  In this manuscript, we consider the modelling of cellular adhesions, which is
a key interaction between biological cells. Continuum models of the
diffusion-advection-reaction type have long been used in tissue modelling. In
2006, Armstrong, Painter, and Sherratt proposed an extension to take adhesion
effects into account. The resulting equation is a non-local advection-diffusion
equation. While immensely successful in applications, the development of
mathematical theory pertaining to steady states and pattern formation is
lacking.
  The mathematical analysis of the non-local adhesion model is challenging. In
this monograph, we contribute to the analysis of steady states and their
bifurcation structure. The importance of steady-states is that these are the
patterns observed in nature and tissues (e.g. cell-sorting experiments). In the
case of periodic boundary conditions, we combine global bifurcation results
pioneered by Rabinowitz, equivariant bifurcation theory, and the mathematical
properties (maximum principle) of the non-local term to obtain a global
bifurcation result for the branches of non-trivial solutions.
