Using numerical bifurcation analysis to study pattern formation in mussel beds. (English) Zbl 1384.92014
Summary: 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 Q.-X. Liu et al. [“Alternative mechanisms alter the emergent properties of self-organization in mussel beds”, Proc. R. Soc. Lond., Ser. B 279, No. 1739, 2744–2753 (2012; doi:10.1098/rspb.2012.0157)]. 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.
MSC:
| 92C15 | Developmental biology, pattern formation |
| 92D40 | Ecology |
| 35M10 | PDEs of mixed type |
| 35C07 | Traveling wave solutions |
Keywords:
mussels; pattern formation; periodic travelling wave; reaction-diffusion-advection; numerical continuation; wavetrainReferences:
| [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. 12 (2009), 237-249. |
| [2] | F.H. Busse. Non-linear properties of thermal convection. Rep. Prog. Phys. 41 (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. 70 (2015), 1249-1294. · Zbl 1456.35028 |
| [4] | W. Chen, M.J. Ward. Oscillatory instabilities and dynamics of multispike patterns for the one-dimensional Gray-Scott model. Eur. J. Appl. Math. 20 (2009), 187-214. · Zbl 1197.35021 |
| [5] | I.M. Côté, E. Jelnikar. Predator-induced clumping behaviour in mussels (Mytilus edulis Linnaeus). J. Exp. Mar. Biol. Ecol. 235 (1999), 201-211. |
| [6] | A.S. Dagbovie, J.A. Sherratt. Absolute stability and dynamical stabilisation in predator-prey systems. J. Math. Biol. 68 (2014), 1403-1421. · Zbl 1284.92109 |
| [7] | V. Deblauwe, P. Couteron, O. Lejeune, J. Bogaert, N. Barbier. Environmental modulation of self-organized periodic vegetation patterns in Sudan. Ecography 34 (2011), 990-1001. |
| [8] | E.J. Doedel. AUTO, a program for the automatic bifurcation analysis of autonomous systems. Cong. Numer. 30 (1981), 265-384. · Zbl 0511.65064 |
| [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 1 (1991), 493-520. · Zbl 0876.65032 |
| [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. · Zbl 1081.37054 |
| [11] | A. Doelman, T.J. Kaper, P. Zegeling. Pattern formation in the one dimensional Gray-Scott model. Nonlinearity 10 (1997), 523-563. · Zbl 0905.35044 |
| [12] | J.J. Donker, M. van der Vegt, P. Hoekstra. Erosion of an intertidal mussel bed by ice- and wave-action. Cont. Shelf Res. 106 (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 11 (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. 200 (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. 28 (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 145 (2005), 371-381. |
| [17] | A. Ghazaryan, V. Manukian. Coherent structures in a population model for mussel-algae interaction. SIAM J. Appl. Dyn. Syst. 14 (2015), 893-913. · Zbl 1314.35196 |
| [18] | P. Gray, S.K. Scott. Autocatalytic reactions in the isothermal, continuous stirred tank reactor: oscillations and instabilities in the system A+2B→ 3B; B→ C. Chem. Eng. Sci. 39 (1984), 1087-1097. |
| [19] | C.A. Klausmeier. Regular and irregular patterns in semiarid vegetation. Science 284 (1999), 1826-1828. |
| [20] | O. Lejeune, M. Tlidi, P. Couteron. Localized vegetation patches: a self-organized response to resource scarcity. Phys. Rev. E 66 (2002), 010901. |
| [21] | S.A. Levin, R.T. Paine. Disturbance, patch formation, and community structure. Proc. Natl. Acad. Sci. USA 71 (1974), 2744-2747. · Zbl 0289.92008 |
| [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. B 14 (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 110 (2013), 11905-11910. |
| [24] | H. Malchow. Motional instabilities in predator-prey systems. J. Theor. Biol. 204 (2000), 639-647. |
| [25] | S.M. Merchant, W. Nagata. Instabilities and spatiotemporal patterns behind predator invasions with nonlocal prey competition. Theor. Pop. Biol. 80 (2011), 289-297. · Zbl 1323.92177 |
| [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. 29 (2010), 444-453. |
| [27] | R.T. Paine, S.A. Levin. Intertidal landscapes: disturbance and the dynamics of pattern. Ecol. Monogr. 51 (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. 55 (1995), 19-33. · Zbl 0843.35041 |
| [29] | J.D.M. Rademacher, A. Scheel. Instabilities of wave trains and Turing patterns in large domains. Int. J. Bifur. Chaos 17 (2007), 2679-2691. · Zbl 1141.37359 |
| [30] | J.D.M. Rademacher, B. Sandstede, A. Scheel. Computing absolute and essential spectra using continuation. Physica D 229 (2007), 166-183. · Zbl 1119.65114 |
| [31] | A.B. Rovinsky, M. Menzinger. Chemical instability induced by a differential flow. Phys. Rev. Lett. 69 (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. · Zbl 1056.35022 |
| [33] | B. Sandstede, A. Scheel. Absolute and convective instabilities of waves on unbounded and large bounded domains. Physica D 145 (2000), 233-277. · Zbl 0963.34072 |
| [34] | J.A. Sherratt. Numerical continuation methods for studying periodic travelling wave (wavetrain) solutions of partial differential equations. Appl. Math. Computation 218 (2012), 4684-4694. · Zbl 1244.65155 |
| [35] | J.A. Sherratt. History-dependent patterns of whole ecosystems. Ecol. Complex. 14 (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. 39 (2013), 175-192. · Zbl 1271.65132 |
| [37] | J.A. Sherratt. Using wavelength and slope to infer the historical origin of semi-arid vegetation bands. Proc. Natl. Acad. Sci. USA 112 (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. 71 (2007), 1-11 · Zbl 1118.92064 |
| [39] | J.A. Sherratt, J.J. Mackenzie. How does tidal flow affect pattern formation in mussel beds? J. Theor. Biol. 406 (2016), 83-92. · Zbl 1344.92189 |
| [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 106 (2009), 10890-10895. · Zbl 1203.37062 |
| [41] | J.A. Sherratt, A.S. Dagbovie, F.M. Hilker. A mathematical biologist’s guide to absolute and convective instability. Bull. Math. Biol. 76 (2014), 1-26. · Zbl 1283.92007 |
| [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 25 (2015), 036411. · Zbl 1374.92167 |
| [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. 20 (2014), 81-96. · Zbl 1301.86005 |
| [44] | M.J. Smith, J.A. Sherratt. Propagating fronts in the complex Ginzburg-Landau equation generate fixed-width bands of plane waves. Phys. Rev. E 80 (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 8 (2009), 1136-1159. · Zbl 1183.35032 |
| [46] | A.C. Staver, S.A. Levin. Integrating theoretical climate and fire effects on savanna and forest systems. Am. Nat. 180 (2012), 211-224. |
| [47] | S.A. Suslov. Numerical aspects of searching convective/absolute instability transition. J. Comp. Phys. 212 (2006), 188-217. · Zbl 1161.76547 |
| [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. 10 (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. 165 (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. 23(2013), 39-95. · Zbl 1270.35093 |
| [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. 36 (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. 85 (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 6 (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 93 (2012), 608-618. |
| [55] | J.T. Wootton. Local interactions predict large-scale pattern in empirically derived cellular automata. Nature 413 (2001), 841-844. |
| [56] | Y.R. Zelnik, E. Meron, G. Bel. Gradual regime shifts in fairy circles. Proc. Natl. Acad. Sci. USA 112 (2015), 12327-12331. |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
