Free Access
Issue
Math. Model. Nat. Phenom.
Volume 11, Number 5, 2016
Bifurcations and Pattern Formation in Biological Applications
Page(s) 86 - 102
DOI https://doi.org/10.1051/mmnp/201611506
Published online 07 December 2016
  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. [CrossRef] [Google Scholar]
  2. F.H. Busse. Non-linear properties of thermal convection. Rep. Prog. Phys. 41 (1978), 1929–1967. [Google Scholar]
  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. [CrossRef] [MathSciNet] [PubMed] [Google Scholar]
  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. [Google Scholar]
  5. I.M. Côté, E. Jelnikar. Predator-induced clumping behaviour in mussels (Mytilus edulis Linnaeus). J. Exp. Mar. Biol. Ecol. 235 (1999), 201–211. [Google Scholar]
  6. A.S. Dagbovie, J.A. Sherratt. Absolute stability and dynamical stabilisation in predator-prey systems. J. Math. Biol. 68 (2014), 1403–1421. [CrossRef] [MathSciNet] [PubMed] [Google Scholar]
  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. [Google Scholar]
  8. E.J. Doedel. AUTO, a program for the automatic bifurcation analysis of autonomous systems. Cong. Numer. 30 (1981), 265–384. [Google Scholar]
  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. [CrossRef] [Google Scholar]
  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. [CrossRef] [Google Scholar]
  11. A. Doelman, T.J. Kaper, P. Zegeling. Pattern formation in the one dimensional Gray-Scott model. Nonlinearity 10 (1997), 523–563. [Google Scholar]
  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. [Google Scholar]
  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. [Google Scholar]
  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. [Google Scholar]
  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. [Google Scholar]
  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. [PubMed] [Google Scholar]
  17. A. Ghazaryan, V. Manukian. Coherent structures in a population model for mussel-algae interaction. SIAM J. Appl. Dyn. Syst. 14 (2015), 893–913. [Google Scholar]
  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. [Google Scholar]
  19. C.A. Klausmeier. Regular and irregular patterns in semiarid vegetation. Science 284 (1999), 1826–1828. [CrossRef] [PubMed] [Google Scholar]
  20. O. Lejeune, M. Tlidi, P. Couteron. Localized vegetation patches: a self-organized response to resource scarcity. Phys. Rev. E 66 (2002), 010901. [Google Scholar]
  21. S.A. Levin, R.T. Paine. Disturbance, patch formation, and community structure. Proc. Natl. Acad. Sci. USA 71 (1974), 2744–2747. [Google Scholar]
  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. [Google Scholar]
  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. [CrossRef] [MathSciNet] [PubMed] [Google Scholar]
  24. H. Malchow. Motional instabilities in predator-prey systems. J. Theor. Biol. 204 (2000), 639–647. [PubMed] [Google Scholar]
  25. S.M. Merchant, W. Nagata. Instabilities and spatiotemporal patterns behind predator invasions with nonlocal prey competition. Theor. Pop. Biol. 80 (2011), 289–297. [CrossRef] [PubMed] [Google Scholar]
  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. [CrossRef] [Google Scholar]
  27. R.T. Paine, S.A. Levin. Intertidal landscapes: disturbance and the dynamics of pattern. Ecol. Monogr. 51 (1981), 145–178. [Google Scholar]
  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. [Google Scholar]
  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. [CrossRef] [Google Scholar]
  30. J.D.M. Rademacher, B. Sandstede, A. Scheel. Computing absolute and essential spectra using continuation. Physica D 229 (2007), 166–183. [Google Scholar]
  31. A.B. Rovinsky, M. Menzinger. Chemical instability induced by a differential flow. Phys. Rev. Lett. 69 (1992), 1193–1196. [CrossRef] [PubMed] [Google Scholar]
  32. B. Sandstede, Stability of travelling waves. In: B. Fiedler (ed.) Handbook of Dynamical Systems II. North-Holland, Amsterdam (2002), pp. 983–1055. [CrossRef] [Google Scholar]
  33. B. Sandstede, A. Scheel. Absolute and convective instabilities of waves on unbounded and large bounded domains. Physica D 145 (2000), 233–277. [Google Scholar]
  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. [CrossRef] [MathSciNet] [Google Scholar]
  35. J.A. Sherratt. History-dependent patterns of whole ecosystems. Ecol. Complex. 14 (2013), 8–20. [CrossRef] [Google Scholar]
  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. [Google Scholar]
  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. [CrossRef] [Google Scholar]
  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 [CrossRef] [Google Scholar]
  39. J.A. Sherratt, J.J. Mackenzie. How does tidal flow affect pattern formation in mussel beds? J. Theor. Biol. 406 (2016), 83–92. [PubMed] [Google Scholar]
  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. [CrossRef] [Google Scholar]
  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. [CrossRef] [MathSciNet] [PubMed] [Google Scholar]
  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. [CrossRef] [MathSciNet] [PubMed] [Google Scholar]
  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. [CrossRef] [Google Scholar]
  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. [Google Scholar]
  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. [CrossRef] [Google Scholar]
  46. A.C. Staver, S.A. Levin. Integrating theoretical climate and fire effects on savanna and forest systems. Am. Nat. 180 (2012), 211–224. [CrossRef] [PubMed] [Google Scholar]
  47. S.A. Suslov. Numerical aspects of searching convective/absolute instability transition. J. Comp. Phys. 212 (2006), 188–217. [CrossRef] [Google Scholar]
  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. [CrossRef] [Google Scholar]
  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. [CrossRef] [PubMed] [Google Scholar]
  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. [CrossRef] [MathSciNet] [Google Scholar]
  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. [Google Scholar]
  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. [CrossRef] [Google Scholar]
  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. [CrossRef] [PubMed] [Google Scholar]
  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. [PubMed] [Google Scholar]
  55. J.T. Wootton. Local interactions predict large-scale pattern in empirically derived cellular automata. Nature 413 (2001), 841–844. [CrossRef] [PubMed] [Google Scholar]
  56. Y.R. Zelnik, E. Meron, G. Bel. Gradual regime shifts in fairy circles. Proc. Natl. Acad. Sci. USA 112 (2015), 12327–12331. [CrossRef] [Google Scholar]

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