Lyapunov-Schmidt and Centre Manifold reduction methods for nonlocal PDEs modelling animal aggregations

P.-L. Buono (Lead / Corresponding author), R. Eftimie

Research output: Chapter in Book/Report/Conference proceedingChapter

4 Citations (Scopus)

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.
Original languageEnglish
Title of host publicationMathematical sciences with multidisciplinary applications
Subtitle of host publicationin honor of Professor Christiane Rousseau. And in recognition of the Mathematics for Planet Earth Initiative
EditorsBourama Toni
PublisherSpringer International Publishing
Pages29-59
Number of pages31
Volume157
ISBN (Electronic)9783319313238
ISBN (Print)9783319313214
DOIs
Publication statusPublished - 2016

Publication series

NameSpringer proceedings in mathematics & statistics
PublisherSpringer International Publishing
Volume157
ISSN (Print)2194-1009

Keywords

  • Hyperbolic PDE
  • Animal aggregation
  • Centre Manifold reduction
  • Lyapunov–Schmidt reduction
  • Symmetry Fredholm property

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    Buono, P-L., & Eftimie, R. (2016). Lyapunov-Schmidt and Centre Manifold reduction methods for nonlocal PDEs modelling animal aggregations. In B. Toni (Ed.), Mathematical sciences with multidisciplinary applications: in honor of Professor Christiane Rousseau. And in recognition of the Mathematics for Planet Earth Initiative (Vol. 157, pp. 29-59). (Springer proceedings in mathematics & statistics; Vol. 157). Springer International Publishing. https://doi.org/10.1007/978-3-319-31323-8_3