Eckhaus instability of stationary patterns in hyperbolic reaction-diffusion models on large finite domains. (English) Zbl 1498.35033
Summary: We have theoretically investigated the phenomenon of Eckhaus instability of stationary patterns arising in hyperbolic reaction-diffusion models on large finite domains, in both supercritical and subcritical regime. Adopting multiple-scale weakly-nonlinear analysis, we have deduced the cubic and cubic-quintic real Ginzburg-Landau equations ruling the evolution of pattern amplitude close to criticality. Starting from these envelope equations, we have provided the explicit expressions of the most relevant dynamical features characterizing primary and secondary quantized branches of any order: stationary amplitude, existence and stability thresholds and linear growth rate. Particular emphasis is given on the subcritical regime, where cubic and cubic-quintic Ginzburg-Landau equations predict qualitatively different dynamical pictures. As an illustrative example, we have compared the above-mentioned analytical predictions to numerical simulations carried out on the hyperbolic modified Klausmeier model, a conceptual tool used to describe the generation of stationary vegetation stripes over flat arid environments. Our analysis has also allowed to elucidate the role played by inertia during the transient regime, where an unstable patterned state evolves towards a more favorable stable configuration through sequences of phase-slips. In particular, we have inspected the functional dependence of time and location at which wavelength adjustment takes place as well as the possibility to control these quantities, independently of each other.
MSC:
| 35B32 | Bifurcations in context of PDEs |
| 35B36 | Pattern formations in context of PDEs |
| 35L51 | Second-order hyperbolic systems |
| 35L60 | First-order nonlinear hyperbolic equations |
| 35Q56 | Ginzburg-Landau equations |
| 35Q92 | PDEs in connection with biology, chemistry and other natural sciences |
Keywords:
hyperbolic reaction-diffusion models; inertial effects; pattern dynamics; Ginzburg-Landau equation; Eckhaus instability; phase slips; hyperbolic modified Klausmeier modelReferences:
| [1] | Murray, JD, Mathematical Biology: I (2002), New York: An Introduction. Springer-Verlag, New York · Zbl 1006.92001 · doi:10.1007/b98868 |
| [2] | Murray, JD, Mathematical Biology II: Spatial Models and Biomedical Applications (2003), Berlin: Springer, Berlin · Zbl 1006.92002 · doi:10.1007/b98869 |
| [3] | Eckhaus, W.; Iooss, G., Strong selection or rejection of spatially periodic patterns in degenerate bifurcations, Physica D, 39, 124-146 (1989) · Zbl 0677.76046 · doi:10.1016/0167-2789(89)90043-2 |
| [4] | Tuckermann, LS; Barkley, D., Bifurcation analysis of the Eckhaus instability, Physica D, 46, 57-86 (1990) · Zbl 0721.35008 · doi:10.1016/0167-2789(90)90113-4 |
| [5] | Eckmann, JP; Gallay, T.; Wayne, CE, Phase slips and the Eckhaus instability, Nonlinearity, 8, 943-961 (1995) · Zbl 0846.35032 · doi:10.1088/0951-7715/8/6/004 |
| [6] | Hoyle, R., Pattern Formation (2007), New York: An Introduction to Methods. Cambridge University Press, New York |
| [7] | Knobloch, E.; Krechetnikov, R., Stability on time-dependent domains, J. Nonlinear Sci., 24, 493-523 (2014) · Zbl 1302.70057 · doi:10.1007/s00332-014-9197-6 |
