When cells in a tissue exert traction forces, they pull on one another and on the underlying matrix, much like a tug of war. The side that exerts the largest force ``wins’’ and determines the direction of cell movement. Interestingly, such effects operate over long distances (nonlocal), not simply by neighbour-to-neighbour (local) contact. But most traditional continuum models (PDEs) can only describe local interactions. Hence, the novelty of integro-PDEs (iPDEs), namely their integral terms, is that they capture the nonlocal mechanical stresses in a tissue, and are a much more accurate description. The scalar non-local adhesion model is written as:
where $u(x,t)$ is the density of a cell population, $R$ the sensing radius of a cell, $d$ the diffusion constant, $\alpha$ the strength of homotypic adhesion between members of population $u(x,t)$, $\Omega(r)$ a odd function, and $h(u)$ a function modelling the adhesion bond kinetics.