Developing hybrid frameworks for modelling reaction-diffusion systems

: (Alternative Format Thesis)

Abstract

Reaction-diffusion systems are of importance in both a biological and physical context, across a multitude of length scales. From the movement and interaction of calcium ions in an intracellular environment to the spread of a contagious disease through a population, reaction-diffusion systems are flexible and can provide, at least to a first approximation, a modelling framework for the evaluation of real-world problems. Mathematically, there are several ways in which we can model reaction-diffusion systems, three of which form the focus of this thesis. At the coarsest scale lie macroscopic models such as partial differential equations (PDEs), which contain no stochasticity but for whose solution there exists a wealth of analytical and numerical techniques. Whilst they can be relatively quick to simulate, they can, however, be inaccurate if complex interactions are present in the system. At a finer level of representation, we have the mesoscale, represented by an on-lattice position jump process, coupled with interaction rules, typically simulated using the Gillespie algorithm or its variants. This method allows for stochastic fluctuations but can be prohibitively slow if there are many particles present. At the finest level we have the microscale, where individual particle locations are tracked and used for the purposes of interaction. This is our most accurate representation, but it is also typically the slowest of the three.

Hybrid methods combine these different representations in order to exploit the advantages, whilst limiting the disadvantages of using each one individually. In particular, this thesis is concerned with so-called "spatially coupled" hybrid methods - those in which the spatial domain is split into two or more regions within which different modelling paradigms are employed, the regions interacting through either an interface or overlap region. Such methods are important when the system under consideration has large spatial variation in particle numbers, or when a particular region of the spatial domain requires more detail. In this thesis, we develop four new hybrid methods, with one on a static domain and the three on growing domains. We also look at developing modelling methods for some of the individual paradigms, focussing on forming equivalence frameworks between different modelling regimes.

Date of Award	16 Jun 2021
Original language	English
Awarding Institution	University of Bath
Supervisor	Kit Yates (Supervisor) & Ben Adams (Supervisor)