| [8] | Doelman, A.: Chapter 4: Pattern formation in reaction-diffusion systems-an explicit approach. In: Peletier, M.A., van Santen, R.A., Siteur, E. (eds.) Complexity Science. An Introduction, pp. 129-182. World Scientific, Singapore (2018) |
| [9] | Mielke, A.; Schneider, G.; Deift, P.; Levermore, CD; Wayne, CE, Derivation and justification of the complex Ginzburg-Landau equation as a modulation equation, Lecture in Applied Mathematics, 191-216 (1994), Providence: American Mathematical Society, Providence · Zbl 0856.35060 |
| [10] | Van Saarloos, W.; Hohenberg, PC, Fronts, pulses, sources and sinks in generalized complex Ginzburg-Landau equations, Physica D, 56, 303-367 (1992) · Zbl 0763.35088 · doi:10.1016/0167-2789(92)90175-M |
| [11] | Aranson, IS; Kramer, L., The world of the complex Ginzburg-Landau equation, Rev. Mod. Phys., 74, 99-143 (2002) · Zbl 1205.35299 · doi:10.1103/RevModPhys.74.99 |
| [12] | Mielke, A.; Fiedler, B., The Ginzburg-Landau equation in its role as a modulation equation, Handbook of Dynamical Systems, 759-834 (2002), Amsterdam: Elsevier Science B.V, Amsterdam · Zbl 1041.37037 |
| [13] | Doelman, A.; Eckhaus, W., Periodic and quasi-periodic solutions of degenerate modulation equations, Physica D, 53, 249-266 (1991) · Zbl 0744.35058 · doi:10.1016/0167-2789(91)90065-H |
| [14] | Brand, HR; Deissler, RJ, Eckhaus and Benjamin-Feir instabilities near a weakly inverted bifurcation, Phys. Rev. A, 45, 3732-3736 (1992) · doi:10.1103/PhysRevA.45.3732 |
| [15] | Dawes, JHP, Modulated and localized states in a finite domain, SIAM J. Appl. Dyn. Syst., 8, 909-930 (2009) · Zbl 1167.76016 · doi:10.1137/080724344 |
| [16] | Kao, HC; Knobloch, E., Weakly subcritical stationary patterns: Eckhaus instability and homoclinic snaking, Phys. Rev. E, 85 (2012) · doi:10.1103/PhysRevE.85.026211 |
| [17] | Morgan, D.; Dawes, JHP, The Swift-Hohenberg equation with a nonlocal nonlinearity, Physica D, 270, 60-80 (2014) · Zbl 1288.35067 · doi:10.1016/j.physd.2013.11.018 |
| [18] | Kao, HC; Knobloch, E., Instabilities and dynamics of weakly subcritical patterns, Math. Model. Nat. Phenom., 8, 131-154 (2013) · Zbl 1294.35151 · doi:10.1051/mmnp/20138509 |
| [19] | Consolo, G.; Curró, C.; Valenti, G., Supercritical and subcritical Turing pattern formation in a hyperbolic vegetation model for flat arid environments, Physica D, 398, 141-163 (2019) · Zbl 1453.37082 · doi:10.1016/j.physd.2019.03.006 |
| [20] | Consolo, G.; Curró, C.; Valenti, G., Turing vegetation patterns in a generalized hyperbolic Klausmeier model, Math. Meth. Appl. Sci., 43, 10474 (2020) · Zbl 1467.35218 · doi:10.1002/mma.6518 |
| [21] | Consolo, G.; Curró, C.; Valenti, G., Pattern formation and modulation in a hyperbolic vegetation model for semiarid environments, Appl. Math. Modell., 43, 372-392 (2017) · Zbl 1446.92011 · doi:10.1016/j.apm.2016.11.031 |
| [22] | Horsthemke, W., Spatial instabilities in reaction random walks with direction-independent kinetics, Phys. Rev. E, 60, 2651-2663 (1999) · doi:10.1103/PhysRevE.60.2651 |
| [23] | Al-Ghoul, M.; Eu, BC, Hyperbolic reaction-diffusion equations and irreversible thermodynamics: cubic reversible reaction model, Physica D, 90, 119-153 (1996) · Zbl 0883.92031 · doi:10.1016/0167-2789(95)00231-6 |
| [24] | Zemskov, EP; Horsthemke, W., Diffusive instabilities in hyperbolic reaction-diffusion equations, Phys. Rev. E, 93 (2016) · doi:10.1103/PhysRevE.93.032211 |
| [25] | Curro’, C.; Valenti, G., Pattern formation in hyperbolic models with cross-diffusion: theory and applications, Physica D, 418 (2021) · Zbl 1484.35038 · doi:10.1016/j.physd.2021.132846 |
