Using Numerical Bifurcation Analysis to Study Pattern Formation in Mussel Beds

J.A. Sherratt¹

¹ Department of Mathematics and Maxwell Institute for Mathematical Sciences, Heriot-Watt University, Edinburgh EH14 4AS, UK

Mathematical modelling of natural phenomena, Tome 11 (2016) no. 5, pp. 86-102.

Résumé

Soft-bottomed mussel beds provide an important example of ecosystem-scale self-organisation. Field data from some intertidal regions shows banded patterns of mussels, running parallel to the shore. This paper demonstrates the use of numerical bifurcation methods to investigate in detail the predictions made by mathematical models concerning these patterns. The paper focusses on the “sediment accumulation model” proposed by Liu et al (Proc. R. Soc. Lond. B 14 (2012), 20120157). The author calculates the parameter region in which patterns exist, and the sub-region in which these patterns are stable as solutions of the original model. He then shows how his results can be used to explain numerical observations of history-dependent wavelength selection as parameters are varied slowly.

Détail

DOI : 10.1051/mmnp/201611506

Bibliographie

[1] M.F. Bekker, J.T. Clark, M.W. Jackson Landscape metrics indicate differences in patterns and dominant controls of ribbon forests in the Rocky Mountains, USA Appl. Veg. Sci 2009 237 249

[2] F.H. Busse Non-linear properties of thermal convection Rep. Prog. Phys. 1978 1929 1967

[3] R.A. Cangelosi, D.J. Wollkind, B.J. Kealy-Dichone, I. Chaiya Nonlinear stability analyses of Turing patterns for a mussel-algae model J. Math. Biol. 2015 1249 1294

[4] W. Chen, M.J. Ward Oscillatory instabilities and dynamics of multispike patterns for the one-dimensional Gray-Scott model Eur. J. Appl. Math. 2009 187 214

[5] I.M. Côté, E. Jelnikar Predator-induced clumping behaviour in mussels (Mytilus edulis Linnaeus) J. Exp. Mar. Biol. Ecol. 1999 201 211

[6] A.S. Dagbovie, J.A. Sherratt Absolute stability and dynamical stabilisation in predator-prey systems J. Math. Biol. 2014 1403 1421

[7] V. Deblauwe, P. Couteron, O. Lejeune, J. Bogaert, N. Barbier Environmental modulation of self-organized periodic vegetation patterns in Sudan Ecography 2011 990 1001

[8] E.J. Doedel AUTO, a program for the automatic bifurcation analysis of autonomous systems Cong. Numer. 1981 265 384

[9] E.J. Doedel, H.B. Keller, J.P. Kernévez Numerical analysis and control of bifurction problems: (I) bifurcation in finite dimensions Int. J. Bifurcation Chaos 1991 493 520

[10] E.J. Doedel, W. Govaerts, Y.A. Kuznetsov, A. Dhooge. Numerical continuation of branch points of equilibria and periodic orbits. In: E.J. Doedel, G. Domokos, I.G. Kevrekidis (eds.) Modelling and Computations in Dynamical Systems. World Scientific, Singapore (2006), pp. 145–164.

[11] A. Doelman, T.J. Kaper, P. Zegeling Pattern formation in the one dimensional Gray-Scott model Nonlinearity 1997 523 563

[12] J.J. Donker, M. Van Der Vegt, P. Hoekstra Erosion of an intertidal mussel bed by ice- and wave-action Cont. Shelf Res. 2015 60 69

[13] M.B. Eppinga, M. Rietkerk, W. Borren, E.D. Lapshina, W. Bleuten, M.J. Wassen Regular surface patterning of peatlands: confronting theory with field data Ecosystems 2008 520 536

[14] M.B. Eppinga, M. Rietkerk, M.J. Wassen, P.C. De Ruiter Linking habitat modification to catastrophic shifts and vegetation patterns in bogs Plant Ecol 2009 53 68

