Spatially Periodic Multipulse Patterns in a Generalized Klausmeier-Gray-Scott Model

Semiarid ecosystems form the stage for a plethora of vegetation patterns, a feature that has been captured in terms of mathematical models since the beginning of this millennium. To study these patterns, we use a reaction-advection-diffusion model that describes the interaction of vegetation and water supply on gentle slopes. As water diffuses much faster than vegetation, this model operates on multiple timescales. While many types of patters are observed in the field, our focus is on two-dimensional stripe patterns. These patterns are typically observed on sloped terrains. In the present idealized setting they correspond to solutions of the model that are spatially periodic in one direction---the $x$-direction---and are extended trivially in the perpendicular $y$-direction. The existence of long wavelength patterns in our model is established analytically using methods from geometric singular perturbation theory, in which a correct parameter scaling is crucial. Subsequently, an Evans function approach yi...

[1]  Neil Fenichel.  Persistence and Smoothness of Invariant Manifolds for Flows , 1971 .

[2]  R. Gardner,et al.  Spectral analysis of long wavelength periodic waves and applications. , 1997 .

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

[4]  E. Meron.  Modeling Dryland Landscapes , 2011 .

[5]  Michael J. Ward,et al.  Hopf bifurcation of spike solutions for the shadow Gierer–Meinhardt model , 2003, European Journal of Applied Mathematics.

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

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

[8]  A. Doelman,et al.  Ecosystems off track: rate‐induced critical transitions in ecological models , 2016 .

[9]  R. Gardner,et al.  ON THE STRUCTURE OF THE SPECTRA OF PERIODIC TRAVELING WAVES , 1993 .

[10]  Mary Silber,et al.  Assessing the robustness of spatial pattern sequences in a dryland vegetation model , 2015, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[11]  J. Tzou,et al.  Patterned vegetation, tipping points, and the rate of climate change , 2015, European Journal of Applied Mathematics.

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

[13]  Christopher Jones,et al.  Geometric singular perturbation theory , 1995 .

[14]  J. van de Koppel,et al.  Alternate stable states and threshold effects in semi-arid grazing systems. , 1997 .

[15]  Un Desa.  Transforming our world : The 2030 Agenda for Sustainable Development , 2016 .

[16]  H. Prins,et al.  VEGETATION PATTERN FORMATION IN SEMI-ARID GRAZING SYSTEMS , 2001 .

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

[18]  Juncheng Wei,et al.  The existence and stability of spike equilibria in the one-dimensional Gray-Scott model on a finite domain , 2005, Appl. Math. Lett..

[19]  S. Carpenter,et al.  Early-warning signals for critical transitions , 2009, Nature.

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

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

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

[23]  A. M. Turing,et al.  The chemical basis of morphogenesis , 1952, Philosophical Transactions of the Royal Society of London. Series B, Biological Sciences.

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

[25]  J. Pearson,et al.  Turing instabilities with nearly equal diffusion coefficients , 1989 .

[26]  L. P. White.  Brousse tigree patterns in southern Niger. , 1970 .

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

[28]  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..

[29]  Michael J. Ward,et al.  The stability of spike solutions to the one-dimensional Gierer—Meinhardt model , 2001 .

[30]  M. Rietkerk,et al.  Self-Organized Patchiness and Catastrophic Shifts in Ecosystems , 2004, Science.

[31]  P. Couteron,et al.  Case Study of Self-Organized Vegetation Patterning in Dryland Regions of Central Africa , 2014 .

[32]  J. Alexander,et al.  A topological invariant arising in the stability analysis of travelling waves. , 1990 .

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

[34]  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 .

[35]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[36]  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..

[37]  Pattern formation in the 1-D Gray-Scott model , 1996 .

[38]  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..

[39]  G. Lord,et al.  Nonlinear dynamics and pattern bifurcations in a model for vegetation stripes in semi-arid environments. , 2007, Theoretical population biology.

[40]  W. A. Macfadyen.  Vegetation patterns in the semi-desert plains of British Somaliland. , 1950 .

[41]  Arjen Doelman,et al.  Stationary periodic patterns in the 1D Gray–Scott model , 2000 .

[42]  Paul Nicholas,et al.  Pattern in(formation) , 2012 .

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

[44]  Yuval R. Zelnik,et al.  Gradual regime shifts in fairy circles , 2015, Proceedings of the National Academy of Sciences.

[45]  Rebecca B. Hoyle.  Pattern Formation: An Introduction to Methods , 2006 .

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

[47]  H. Bhadeshia.  Diffusion , 1995, Theory of Transformations in Steels.

[48]  Daniel B. Henry.  Geometric Theory of Semilinear Parabolic Equations , 1989 .

[49]  A. Doelman,et al.  Striped pattern selection by advective reaction-diffusion systems: resilience of banded vegetation on slopes. , 2015, Chaos.

[50]  Arjen Doelman,et al.  Spectra and Stability of Spatially Periodic Pulse Patterns: Evans Function Factorization via Riccati Transformation , 2015, SIAM J. Math. Anal..

[51]  Michael J. Ward,et al.  Explicitly Solvable Nonlocal Eigenvalue Problems and the Stability of Localized Stripes in Reaction‐Diffusion Systems , 2016 .

[52]  P ? ? ? ? ? ? ? % ? ? ? ? , 1991 .

[53]  Jungcheng Wei,et al.  Pattern formations in two-dimensional Gray-Scott model: existence of single-spot solutions and their stability , 2001 .

[54]  Christopher K. R. T. Jones,et al.  Tracking invariant manifolds up to exponentially small errors , 1996 .

[55]  Peter W. Bates,et al.  Invariant Manifolds for Semilinear Partial Differential Equations , 1989 .

[56]  EXISTENCE AND STABILITY OF MULTIPLE SPOT SOLUTIONS FOR THE GRAY-SCOTT MODEL IN R 2 JUNCHENG , 2007 .

[57]  J. Sherratt.  History-dependent patterns of whole ecosystems , 2013 .

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

[59]  E. Meron,et al.  Ecosystem engineers: from pattern formation to habitat creation. , 2004, Physical review letters.

[60]  P. Holmes,et al.  Homoclinic explosions and implosions , 1996, Philosophical Transactions of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.