| [26] | Buono, PL; Eftimie, R., Symmetries and pattern formation in hyperbolic versus parabolic models of self-organised aggregation, J. Math. Biol., 71, 847-881 (2015) · Zbl 1329.35052 · doi:10.1007/s00285-014-0842-3 |
| [27] | Consolo, G.; Curró, C.; Grifó, G.; Valenti, G., Oscillatory periodic pattern dynamics in hyperbolic reaction-advection-diffusion models, Phys. Rev. E, 105 (2022) · doi:10.1103/PhysRevE.105.034206 |
| [28] | Klausmeier, CA, Regular and irregular patterns in semiarid vegetation, Science, 284, 1826-1828 (1999) · doi:10.1126/science.284.5421.1826 |
| [29] | Zelnik, Y.; Kinast, S.; Yizhaq, H.; Bel, G.; Meron, E., Regime shifts in models of dryland vegetation, Phil. Trans. R. Soc. A, 371, 20120358 (2013) · Zbl 1353.86006 · doi:10.1098/rsta.2012.0358 |
| [30] | Kealy, BJ; Wollkind, DJ, A nonlinear stability analysis of vegetative turing pattern formation for an interaction-diffusion plant-surface water model system in an arid flat environment, Bull. Math. Biol., 74, 803-833 (2012) · Zbl 1235.92009 · doi:10.1007/s11538-011-9688-7 |
| [31] | Sun, GQ; Li, L.; Zhang, ZK, Spatial dynamics of a vegetation model in an arid flat environment, Nonlinear Dyn., 73, 2207-2219 (2013) · doi:10.1007/s11071-013-0935-3 |
| [32] | Ruggeri, T.; Sugiyama, M., Classical and Relativistic Rational Extended Thermodynamics of Gases (2021), Cham: Springer, Cham · Zbl 1461.82002 · doi:10.1007/978-3-030-59144-1 |
| [33] | Barbera, E.; Consolo, G.; Valenti, G., Spread of infectious diseases in a hyperbolic reaction-diffusion susceptible-infected-removed model, Phys. Rev. E, 88 (2013) · doi:10.1103/PhysRevE.88.052719 |
| [34] | Barbera, E.; Curró, C.; Valenti, G., A hyperbolic model for the effects of urbanization on air pollution, Appl. Math. Modell., 34, 2192-2202 (2010) · Zbl 1193.76027 · doi:10.1016/j.apm.2009.10.030 |
| [35] | Ai-Ghoul, M.; Eu, BC, Hyperbolic reaction-diffusion equations and irreversible thermodynamics: cubic reversible reaction model, Physica D, 90, 119-153 (1996) · Zbl 0883.92031 · doi:10.1016/0167-2789(95)00231-6 |
| [36] | Gambino, G.; Lombardo, MC; Lupo, S.; Sammartino, M., Super-critical and sub-critical bifurcations in a reaction-diffusion Schnakenberg model with linear cross-diffusion, Ricer Mate, 65, 449-467 (2016) · Zbl 1361.35086 · doi:10.1007/s11587-016-0267-y |
| [37] | van der Stelt, S.; Doelman, A.; Hek, G.; Rademacher, JDM, Rise and fall of periodic patterns for a generalized Klausmeier-Gray-Scott model, J. Nonlinear Sci., 23, 39-95 (2013) · Zbl 1270.35093 · doi:10.1007/s00332-012-9139-0 |
| [38] | Rameshwar, Y.; Anuradha, V.; Srinivas, G.; Perez, LM; Laroze, D.; Pleiner, H., Nonlinear convection of binary liquids in a porous medium, Chaos, 28 (2018) · doi:10.1063/1.5027468 |
| [39] | Mohammed, WW, Modulation equation for the stochastic Swift-Hohenberg equation with cubic and quintic nonlinearities on the real line, Mathematics, 7, 1217 (2019) · doi:10.3390/math7121217 |
| [40] | Bilotta, E.; Gargano, F.; Giunta, V.; Lombardo, MC; Pantano, P.; Sammartino, M., Eckhaus and zigzag instability in a chemotaxis model of multiple sclerosis, Atti dell’Accad. Peloritana Pericolanti, 93, S3, A9 (2018) |
| [41] | Krechetnikov, R.; Knobloch, E., Stability on time-dependent domains: convective and dilution effects, Physica D, 342, 16-23 (2017) · Zbl 1378.35030 · doi:10.1016/j.physd.2016.10.003 |