[15] B. Flemming, M. Delafontaine Biodeposition in a juvenile mussel bed of the east Frisian Wadden Sea (Southern North Sea) Aqua. Eco. 1994 289 297

[16] J.C. Gascoigne, H.A. Beadman, C. Saurel, M.J. Kaiser Density dependence, spatial scale and patterning in sessile biota Oecologia 2005 371 381

[17] A. Ghazaryan, V. Manukian Coherent structures in a population model for mussel-algae interaction SIAM J. Appl. Dyn. Syst. 2015 893 913

[18] P. Gray, S.K. Scott Autocatalytic reactions in the isothermal, continuous stirred tank reactor: oscillations and instabilities in the system A+2B→ 3B Chem. Eng. Sci. 1984 1087 1097

[19] C.A. Klausmeier Regular and irregular patterns in semiarid vegetation Science 1999 1826 1828

[20] O. Lejeune, M. Tlidi, P. Couteron Localized vegetation patches: a self-organized response to resource scarcity Phys. Rev. 2002 010901

[21] S.A. Levin, R.T. Paine Disturbance, patch formation, and community structure Proc. Natl. Acad. Sci. USA 1974 2744 2747

[22] Q.-X. Liu, E.J. Weerman, P.M. Herman, H. Olff, J. Van De Koppel Alternative mechanisms alter the emergent properties of self-organization in mussel beds Proc. R. Soc. Lond. 2012 20120157

[23] Q.-X. Liu, A. Doelman, V. Rottschäfer, M. De Jager, P.M. Herman, M. Rietkerk, J. Van De Koppel Phase separation explains a new class of self-organized spatial patterns in ecological systems Proc. Natl. Acad. Sci. USA 2013 11905 11910

[24] H. Malchow Motional instabilities in predator-prey systems J. Theor. Biol. 2000 639 647

[25] S.M. Merchant, W. Nagata Instabilities and spatiotemporal patterns behind predator invasions with nonlocal prey competition Theor. Pop. Biol. 2011 289 297

[26] R. Naddafi, K. Pettersson, P. Eklöv Predation and physical environment structure the density and population size structure of zebra mussels J. N. Am. Benthol. Soc. 2010 444 453

[27] R.T. Paine, S.A. Levin Intertidal landscapes: disturbance and the dynamics of pattern Ecol. Monogr. 1981 145 178

[28] A.J. Perumpanani, J.A. Sherratt, P.K. Maini Phase differences in reaction-diffusion-advection systems and applications to morphogenesis IMA J. Appl. Math. 1995 19 33

[29] J.D.M. Rademacher, A. Scheel Instabilities of wave trains and Turing patterns in large domains Int. J. Bifur. Chaos 2007 2679 2691

[30] J.D.M. Rademacher, B. Sandstede, A. Scheel Computing absolute and essential spectra using continuation Physica 2007 166 183

[31] A.B. Rovinsky, M. Menzinger Chemical instability induced by a differential flow Phys. Rev. Lett. 1992 1193 1196

[32] B. Sandstede, Stability of travelling waves. In: B. Fiedler (ed.) Handbook of Dynamical Systems II. North-Holland, Amsterdam (2002), pp. 983–1055.

[33] B. Sandstede, A. Scheel Absolute and convective instabilities of waves on unbounded and large bounded domains Physica 2000 233 277

[34] J.A. Sherratt Numerical continuation methods for studying periodic travelling wave (wavetrain) solutions of partial differential equations Appl. Math. Computation 2012 4684 4694

[35] J.A. Sherratt History-dependent patterns of whole ecosystems Ecol. Complex. 2013 8 20

[36] J.A. Sherratt Numerical continuation of boundaries in parameter space between stable and unstable periodic travelling wave (wavetrain) solutions of partial differential equations Adv. Comput. Math. 2013 175 192

[37] J.A. Sherratt Using wavelength and slope to infer the historical origin of semi-arid vegetation bands Proc. Natl. Acad. Sci. USA 2015 4202 4207

