Some non-local problems arising in mathematical biology

7 May 2015
Graeme Wake

This talk covers two topics: (1) Phenotype change, where we consider the steady-fitness states, in a model developed by Korobeinikov and Dempsey (2014), in which the phenotype is modelled on a continuous scale providing a structured variable to quantify the phenotype state. This enables thresholds for survival/extinction to be established in terms of fitness.

Topic (2) looks at the steady-size distribution of an evolving cohort of cells, such as tumour cells in vitro, and therein establishes thresholds for growth or decay of the cohort. This is established using a new class of non-local (but linear) singular eigenvalue problems which have point spectra, like the traditional Sturm-Liouville problems.  The first eigenvalue gives the threshold required. But these problems are first order unless dispersion is added to incorporate random perturbations. But the same idea will apply here also.  Current work involves binary asymmetrical division of cells, simultaneous with growth. It has implications to cancer biology, helping biologists to conceptualise non-local effects and the part they may play in cancer. This is developed in Zaidi et al (2015).

Acknowledgement. The support of Gravida (NCGD) is gratefully acknowledged.


Korobeinikov A & Dempsey C. A continuous phenotype space model of RNA virus evolution within a host. Mathematical Biosciences and Engineering 11, (2014), 919-927.

Zaidi AA, van-Brunt B, & Wake GC. A model for asymmetrical cell division Mathematical Biosciences and Engineering (June 2015).

  • Industrial and Applied Mathematics Seminar