| [42] | Stich, M.; Mikhailov, AS, Complex pacemakers and wave sinks in heterogeneous oscillatory chemical systems, Z. Phys. Chem., 216, 521-533 (2002) · doi:10.1524/zpch.2002.216.4.521 |
| [43] | Granzow, GD; Riecke, H., Double phase slips and spatiotemporal chaos in a model for parametrically excited standing waves, SIAM J. Appl. Math., 59, 900-919 (1999) · Zbl 0937.35008 · doi:10.1137/S0036139996313861 |
| [44] | Sherratt, JA, Pattern solutions of the Klausmeier model for banded vegetation in semi-arid environments III: The transition between homoclinic solutions, Physica D, 242, 30-41 (2013) · Zbl 1261.35144 · doi:10.1016/j.physd.2012.08.014 |
| [45] | Bennett, JJR; Sherratt, JA, Long-distance seed dispersal affects the resilience of banded vegetation patterns in semi-deserts, J. Theor. Biol., 481, 151-161 (2019) · Zbl 1422.92183 · doi:10.1016/j.jtbi.2018.10.002 |
| [46] | Milchunas, DG; Lauenroth, WK, Inertia in plant community structure: state changes after cessation of nutrient-enrichment stress, Ecol. Appl., 5, 452-458 (1995) · doi:10.2307/1942035 |
| [47] | Valentin, C.; d’Herbes, JM, Niger tiger bush as a natural water harvesting system, Catena, 37, 231-256 (1999) · doi:10.1016/S0341-8162(98)00061-7 |
| [48] | Garcia-Fayos, P.; Gasque, M., Consequences of a severe drought on spatial patterns of woody plants in a two-phase mosaic steppe of Stipa tenacissima L, J. Arid Environ., 52, 199-208 (2002) · doi:10.1006/jare.2002.0987 |
| [49] | Deblauwe, V.; Couteron, P.; Lejeune, O.; Bogaert, J.; Barbier, N., Environmental modulation of self-organized periodic vegetation patterns in Sudan, Ecography, 34, 990-1001 (2011) · doi:10.1111/j.1600-0587.2010.06694.x |
| [50] | Deblauwe, V.; Couteron, P.; Bogaert, J.; Barbier, N., Determinants and dynamics of banded vegetation pattern migration in arid climates, Ecol. Monogr., 82, 3-21 (2012) · doi:10.1890/11-0362.1 |
| [51] | Brown, J.; Whitham, T.; Morgan, E.; Gehring, C., Complex species interactions and the dynamics of ecological systems: long-term experiments, Science, 293, 643-650 (2001) · doi:10.1126/science.293.5530.643 |
| [52] | Hastings, A., Transients: the key to long-term ecological understanding?, Trends Ecol. Evol., 19, 39-45 (2004) · doi:10.1016/j.tree.2003.09.007 |
| [53] | Hastings, A., Abbott, K.C., Cuddington, K., Francis, T., Gellner, G., Lai, Y.C., Morozov, A., Petrovskii, S., Scranton, K., Zeeman, M.L.: Transient phenomena in ecology. Science 361, eaat6412 (2018) |
| [54] | Sherratt, JA, Pattern solutions of the Klausmeier Model for banded vegetation in semi-arid environments, I. Nonlinearity, 23, 2657-2675 (2010) · Zbl 1197.92047 · doi:10.1088/0951-7715/23/10/016 |
| [55] | COMSOL Multiphysics ® v. 5.6. COMSOL AB, Stockholm, Sweden |
| [56] | Uecker, H.; Wetzel, D.; Rademacher, J., pde2path-a Matlab package for continuation and bifurcation in 2D elliptic systems, Numer. Math. Theory Methods Appl., 7, 58-106 (2014) · Zbl 1313.65311 · doi:10.4208/nmtma.2014.1231nm |
| [57] | Lefever, R.; Lejeune, O., On the origin of tiger bush, Bull. Math. Biol., 59, 263-294 (1997) · Zbl 0903.92031 · doi:10.1007/BF02462004 |
| [58] | Barbier, N.; Couteron, P.; Lejoly, J.; Deblauwe, V.; Lejeune, O., Self-organized vegetation patterning as a fingerprint of climate and human impact on semi-arid ecosystems, J. Ecol., 94, 537-547 (2006) · doi:10.1111/j.1365-2745.2006.01126.x |
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