maarten B. Eppinga,et al.  Beyond Turing: The response of patterned ecosystems to environmental change , 2014 .

[40]  J. E. Pearson.  Complex Patterns in a Simple System , 1993, Science.

[41]  Arjen Doelman,et al.  Homoclinic Stripe Patterns , 2002, SIAM J. Appl. Dyn. Syst..

[42]  Greg Hancock,et al.  Eco-geomorphology of banded vegetation patterns in arid and semi-arid regions , 2006 .

[43]  D. Horváth,et al.  Lateral instability controlled by constant electric field in an acid-catalyzed reaction , 2004 .

[44]  Michael J. Ward,et al.  Zigzag and Breakup Instabilities of Stripes and Rings in the Two-Dimensional Gray–Scott Model , 2006 .

[45]  Björn Sandstede,et al.  Computing absolute and essential spectra using continuation , 2007 .

[46]  Stephen K. Scott,et al.  Autocatalytic reactions in the isothermal, continuous stirred tank reactor: Oscillations and instabilities in the system A + 2B → 3B; B → C , 1984 .

[47]  M. Meyries,et al.  Quasi-Linear Parabolic Reaction-Diffusion Systems: A User's Guide to Well-Posedness, Spectra, and Stability of Travelling Waves , 2013, SIAM J. Appl. Dyn. Syst..

[48]  S. Scott,et al.  Spatio-temporal structures in a differential flow reactor with cubic autocatalator kinetics , 1998 .

[49]  C. Klausmeier,et al.  Regular and irregular patterns in semiarid vegetation , 1999, Science.

[50]  Robert Gardner,et al.  Stability analysis of singular patterns in the 1-D Gray-Scott model I: a matched asymptotics approach , 1998 .

[51]  Arjen Doelman,et al.  Slowly Modulated Two-Pulse Solutions in the Gray--Scott Model I: Asymptotic Construction and Stability , 2000, SIAM J. Appl. Math..

[52]  Michael J. Ward,et al.  The existence and stability of spike equilibria in the one-dimensional Gray-Scott model: the pulse-splitting regime , 2005 .

[53]  Daniel Wetzel,et al.  pde2path - A Matlab package for continuation and bifurcation in 2D elliptic systems , 2012, 1208.3112.

[54]  Robert Gardner,et al.  Large stable pulse solutions in reaction-diffusion equations , 2001 .