Symmetries, exact solutions, and reaction-diffusion models in biology

Date: Thu 30th May 2024

Abstract

Nonlinear reaction-diffusion equations are used widely to model many different systems and processes, particularly in biology. While exact analytic solutions are often extremely useful, they can be particularly difficult and sometimes impossible to construct for nonlinear PDEs. In this talk, I'll show how Lie symmetry methods can be used to construct exact analytic solutions to nonlinear reaction-diffusion equations.

First, I'll talk about a family of exact solutions to a nonlinear reaction-diffusion model that is analogous to the well-known Fisher-KPP model in a particular limit. These exact solutions are interesting since exact solutions of the Fisher-KPP model are rare, and often restricted to long-time travelling wave solutions for special values of the travelling wave speed.

Second, I’ll talk about an exact implicit solution to a nonlinear reaction-diffusion equation where the diffusivity is negative for some values of the dependent variable. Negative diffusivities have been used to model aggregation phenomena whereby members of a population move up concentration gradients from regions of low to high concentrations. The implicit exact solution is multivalued in a narrow region which can be resolved by inserting a shock – but where should the shock be?