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Lyapunov–Schmidt and Centre Manifold Reduction Methods for Nonlocal PDEs Modelling Animal Aggregations

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Mathematical Sciences with Multidisciplinary Applications

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 157))

Abstract

The goal of this paper is to establish the applicability of the Lyapunov–Schmidt reduction and the Centre Manifold Theorem (CMT) for a class of hyperbolic partial differential equation models with nonlocal interaction terms describing the aggregation dynamics of animals/cells in a one-dimensional domain with periodic boundary conditions. We show the Fredholm property for the linear operator obtained at a steady-state and from this establish the validity of Lyapunov–Schmidt reduction for steady-state bifurcations, Hopf bifurcations and mode interactions of steady-state and Hopf. Next, we show that the hypotheses of the CMT of Vanderbauwhede and Iooss (Center manifold theory in infinite dimensions. In: Jones, C., Kirchgraber, U., Walther, H.O. (eds.) Dynamics Reported, vol. 1, pp. 125–163. Springer, Berlin, 1992) hold for any type of local bifurcation near steady-state solutions with SO(2) and O(2) symmetry. To put our results in context, we review applications of hyperbolic partial differential equation models in physics and in biology. Moreover, we also survey recent results on Fredholm properties and Centre Manifold reduction for hyperbolic partial differential equations and equations with nonlocal terms.

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References

  1. Armstrong, N.J., Painter, K.J., Sherratt, J.A.: A continuum approach to modelling cell–cell adhesion. J. Theor. Biol. 243, 98–113 (2006)

    Article  MathSciNet  Google Scholar 

  2. Barbera, E., Currò, C., Valenti, G.: Wave features of a hyperbolic prey–predator model. Math. Methods Appl. Sci. 33 (12), 1504–1515 (2010)

    MathSciNet  MATH  Google Scholar 

  3. Barbera, E., Consolo, G., Valenti, G.: A two or three compartments hyperbolic reaction-diffusion model for the aquatic food chain. Math. Biosci. Eng. 12 (3), 451–472 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  4. Belleni-Morante, A., McBride, A.C.: Applied Nonlinear Semigroups: An Introduction. Wiley, New York (1998)

    MATH  Google Scholar 

  5. Bröcker, T., tom Dieck, T.: Representations of Compact Lie Groups, vol. 98. Springer-Verlag, New-York (1985)

    Google Scholar 

  6. Buono, P.-L., Eftimie, R.: Analysis of Hopf/Hopf bifurcations in nonlocal hyperbolic models for self-organised aggregations. Math. Models Methods Appl. Sci. 24 (2), 327–357 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  7. Buono, P.-L., Eftimie, R.: Codimension-two bifurcations in animal aggregation models with symmetry. SIAM J. Appl. Dyn. Syst. 13 (4), 1542–1582 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  8. Carr, J., Muncaster, R.G.: The application of centre manifolds to amplitude expansions. II. Infinite dimensional problems. J. Differ. Equ. 50, 260–279 (1983)

    MathSciNet  MATH  Google Scholar 

  9. Chertock, A., Kurganov, A., Polizzi, A., Timofeyev, I.: Pedestrian flow models with slowdown interactions. Math. Models Methods Appl. Sci. 24, 249–275 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  10. Chicone, C.: Ordinary Differential Equations with Applications. Texts in Applied Mathematics, vol. 34 Springer-Verlag, New-York (2006)

    Google Scholar 

  11. Chossat, P., Golubitsky, M.: Hopf bifurcation in the presence of symmetry, center manifold and Liapunov-Schmidt reduction. In: Atkinson, F.V., Langford, W.F., Mingarelli, A.B. (eds.) Oscillation, Bifurcation and Chaos. CMS-AMS Conference Proceedings Series, vol. 8, pp. 343–352. AMS, Providence (1987)

    Google Scholar 

  12. Chossat, P., Lauterbach, R.: Methods in Equivariant Bifurcation and Dynamical Systems. World Scientific, Singapore River Edge, NJ (2000)

