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Master stability functions reveal diffusion-driven pattern formation in networks

Andreas Brechtel1, Philipp Gramlich1, Daniel Ritterskamp2, Barbara Drossel1, and Thilo Gross2

  • 1Institute of Condensed Matter Physics, Technische Universität Darmstadt, 64289 Darmstadt, Germany
  • 2Department of Engineering Mathematics, Merchant Venturers School of Engineering, University of Bristol, Woodland Road, Bristol BS8 1UB, United Kingdom

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Phys. Rev. E 97, 032307 – Published 19 March, 2018

DOI: https://doi.org/10.1103/PhysRevE.97.032307

Abstract

We study diffusion-driven pattern formation in networks of networks, a class of multilayer systems, where different layers have the same topology, but different internal dynamics. Agents are assumed to disperse within a layer by undergoing random walks, while they can be created or destroyed by reactions between or within a layer. We show that the stability of homogeneous steady states can be analyzed with a master stability function approach that reveals a deep analogy between pattern formation in networks and pattern formation in continuous space. For illustration, we consider a generalized model of ecological meta-food webs. This fairly complex model describes the dispersal of many different species across a region consisting of a network of individual habitats while subject to realistic, nonlinear predator-prey interactions. In this example, the method reveals the intricate dependence of the dynamics on the spatial structure. The ability of the proposed approach to deal with this fairly complex system highlights it as a promising tool for ecology and other applications.

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References (58)

