Singular limits of elliptic and parabolic systems

Analysis of parabolic systems on a changing environment

This project is concerned with the rigorous mathematical analysis of reaction-diffusion systems describing population dynamics, in circumstances where the domain (environment) is changing over time. Such questions may be of interest to applications in terms of modelling the effects of climate change, which may affect the size or shape of habitats over time, and also their ability to support a species.

The overall objective of the research is to investigate the effect on the behaviour of solutions of such reaction-diffusion systems of:
prescribed movement of the boundary (e.g., for bounded domains, the growth or shrinking of the domain, or for a cylindrical domain, the movement of the walls);
the progressive movement of a favourable habitat within an overall domain;
invasion fronts of a species (i.e. travelling waves in a domain which is unbounded in at least one dimension) in a changing environment.

Properties of solutions will be studied both for single species dynamics, via single reaction-diffusion equations, and for interacting systems of multiple species, via coupled systems of reaction-diffusion systems. The specific types of questions that will be addressed mathematically are:

What is the long-time behaviour of the populations in a time-varying domain?
Can a species invade unoccupied territory?
Can one species invade a region occupied by another?
How do the invasion speeds vary according to the environment and according to the rate at which the environment is changing?
What are the existence and stability properties of positive steady states?

It is expected that for each of these questions, different regimes will be possible depending on the rate at which the domain changes relative to the typical rate of species migration. The different possiblilites will be quantified by applying and adapting tools from the analysis of nonlinear partial differential equations to establish rigorous results about solutions of reaction-diffusion systems in various settings.

Novel methods of analysis will be devised to deal with the challenges caused by changing domain and boundaries, when the standard comparison principles and sub-/super-solution arguments (on a fixed domain) do not immediately apply.

EPSRC Research areas: Mathematical Analysis (primary), Mathematical Biology (secondary