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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 onedimensional domain with periodic boundary conditions. We show the Fredholm property for the linear operator obtained at a steadystate and from this establish the validity of Lyapunov–Schmidt reduction for steadystate bifurcations, Hopf bifurcations and mode interactions of steadystate 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 steadystate 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 language  English 

Title of host publication  Mathematical sciences with multidisciplinary applications 
Subtitle of host publication  in honor of Professor Christiane Rousseau. And in recognition of the Mathematics for Planet Earth Initiative 
Editors  Bourama Toni 
Publisher  Springer International Publishing 
Pages  2959 
Number of pages  31 
Volume  157 
ISBN (Electronic)  9783319313238 
ISBN (Print)  9783319313214 
DOIs  
Publication status  Published  2016 
Publication series
Name  Springer proceedings in mathematics & statistics 

Publisher  Springer International Publishing 
Volume  157 
ISSN (Print)  21941009 
Keywords
 Hyperbolic PDE
 Animal aggregation
 Centre Manifold reduction
 Lyapunov–Schmidt reduction
 Symmetry Fredholm property
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Profiles

Eftimie, Raluca
 Science and Engineering Office  Honorary Professor
 Mathematics  Associate Staff
Person: Associate Staff, Honorary