  We study dynamics emergent from a two-dimensional reaction--diffusion process
modelled 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 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.