  1. M. E. W. Newman, D. J. Watts, and L. Barabasi, The Structure and Dynamics of Networks (Princeton University Press, Princeton, 2006).
  2. A. M. Turing, The chemical basis of morphogenesis, Philos. Trans. R. Soc. London 237, 37 (1952).
  3. A. Gierer and H. Meinhardt, A theory of biological pattern formation, Kybernetik 12, 30 (1972).
  4. H. Nakao and A. S. Mikhailov, Turing patterns in network-organized activator-inhibitor systems, Nat. Phys. 6, 544 (2010).
  5. L. D. Fernandes and M. A. M. de Aguiar, Turing patterns and apparent competition in predator-prey food webs on networks, Phys. Rev. E 86, 056203 (2012).
  6. L. M. Pecora and T. L. Carroll, Master Stability Functions for Synchronized Coupled Systems, Phys. Rev. Lett. 80, 2109 (1998).
  7. A. Arenas, A. Díaz-Guilera, J. Kurths, Y. Moreno, and C. Zhou, Synchronization in complex networks, Phys. Rep. 469, 93 (2008).
  8. A. Pikovsky, J. Kurths, and M. Rosenblum, Synchronization (Cambridge University Press, Cambridge, 2001).
  9. R. Parshani, S. V. Buldyrev, and S. Havlin, Interdependent Networks: Reducing the Coupling Strength Leads to a Change from a First to Second Order Percolation Transition, Phys. Rev. Lett. 105, 048701 (2010).
  10. S. V. Buldyrev, R. Parshani, G. Paul, H. E. Stanley, and S. Havlin, Catastrophic cascade of failures in interdependent networks, Nature (London) 464, 1025 (2010).
  11. A. Bashan, Y. Berezin, S. V. Buldyrev, and S. Havlin, The extreme vulnerability of interdependent spatially embedded networks, Nat. Phys. 9, 667 (2013).
  12. M. De Domenico, A. Sole-Ribalta, E. Omodei, S. Gomez, and A. Arenas, Ranking in interconnected multilayer networks reveals versatile nodes, Nat. Commun. 6, 6868 (2015).
  13. J. Gao, S. V. Buldyrev, H. E. Stanley, and S. Havlin, Networks formed from interdependent networks, Nat. Phys. 8, 40 (2011).
  14. M. Kivelä, A. Arenas, M. Barthelemy, J. P. Gleeson, Y. Moreno, and M. A. Porter, Multilayer networks, J. Complex Networks 2, 203 (2014).
  15. M. De Domenico, A. Solé-Ribalta, E. Cozzo, M. Kivelä, Y. Moreno, M. A. Porter, S. Gómez, and A. Arenas, Mathematical Formulation of Multilayer Networks, Phys. Rev. X 3, 041022 (2013).
  16. S. Boccaletti, G. Bianconi, R. Criado, C. I. del Genio, J. Gómez-Gardeñes, M. Romance, I. Sendiña-Nadal, Z. Wang, and M. Zanin, The structure and dynamics of multilayer networks, Phys. Rep. 544, 1 (2014).
  17. E. Cozzo, G. F. de Arruda, F. A. Rodrigues, and Y. Moreno, in Interconnected Networks, edited by F. Schweitzer and A. Garas (Springer, Heidelberg, 2015).
  18. M. De Domenico, C. Granell, M. A. Porter, and A. Arenas, The physics of spreading processes in multilayer networks, Nat. Phys. 12, 901 (2016).
  19. M. Asllani, D. M. Busiello, T. Carletti, D. Fanelli, and G. Planchon, Turing patterns in multiplex networks, Phys. Rev. E 90, 042814 (2014).
  20. X. Zhang, S. Boccaletti, S. Guan, and Z. Liu, Explosive Synchronization in Adaptive and Multilayer Networks, Phys. Rev. Lett. 114, 038701 (2015).
  21. N. E. Kouvaris, S. Hata, and A. D. Guilera, Pattern formation in multiplex networks, Sci. Rep. 5, 10840 (2015).
  22. C. I. del Genio, J. Gómez-Gardeñes, I. Bonamassa, and S. Boccaletti, Synchronization in networks with multiple interaction layers, Sci. Adv. 2, e1601679 (2016).
  23. R. Sevilla-Escoboza, I. Sendiña-Nadal, I. Leyva, R. Gutiérrez, J. M. Buldú, and S. Boccaletti, Inter-layer synchronization in multiplex networks of identical layers, Chaos 26, 065304 (2016).
  24. L. Tang, X. Wu, J. Lü, J.-a. Lu, and R. M. D'Souza, Master stability functions for multiplex networks, arXiv:1611.09110.
  25. I. Leyva, R. Sevilla-Escoboza, I. Sendiña-Nadal, R. Gutiérrez, J. Buldú, and S. Boccaletti, Inter-layer synchronization in non-identical multi-layer networks, Sci. Rep. 7, 45475 (2017).
  26. G. E. Hutchinson, Homage to Santa Rosalia or why are there so many kinds of animals? Am. Nat. 93, 145 (1959).
  27. K. S. McCann, The diversity-stability debate, Nature (London) 405, 228 (2000).
  28. T. Gross, L. Rudolf, S. A. Levin, and U. Dieckmann, Generalized models reveal stabilizing factors in food webs, Science 325, 747 (2009).
  29. B. Kartascheff, L. Heckmann, B. Drossel, and C. Guill, Why allometric scaling enhances stability in food web models, Theor. Ecol. 3, 195 (2010).
