Experimentalists who study cancer invasion and wound repair often employ simple assays such as the in vitro scratch assay to quantify the combined effects of cell proliferation and cell migration on the collective motion of cells in two dimensions. In turn, these experiments prove to be fruitful for researchers in mathematical biology to test and explore mathematical models for collective cell motion.

I will discuss some of these ideas from the perspective of an applied mathematician, making reference to PDE models such as the Fisher-KPP equation as well as discrete processes based on random walk models. Then I will spend some time on a hole-closing model for a two-dimensional wound assay that is based on a PDE with nonlinear degenerate diffusion. The mathematics here is interesting as it involves similarity solutions of the second kind.

Finally, I will touch on slightly more complicated PDE models with nonlinear diffusion that can be used to study problems in similar geometries, such as thin tissue growth in printed bioscaffolds.

### About Mathematical biology seminars

We present regular seminars on diverse topics in mathematical biology. The seminars often show how dynamical systems, probability, or other mathematical techniques help us understand and manage biological systems, from microscopic cells to the world's largest ecosystems.

All are welcome, and past audiences have been diverse. The majority of the audience is made up of applied mathematicians, but pure mathematicians, biologists, and other scientists often attend as well.

Talks should be pitched at a level such that HDR students in mathematics and quantitative biology are able to understand the content.

These seminars are held at various times throughout the year.