Blaine van Rensburg 

Research Interests

My main interests are the analysis of PDEs inspired by evolution and ecology.  My research is focused on the analysis of PDE models devised to study biological hypothesis about the evolution of cells in health and disease. I have completed three projects under this theme, specifically concerning:

The role of mutant epithelial cells in Inflammatory Bowel Disease: it involves a detailed numerical and asymptotic analysis of travelling waves of mutant epithelial cells and inflammation. 

The evolutionary dynamics of a phenotypically-structured population with multiple optimal traits in a changing environment by a non-local parabolic integro-differential equation. The main novelty is that the assumption that there are multiple locally optimal traits that shift at different velocities due to the change in the environment. This work reveals that it is not the strongest species that survives, but rather that which needs to adapt the least.

How the dynamic interplay between ecology and evolution (eco-evolutionary dynamics) in a trait-structured predator-prey model determines optimal prey traits. Births via sexual reproduction are modelled using a non-linear, non-local operator called the 'infinitesimal model', and the analysis is focused on determining the dynamics of the moments of the trait-distribution in the limit of small segregational variance. We obtain an autonomous equation for the evolution of mean trait that accounts for ecological dynamics.