  30. R. Levins, Some demographic and genetic consequences of environmental heterogeneity for biological control, Bull. Entomol. Soc. Am. 15, 237 (1969).
  31. J. D. Yeakel, J. W. Moore, P. R. Guimarães, and M. A. M. de Aguiar, Synchronisation and stability in river metapopulation networks, Ecol. Lett. 17, 273 (2014).
  32. E. Tromeur, L. Rudolf, and T. Gross, Impact of dispersal on the stability of metapopulations, J. Theor. Biol. 392, 1 (2016).
  33. P. Pillai, M. Loreau, and A. Gonzalez, A patch-dynamic framework for food web metacommunities, Theor. Ecol. 3, 223 (2010).
  34. P. Pillai, A. Gonzalez, and M. Loreau, Metacommunity theory explains the emergence of food web complexity, Proc. Natl. Acad. Sci. USA 108, 19293 (2011).
  35. K. Ristl, S. J. Plitzko, and B. Drossel, Complex response of a food-web module to symmetric and asymmetric migration between several patches, J. Theor. Biol. 354, 54 (2014).
  36. P. Gramlich, S. J. Plitzko, L. Rudolf, B. Drossel, and T. Gross, The influence of dispersal on a predator-prey system with two habitats, J. Theor. Biol. 398, 150 (2015).
  37. E. Barter and T. Gross, Meta-food-chains as a many-layer epidemic process on networks, Phys. Rev. E 93, 022303 (2016).
  38. A. Mougi and M. Kondoh, Food-web complexity, meta-community complexity and community stability, Sci. Rep. 6, 24478 (2016).
  39. S. Pilosof, M. A. Porter, M. Pascual, and S. Kéfi, The multilayer nature of ecological networks, Nat. Ecol. Evolution 1, 0101 (2015).
  40. D. Gravel, F. Massol, and M. A. Leibold, Stability and complexity in model meta-ecosystems, Nat. Commun. 7, 12457 (2016).
  41. T. Gross and U. Feudel, Generalized models as a universal approach to the analysis of nonlinear dynamical systems, Phys. Rev. E 73, 016205 (2006).
  42. M. R. Hirt, W. Jetz, B. C. Rall, and U. Brose, A general scaling law reveals why the largest animals are not the fastest, Nat. Ecol. Evolution 1, 1116 (2017).
  43. R. Merris, Laplacian graph eigenvectors, Lin. Algebra Applicat. 278, 221 (1998).
  44. B. Mohar, Y. Alavi, G. Chartrand, and O. R. Oellermann, The Laplacian spectrum of graphs, Graph Theory, Combinatorics, Applicat. 2, 871 (1991).
  45. R. Agaev and P. Chebotarev, in Linear Algebra and Its Applications (Elsevier, Amsterdam, 2005), Vol. 399, pp. 157–168.
  46. M. Baurmann, T. Gross, and U. Feudel, Instabilities in spatially extended predator-prey systems: Spatio-temporal patterns in the neighborhood of Turing-Hopf bifurcations, J. Theor. Biol. 245, 220 (2007).
  47. J. D. Yeakel, M. M. Pires, L. Rudolf, N. J. Dominy, P. L. Koch, P. R. Guimarães, and T. Gross, Collapse of an ecological network in Ancient Egypt, Proc. Natl. Acad. Sci. USA 111, 14472 (2014).
  48. S. J. Plitzko, B. Drossel, and C. Guill, Complexity-stability relations in generalized food-web models with realistic parameters, J. Theor. Biol. 306, 7 (2012).
  49. C. Kuehn, S. Siegmund, and T. Gross, Dynamical analysis of evolution equations in generalized models, IMA J. Appl. Math. 78, 1051 (2013).
  50. Y. Nievergelt, The concept of elasticity in economics, SIAM Rev. 25, 261 (1983).
  51. R. J. Williams and N. D. Martinez, Simple rules yield complex foodwebs, Nature (London) 404, 180 (2000).
  52. J. D. Yeakel, D. Stiefs, M. Novak, and T. Gross, Generalized modeling of ecological population dynamics, Theor. Ecol. 4, 179 (2011).
  53. A. Stein, K. Gerstner, and H. Kreft, Environmental heterogeneity as a universal driver of species richness across taxa, biomes and spatial scales, Ecol. Lett. 17, 866 (2014).
  54. M. Cucuringu and M. W. Mahoney, Localization on low-order eigenvectors of data matrices, arXiv:1109.1355.
  55. A. Nyberg, T. Gross, and K. E. Bassler, Mesoscopic structures and the Laplacian spectra of random geometric graphs, J. Complex Networks 3, 543 (2014).
  56. C. Dettmann, O. Georgiou, and G. Knight, Spectral statistics of random geometric graphs, Europhys. Lett. 118, 18003 (2017).
  57. J. Dall and M. Christensen, Random geometric graphs, Phys. Rev. E 66, 016121 (2002).
  58. J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems and the Bifurcations of Vector Fields (Springer, New York, 1983).

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Phys. Rev. E 97, 032307– Published 19 March, 2018
  • Received 4 April 2017
  • Revised 8 October 2017
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