Pulse Solutions for an Extended Klausmeier Model with Spatially Varying Coefficients

Motivated by its application in ecology, we consider an extended Klausmeier model, a singularly perturbed reaction-advection-diffusion equation with spatially varying coefficients. We rigorously establish existence of stationary pulse solutions by blending techniques from geometric singular perturbation theory with bounds derived from the theory of exponential dichotomies. Moreover, the spectral stability of these solutions is determined, using similar methods. It is found that, due to the break-down of translation invariance, the presence of spatially varying terms can stabilize or destabilize a pulse solution. In particular, this leads to the discovery of a pitchfork bifurcation and existence of stationary multi-pulse solutions.

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

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

[3]  W. A. Coppel.  Dichotomies in Stability Theory , 1978 .

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

[5]  Takashi Teramoto,et al.  Heterogeneity-induced defect bifurcation and pulse dynamics for a three-component reaction-diffusion system. , 2007, Physical review. E, Statistical, nonlinear, and soft matter physics.

[6]  Neil Fenichel.  Geometric singular perturbation theory for ordinary differential equations , 1979 .

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

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

[9]  Michael J. Ward,et al.  Spot dynamics in a reaction-diffusion model of plant root hair initiation , 2017 .

[10]  Robbin Bastiaansen,et al.  The dynamics of disappearing pulses in a singularly perturbed reaction–diffusion system with parameters that vary in time and space , 2018, Physica D: Nonlinear Phenomena.

[11]  Peter van Heijster,et al.  Pinned fronts in heterogeneous media of jump type , 2010 .

[12]  Hans Meinhardt,et al.  Models of biological pattern formation: from elementary steps to the organization of embryonic axes. , 2008, Current topics in developmental biology.

[13]  Michael J. Ward,et al.  Transition to blow-up in a reaction–diffusion model with localized spike solutions† , 2017, European Journal of Applied Mathematics.

[14]  Philip K. Maini,et al.  Applications of mathematical modelling to biological pattern formation , 2001 .

[15]  Arjen Doelman,et al.  Spatially Periodic Multipulse Patterns in a Generalized Klausmeier-Gray-Scott Model , 2017, SIAM J. Appl. Dyn. Syst..

[16]  C. Kuehn.  Multiple Time Scale Dynamics , 2015 .

[17]  Kei-Ichi Ueda,et al.  Dynamics of traveling pulses in heterogeneous media of jump type , 2006 .

[18]  Jack Xin,et al.  Front Propagation in Heterogeneous Media , 2000, SIAM Rev..

[19]  H. Meinhardt,et al.  Biological pattern formation: fmm basic mechanisms ta complex structures , 1994 .

[20]  Alan R. Champneys,et al.  Stripe to Spot Transition in a Plant Root Hair Initiation Model , 2014, SIAM J. Appl. Math..

[21]  Jonathan A. Sherratt,et al.  Using wavelength and slope to infer the historical origin of semiarid vegetation bands , 2015, Proceedings of the National Academy of Sciences.

[22]  Christopher K. R. T. Jones,et al.  Stability of travelling wave solutions of diffusive predator-prey systems , 1991 .

[23]  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..

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

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

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

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

[28]  Peter van Heijster,et al.  A Geometric Approach to Stationary Defect Solutions in One Space Dimension , 2016, SIAM J. Appl. Dyn. Syst..

[29]  Keith Promislow,et al.  Adiabatic stability under semi-strong interactions: The weakly damped regime , 2013, 1301.4466.

[30]  David Dunkerley,et al.  Vegetation Mosaics of Arid Western New South Wales, Australia: Considerations of Their Origin and Persistence , 2014 .

[31]  Frits Veerman,et al.  Pulses in a Gierer-Meinhardt Equation with a Slow Nonlinearity , 2013, SIAM J. Appl. Dyn. Syst..

[32]  Alan R. Champneys,et al.  Mathematical Modeling of Plant Root Hair Initiation: Dynamics of Localized Patches , 2014, SIAM J. Appl. Dyn. Syst..

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

[34]  Alexandre Bouvet,et al.  Multistability of model and real dryland ecosystems through spatial self-organization , 2018, Proceedings of the National Academy of Sciences.

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

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

[37]  Matthias Winter,et al.  Stable spike clusters for the one-dimensional Gierer–Meinhardt system , 2016, European Journal of Applied Mathematics.

[38]  Matthias Winter,et al.  Stable spike clusters for the precursor Gierer–Meinhardt system in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ma , 2017, Calculus of Variations and Partial Differential Equations.

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

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

[41]  Juncheng Wei,et al.  Existence of Symmetric and Asymmetric Spikes for a Crime Hotspot Model , 2013, SIAM J. Math. Anal..

[42]  Keith Promislow,et al.  Nonlinear Asymptotic Stability of the Semistrong Pulse Dynamics in a Regularized Gierer-Meinhardt Model , 2007, SIAM J. Math. Anal..

[43]  Arjen Doelman,et al.  Stable planar vegetation stripe patterns on sloped terrain in dryland ecosystems , 2018, Nonlinearity.

[44]  Michael J. Ward,et al.  Oscillatory instabilities and dynamics of multi-spike patterns for the one-dimensional Gray-Scott model , 2009, European Journal of Applied Mathematics.

[45]  Kei-Ichi Ueda,et al.  Dynamics of traveling pulses in heterogeneous media. , 2007, Chaos.

[46]  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..

[47]  Frits Veerman,et al.  An Explicit Theory for Pulses in Two Component, Singularly Perturbed, Reaction–Diffusion Equations , 2015 .

[48]  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..