    Book  MATH  Google Scholar 

  13. Colombo, R.M., Rossi, E.: Hyperbolic predators vs. parabolic prey. Commun. Math. Sci. 13 (2), 369–400 (2015)

    Google Scholar 

  14. Diekmann, O., van Gils, S.A., Verduyn Lunel, S.M., Walther, H.O.: Delay Equations, Functional-, Complex-, and Nonlinear Analysis. Springer-Verlag, New-York (1995)

    MATH  Google Scholar 

  15. Eftimie, R.: Hyperbolic and kinetic models for self-organised biological aggregations and movement: a brief review. J. Math. Biol. 65 (1), 35–75 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  16. Eftimie, R.: Simultaneous use of different communication mechanisms leads to spatial sorting and unexpected collective behaviours in animal groups. J. Theor. Biol. 337, 42–53 (2013)

    Article  MathSciNet  Google Scholar 

  17. Eftimie, R., de Vries, G., Lewis, M.A.: Complex spatial group patterns result from different animal communication mechanisms. Proc. Natl. Acad. Sci. 104 (17), 6974–6979 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  18. Eftimie, R., de Vries, G., Lewis, M.A., Lutscher, F.: Modeling group formation and activity patterns in self-organizing collectives of individuals. Bull. Math. Biol. 69 (5), 1537–1566 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  19. Engel, K.-J., Nagel, R.: A Short Course on Operator Semigroups. Springer, Berlin (2006)

    MATH  Google Scholar 

  20. Ermentrout, G.B., McLeod, J.B.: Existence and uniqueness of travelling waves for a neural network. Proc. R. Soc. Edinb. 123A, 461–478 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  21. Faye, G., Scheel, A.: Fredholm properties of nonlocal differential operators via spectral flow. Indiana Univ. Math. J. 63 (5), 1–34 (2013)

    MathSciNet  MATH  Google Scholar 

  22. Fetecau, R.: Collective behaviour of biological aggregations in two dimensions: a nonlocal kinetic model. Math. Models Methods Appl. Sci. 21, 1539–1569 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  23. Filbet, F., Laurencot, P., Perthame, B.: Derivation of hyperbolic models for chemosensitive movement. J. Math. Biol. 50 (2), 189–207 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  24. Fujimura, K.: Methods of centre manifold and multiple scales in the theory of weakly nonlinear stability for fluid motions. Proc. R. Soc. Lond. A 434, 719–733 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  25. Golubitsky, M., Schaeffer, D.G.: Singularities and Groups in Bifurcation Theory, vol. 1. Springer, New York (1985)

    Book  MATH  Google Scholar 

  26. Golubitsky, M., Stewart, I.: The Symmetry Perspective: From Equilibrium to Chaos in Phase Space and Physical Space. Birkhäuser, Basel (2002)

    Book  MATH  Google Scholar 

  27. Golubitsky, M., Stewart, I., Schaeffer, D.G.: Singularities and Groups in Bifurcation Theory, vol. 2. Springer, New York (1988)

    Book  MATH  Google Scholar 

  28. Golubitsky, M., Marsden, J., Stewart, I., Dellnitz, M.: The constrained Liapunov-Schmidt procedure and periodic orbits. In: Normal Forms and Homoclinic Chaos. Fields Institute Communications, vol. 4, pp. 81–127. American Mathematical Society, Providence, RI (1995)

    Google Scholar 

  29. Hackett-Jones, E.J., Landman, K.A., Fellner, K.: Aggregation patterns from non-local interactions: discrete stochastic and continuum modelling. Phys. Rev. E 85, 041912 (2012)

    Article  Google Scholar 

  30. Hadeler, K.P.: Reaction transport equations in biological modeling. Math. Comput. Model. 31 (4–5), 75–81 (2000). Proceedings of the Conference on Dynamical Systems in Biology and Medicine

    Google Scholar 

  31. Hale, J.K., Verduyn Lunel, S.M.: Introduction to Functional Differential Equations. Springer-Verlag, London (1993)

    Book  MATH  Google Scholar 

  32. Haragus, M., Iooss, G.: Local Bifurcations, Centre Manifolds, and Normal Forms in Infinite-Dimensional Systems. Springer-Verlag, London (2010)

