# CAIMS Prize Award

Department of Mathematics and Statistics, University of Guelph

Supervisor: Anna Lawniczak

Thesis title: Spatial and Temporal Patterns in Chemical Systems: Theoretical and Computational Approaches.

Abstract:

Understanding how complicated structures can emerge and be maintained on microscopic scales in chemical systems through natural processes alone requires new mathematical techniques and novel use of traditional methods. This thesis (i) examines how complex structures emerge from the microscopic interaction of two natural molecular processes, chemical reaction and diffusion, (ii) studies spatio-temporal phenomena including Turing pattern formation, excitability, bistability, canard explosion, spirals, and travelling waves, and (iii) investigates how heterogeneity can cause interactions between different kinds of pattern formation phenomena. Heterogeneity is introduced through immobile species. Conservation laws cause spatial dependence in the initial concentrations of immobile species to be incorporated in partial differential equations (PDEs) via spatially dependent parameters. Thus, initial conditions can be interpreted as bifurcation parameters and used to predict emergence of, and interactions between, different spatio-temporal phenomena. Heterogeneity, the basis for the interactions, occurs naturally on molecular scales, where in addition fluctuations affect the emergence of many dynamical phenomena. Because on these scales concentrations are in discrete units of molecules, PDE descriptions break down and new techniques are required to investigate dynamical behaviours.

Thus, this thesis extends lattice gas automata (LGA) for chemical systems to include heterogeneously distributed immobile species. LGA are discrete space and time stochastic interacting particle systems. LGA algorithms are scalable, fully parallel, and have no round-off or floating point errors due to the fully discrete nature of the modelling. LGA link microscopic properties, such as the probability of a reaction occurring, and macroscopic parameters, such as the values of rate coefficients. LGA naturally incorporates effects due to fluctuations and discrete chemical interactions. Hence, we apply LGA to study effects of fluctuations on pattern formation phenomena.

This thesis also investigates the occurrence of multiple periodic orbits, seen in the same models where spatial patterns occur. This investigation requires techniques for analysing Hopf bifurcations which do not satisfy the requirements of Hopf's original theorem. This thesis presents a new adaptable MAPLE code to perform the many matrix operations and differentiations that are required in typical applications.

The digital laboratory constructed in this thesis can be applied to other reaction-diffusion system such as those from ecology, sociology, epidemiology, engineering, pathology, etc.