[38] J.A. Sherratt, G.J. Lord Nonlinear dynamics and pattern bifurcations in a model for vegetation stripes in semi-arid environments Theor. Pop. Biol. 2007 1 11

[39] J.A. Sherratt, J.J. Mackenzie How does tidal flow affect pattern formation in mussel beds? J. Theor. Biol. 2016 83 92

[40] J.A. Sherratt, M.J. Smith, J.D.M. Rademacher Locating the transition from periodic oscillations to spatiotemporal chaos in the wake of invasion Proc. Natl. Acad. Sci. USA 2009 10890 10895

[41] J.A. Sherratt, A.S. Dagbovie, F.M. Hilker A mathematical biologist’s guide to absolute and convective instability Bull. Math. Biol. 2014 1 26

[42] E. Siero, A. Doelman, M.B. Eppinga, J.D.M. Rademacher, M. Rietkerk, K. Siteur Striped pattern selection by advective reaction-diffusion systems: Resilience of banded vegetation on slopes Chaos 2015 036411

[43] K. Siteur, E. Siero, M.B. Eppinga, J. Rademacher, A. Doelman, M. Rietkerk Beyond Turing: the response of patterned ecosystems to environmental change Ecol. Complex. 2014 81 96

[44] M.J. Smith, J.A. Sherratt Propagating fronts in the complex Ginzburg-Landau equation generate fixed-width bands of plane waves Phys. Rev. 2009 046209

[45] M.J. Smith, J.D.M. Rademacher, J.A. Sherratt Absolute stability of wavetrains can explain spatiotemporal dynamics in reaction-diffusion systems of lambda-omega type SIAM J. Appl. Dyn. Systems 2009 1136 1159

[46] A.C. Staver, S.A. Levin Integrating theoretical climate and fire effects on savanna and forest systems Am. Nat. 2012 211 224

[47] S.A. Suslov Numerical aspects of searching convective/absolute instability transition J. Comp. Phys. 2006 188 217

[48] J.C. Tam, R.A. Scrosati Distribution of cryptic mussel species (Mytilus edulis and M. trossulus) along wave exposure gradients on northwest Atlantic rocky shores Mar. Biol. Res. 2014 51 60

[49] J. Van De Koppel, M. Rietkerk, N. Dankers, P.M. Herman Scale-dependent feedback and regular spatial patterns in young mussel beds Am. Nat. 2005 E66 77

[50] S. Van Der Stelt, A. Doelman, G. Hek, J.D.M. Rademacher Rise and fall of periodic patterns for a generalized Klausmeier-Gray-Scott model J. Nonlinear Sci. 2013 39 95

[51] B. Van Leeuwen, D.C. Augustijn, B.K. Van Wesenbeeck, S.J. Hulscher, M.B. De Vries Modeling the influence of a young mussel bed on fine sediment dynamics on an intertidal flat in the Wadden Sea Ecol. Eng. 2010 145 153

[52] A.K. Wa Kangeri, J.M. Jansen, B.R. Barkman, J.J. Donker, D.J. Joppe, N.M. Dankers Perturbation induced changes in substrate use by the blue mussel, Mytilus edulis, in sedimentary systems J. Sea Res. 2014 233 240

[53] R.H. Wang, Q.-X. Liu, G.Q. Sun, Z. Jin, J. Van De Koppel Nonlinear dynamic and pattern bifurcations in a model for spatial patterns in young mussel beds J. R. Soc. Interface 2009 705 718

[54] E.J. Weerman, J. Van Belzen, M. Rietkerk, S. Temmerman, S. Kefi, P.W.J. Herman, J. Van De Koppel Changes in diatom patch-size distribution and degradation in a spatially self-organized intertidal mudflat ecosystem Ecology 2012 608 618

[55] J.T. Wootton Local interactions predict large-scale pattern in empirically derived cellular automata Nature 2001 841 844

[56] Y.R. Zelnik, E. Meron, G. Bel Gradual regime shifts in fairy circles Proc. Natl. Acad. Sci. USA 2015 12327 12331