Bifurcations and Dynamics Emergent from Lattice and Continuum Models of Bioactive Porous Media

Andrew L. Krause, Dmitry Beliaev, Robert A. Van Gorder, Sarah L. Waters

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We study the dynamics emergent from a two-dimensional reaction-diffusion process modeled via a finite lattice dynamical system, as well as an analogous PDE system, involving spatially nonlocal interactions. These models govern the evolution of cells in a bioactive porous medium, with the evolution of the local cell density depending on a coupled quasi-static fluid flow problem. We demonstrate differences emergent from the choice of a discrete lattice or a continuum for the spatial domain of such a process. We find long-time oscillations and steady states in cell density in both lattice and continuum models, but that the continuum model only exhibits solutions with vertical symmetry, independent of initial data, whereas the finite lattice admits asymmetric oscillations and steady states arising from symmetry-breaking bifurcations. We conjecture that it is the structure of the finite lattice which allows for more complicated asymmetric dynamics. Our analysis suggests that the origin of both types of oscillations is a nonlocal reaction-diffusion mechanism mediated by quasi-static fluid flow.

Original languageEnglish
Article number1830037
JournalInternational Journal of Bifurcation and Chaos
Issue number11
Publication statusPublished - 1 Oct 2018
Externally publishedYes


  • bifurcation analysis
  • Bioactive porous media
  • lattice and continuum models
  • nonlocal reaction-diffusion equations


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