    MATH  Google Scholar 

  33. Härterich, J., Sandstede, B., Scheel, A.: Exponential dichotomies for linear non-autonomous functional differential equations of mixed type. Indiana Univ. Math. J. 51, 1081–1109 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  34. Hillen, T.: Invariance principles for hyperbolic random walk systems. J. Math. Anal. Appl. 210 (1), 360–374 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  35. Hillen, T.: Hyperbolic models for chemosensitive movement. Math. Models Methods Appl. Sci. 12 (07), 1007–1034 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  36. Hillen, T.: Existence theory for correlated random walks on bounded domains. Canad. Appl. Math. Quart. 18 (1), 1–40 (2010)

    MathSciNet  MATH  Google Scholar 

  37. Hillen, T., Hadeler, K.P.: Hyperbolic systems and transport equations in mathematical biology. In: Warnecke, G. (ed.) Analysis and Numerics for Conservation Laws, pp. 257–279. Springer, Berlin/Heidelberg (2005)

    Chapter  Google Scholar 

  38. Hupkes, H.J., Verduyn Lunel, S.M.: Center manifold theory for functional differential equations of mixed type. J. Dynam. Differ. Equ. 19, 497–560 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  39. Inaba, H.: Threshold and stability results for an age-structured epidemic model. J. Math. Biol. 28, 411–434 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  40. Iooss, G., Kirchgässner, K.: Travelling waves in a chain of coupled nonlinear oscillators. Commun. Math. Phys. 211, 439–464 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  41. Kato, T.: Perturbation Theory for Linear Operators. Springer-Verlag, New-York (1995)

    MATH  Google Scholar 

  42. Keyfitz, B.L., Keyfitz, N.: The Mckendrick partial differential equation and its uses in epidemiology and population study. Math. Comput. Model. 26 (6), 1–9 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  43. Kmit, I.: Fredholm solvability of a periodic Neumann problem for a linear telegraph equation. Ukrainian Math. J. 65 (3) (2013)

    Google Scholar 

  44. Kmit, I., Recke, L.: Fredholm alternative for periodic-Dirichlet problems for linear hyperbolic systems. J. Math. Anal. Appl. 335 (1), 355–370 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  45. Kmit, I., Recke, L.: Fredholmness and smooth dependence for linear time-periodic hyperbolic systems. J. Differ. Equ. 252 (2), 1962–1986 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  46. Kmit, I., Recke, L.: Periodic solutions to dissipative hyperbolic systems. I: Fredholm solvability of linear problems. 999:DFG Research Center MATHEON (2013, preprint)

    Google Scholar 

  47. Kmit, I., Recke, L.: Hopf bifurcation for semilinear dissipative hyperbolic systems. J. Differ. Equ. 257, 264–309 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  48. Kovacic, M.: On matrix-free pseudo-arclength continuation methods applied to a nonlocal PDE in 1+1D with pseudo-spectral time-stepping. Master’s thesis, University of Ontario Institute of Technology (2013)

    Google Scholar 

  49. Larkin, R., Szafoni, R.: Evidence for widely dispersed birds migrating together at night. Integr. Comp. Biol. 48 (1), 40–49 (2008)

    Article  Google Scholar 

  50. Latushkin, Y., Tomilov, Y.: Fredholm differential operators with unbounded coefficients. J. Differ. Equ. 208, 388–429 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  51. Lichtner, M.: Exponential Dichotomy and Smooth Invariant Center Manifolds for Semilinear Hyperbolic Systems. Ph.D. thesis, Humboldt-Universität zu Berlin, Berlin (2006)

    Google Scholar 

  52. Lichtner, M., Radziunas, M., Recke, L.: Well-posedness, smooth dependence and centre manifold reduction for a semilinear hyperbolic system from laser dynamics. Math. Methods Appl. Sci. 30, 931–960 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  53. Lutscher, F.: Modeling alignment and movement of animals and cells. J. Math. Biol. 45, 234–260 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  54. Magal, P., Ruan, S.: On integrated semigroups and age structured models in L p spaces. Differ. Integr. Equ. 20 (2), 197–239 (2007)

    MathSciNet  MATH  Google Scholar 

  55. Magal, P., Ruan, S.: Center Manifolds for Semilinear Equations with Non-dense Domain and Applications to Hopf Bifurcation in Age-Structured Models. American Mathematical Society, Providence (2009)

