Mathematics

Systems of activator-inhibitor reaction-diffusion equations posed on an infinite line are studied using a variety of analytical and numerical methods. A canonical form is considered that contains all known models with simple cubic autocatalytic nonlinearity and arbitrary constant and linear kinetics. Restricting attention at first to models that have a unique homogeneous equilibrium, this class includes the classical Schnakenberg and Brusselator models, as well as other systems proposed in the literature to model morphogenesis. Such models are known to feature Turing instability, when activator diffuses more slowly than inhibitor, leading to stable spatially periodic patterns. Conversely in the limit of small feed rates, semi-strong interaction asymptotic analysis as introduced by Michael Ward and his collaborators shows existence of isolated spike-like patterns.

Connecting these two regions, a certain universal two-parameter state diagram is revealed in which the Turing bifurcation becomes sub-critical, leading to the onset of homoclinic snaking. This regime then morphs into the spike regime, with the outer-fold being predicted by the semi-strong asymptotics. A rescaling of parameters and field concentrations shows how this state diagram can be studied independently of the diffusion rates. Temporal dynamics is found to strongly depend on the diffusion ratio though. A Hopf bifurcation occurs along the branch of stable spikes, which is subcritical for small diffusion ratio, leading to collapse to the homogeneous state. As the diffusion ratio increases, this bifurcation typically becomes supercritical, interacts with the homoclinic snaking and also with a supercritical homogeneous Hopf bifurcation, leading to complex spatio-temporal dynamics. The details are worked out for a number of different models that fit the theory using a mixture of weakly nonlinear analysis, semi-strong asymptotics and different numerical continuation algorithms.

The theory is extended include models, such as Gray-Scott, with bistability of homogeneous equilibria. A homotopy is studied that takes a Schnakenberg-like glycolysis model for r = 0 to the Gray-Scott model for r = 1. Numerical continuation is used to understand the complete sequence of transitions to two-parameter bifurcation diagrams within the localised pattern parameter regime as r varies. Several distinct codimension-two bifurcations are dis-covered including cusp and quadruple zero points for homogeneous steady states, a degenerate heteroclinic connection and a change in connectedness of the homoclinic snaking structure. The analysis is repeated for the Gierer-Meinhardt system, which lies outside the canonical framework. Similar transitions are found under homotopy between bifurcation diagrams for the case where there is a constant feed in the active field, to it being in the inactive field. Wider implications of the results are discussed for other kinds of pattern-formation systems as well as to distinguishing between different kinds of observed behaviour in the natural world.

For every non-elementary hyperbolic group, we consider the Manhattan curve, which was originally introduced by M. Burger (1993), associated to any pair of (say) word metrics. It is convex; we show that it is continuously differentiable and moreover is a straight line if and only if the corresponding two metrics are roughly similar, that is, they are within bounded distance after multiplying by a positive constant.

I would like to explain how it is related to central limit theorem for uniform counting measures on spheres, to ergodic theory of topological flows built on general hyperbolic groups, and to multifractal structure of Patterson-Sullivan measures. Furthermore I will present some explicit examples including a hyperbolic triangle group and compute the exact value of the mean distortion for a pair of word metrics by using automatic structures of the group.

Joint work with Stephen Cantrell (University of Chicago).

How organ size is controlled during development has been a subject of scientific study for centuries, but the growth control mechanisms are still poorly understood. The Drosophila wing imaginal disc has widely been used as a model system to study the regulation of growth. Growth control in the Drosophila wing disc involves various local signals, including signaling pathways, mechanical signals, etc. We developed a model of the Hippo pathway, which is the core regulatory pathway that mediates cell proliferation and apoptosis in Drosophila and mammalian cells, and contains a core kinase mechanism that affects control of the cell cycle and growth. We investigated the regulatory role of two upstream components Fat and Ds on the downstream mediator Yki of the pathway, and provide explanations to some of the seemingly contradictory experimental results. We found that a number of non-intuitive experimental results can be explained by subtle changes in the balances between inputs to the Hippo pathway. Since signal transduction and growth control pathways are highly con-served across species and directly involved in tumor growth, much of what is learned about Drosophila will have relevance to growth control in mammalian systems. Our recent work on morphogen transport in the wing disc will also be discussed.

The first passage time (FPT) of a diffusive searcher to a target determines the timescale of many physical, chemical, and biological processes. While most studies focus on the FPT of a given single searcher, another important quantity in some scenarios is the FPT of the first searcher to find a target from a large group of searchers. This fastest FPT is called an extreme FPT and can be orders of magnitude faster than the FPT of a given single searcher. In this talk, we will explain recent results in extreme FPT theory and give special attention to the case of extreme FPTs to small targets. Time permitting, we will also explain results on extreme FPTs of subdiffusion modeled by a fractional time derivative and superdiffusion modeled by a fractional Laplacian.

Collective cell movement occurs throughout biology and medicine and there are many common features shared across different areas. I will review work we have carried out over the past few years on (i) systematically deriving a PDE model for tumour angiogenesis from a discrete formulation and comparing this model with the classical, phenomenological snail-trail model; (ii) agent-based models for cranial neural crest cell migration in a collabo-ration with experimental biologists that has revealed a number of new biological insights.