# Bifurcations and dynamics emergent from lattice and continuum models of bioactive porous media

## TL;DR

This paper compares lattice and continuum models of bioactive porous media, revealing that lattice models exhibit asymmetric behaviors and bifurcations absent in continuum models, due to their structural differences.

## Contribution

It demonstrates how the discrete lattice structure allows for complex asymmetric dynamics and bifurcations not present in the continuum model, advancing understanding of such systems.

## Key findings

- Lattice models show asymmetric oscillations and steady states.
- Continuum models exhibit only symmetric solutions.
- Nonlocal reaction-diffusion mechanisms drive oscillations.

## Abstract

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.