    MATH  Google Scholar 

  56. Mallet-Paret, J.: The Fredholm alternative for functional differential equations of mixed type. J. Dyn. Differ. Equ. 11 (1), 1–47 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  57. Mogilner, A., Edelstein-Keshet, L.: A non-local model for a swarm. J. Math. Biol. 38, 534–570 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  58. Pfistner, B.: A one dimensional model for the swarming behaviour of Myxobacteria. In: Hoffmann, G., Alt, W. (eds.) Biological Motion. Lecture Notes on Biomathematics, pp. 556–563. Springer, Berlin (1990)

    Google Scholar 

  59. Pliny the Elder: The Natural History. Book X. Taylor and Francis, London (1855)

    Google Scholar 

  60. Renardy, M.: A centre manifold theorem for hyperbolic PDEs. Proc. R. Soc. Edinb. Sect. A 122 (3–4), 363–377 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  61. Robinson, J.C.: Infinite-Dimensional Dynamical Systems. Cambridge University Press, Cambridge (2001)

    Book  MATH  Google Scholar 

  62. Sieber, J., Radziunas, M., Scneider, K.R.: Dynamics of multisection lasers. Math. Model. Anal. 9 (1), 51–66 (2004)

    MathSciNet  MATH  Google Scholar 

  63. Sieber, J., Recke, L., Schneider, K.R.: Dynamics of multisection semiconductor lasers. J. Math. Sci. 124 (5), 5298–5309 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  64. Topaz, C.M., Bertozzi, A.L., Lewis, M.A.: A nonlocal continuum model for biological aggregation. Bull. Math. Biol. 68, 1601–1623 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  65. Topaz, C.M., D’Orsogna, M.R., Edelstein-Keshet, L., Bernoff, A.J.: Locust dynamics: behavioral phase change and swarming. PLoS Comput. Biol. 8, e1002642 (2012)

    Article  MathSciNet  Google Scholar 

  66. Topaz, C.M., D’Orsogna, M.R., Edelstein-Keshet, L., Bernoff, A.J.: Locust dynamics: behavioural phase change and swarming. PLoS Comput. Biol. 8 (8), e1002642 (2012)

    Article  MathSciNet  Google Scholar 

  67. Vanderbauwhede, A., Iooss, G.: Center manifold theory in infinite dimensions. In: Jones, C., Kirchgraber, U., Walther, H.O. (eds.) Dynamics Reported, vol. 1, pp. 125–163. Springer, Berlin (1992)

    Chapter  Google Scholar 

  68. Witten, M. (ed.) Hyperbolic Partial Differential Equations. Populations, Reactors, Tides and Waves: Theory and Applications. Pergamon, Elmsford, N.Y. (1983)

    Google Scholar 

  69. Wollkind, D.J.: Applications of linear hyperbolic partial equations: predator–prey systems and gravitational instability of nebulae. Math. Model. 7, 413–428 (1986)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

PLB acknowledges the financial support from NSERC in the form of a Discovery Grant. RE acknowledges support from an Engineering and Physical Sciences Research Council (UK) First Grant number EP/K033689/1. PLB would like to thank Christiane Rousseau for her support and encouragements over the years. PLB is particularly grateful to her for proposing and championing the Mathematics of Planet Earth initiative.

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Authors and Affiliations

  1. Faculty of Science, University of Ontario Institute of Technology, Oshawa, ON, Canada, L1H 7K4

    Pietro-Luciano Buono

  2. Division of Mathematics, University of Dundee, Dundee, DD1 4HN, UK

    R. Eftimie

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  1. Pietro-Luciano Buono
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  2. R. Eftimie
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Correspondence to Pietro-Luciano Buono .

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Editors and Affiliations

  1. Dept of Mathematics and Economics, Virginia State University, Petersburg, Virginia, USA

    Bourama Toni

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Buono, PL., Eftimie, R. (2016). Lyapunov–Schmidt and Centre Manifold Reduction Methods for Nonlocal PDEs Modelling Animal Aggregations. In: Toni, B. (eds) Mathematical Sciences with Multidisciplinary Applications. Springer Proceedings in Mathematics & Statistics, vol 157. Springer, Cham. https://doi.org/10.1007/978-3-319-31323-8_3

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