In both normal tissue and disease states, cells interact with one another, and other tissue components using adhesion proteins. These interactions are fundamental in determining tissue fates, and the outcomes of normal development, and cancer metastasis. Traditionally continuum models (PDEs) of tissues are based on purely local interactions. However, these models ignore important nonlocal effects in tissues, such as long-ranged adhesion forces between cells.

In this talk, I focus on the nonlocal ``Armstrong adhesion model`` (2006) for adhering tissue (an example of an aggregation equation). Since its introduction, this approach has proven popular in applications to embyonic development and cancer modeling. Combining global bifurcation results pioneered by Rabinowitz, equivariant bifurcation theory, and the mathematical properties of the non-local term, we prove a global bifurcation result for the non-trivial solution branches of the scalar Armstrong adhesion model. I will demonstrate how we used the equation's symmetries to classify the solution branches by the nodal properties of the solution's derivative.

Joint work with Thomas Hillen (University of Alberta).

The process of gaining information about presence of other species in the environment is intrinsically non-local and mathematically the non-local sensing of neighboring individuals leads to non-local advection terms in continuum models.

In this talk, I will focus on a class of nonlocal advection-diffusion equations modeling population movements generated by inter- and intra-species interactions. After a brief discussion of the well-posedness of this problem, I will show that the model supports a great variety of spatio-temporal patterns, including stationary aggregations, segregations, oscillatory patterns, and irregular spatio-temporal solutions. However, if populations respond to each other in a symmetric fashion, linear stability analysis shows that the only patterns that emerge from small perturbations of the stable steady state are stationary. In this case, the system admits an energy functional that is decreasing and bounded below, suggesting that patterns remain stationary for all time. I will show how to use this functional to gain insight into the analytic structure of the stable steady state solutions. This procedure reveals a range of possible stationary patterns, including various multi-stable situations, which we validate via comparison with numerical simulations.