Mathematics

We propose an extension of the well-known Klausmeier model of vegetation to two plant species that consume water at different rates. Rather than competing directly, the plants compete through their intake of water, which is a shared resource between them. In semi-arid regions, the Klausmeier model produces vegetation spot patterns. We are interested in how the competition for water affects co-existence and stability of patches of different plant species. We consider two plant types: a thirsty species and a frugal species, that only differ by the amount of water they consume, while being identical in all other aspects. We find that there is a finite range of precipitation rate for which two species can co-exist. Outside of that range, (when the rate is either sufficiently low or high), the frugal species outcompetes the thirsty species. As the precipitation rate is decreased, there is sequence of stability thresholds such that thirsty plant patches are the first to die off, while the frugal spots remain resilient for longer. The pattern consisting of only frugal spots is the most resilient. The next-most-resilient pattern consists of all-thirsty patches, with the mixed pattern being less resilient than either of the homogeneous patterns. We also examine numerically what happens for very large precipitation rate. We find that for sufficiently high rate, the frugal plant takes over the entire range, outcompeting the thirsty plant.

The stability and dynamic properties of spike-type solutions to the Gierer- Meinhart equations are well understood. We examine the effect of adding noise to the equations on the spike-dynamics. We derive a stochastic ordinary differential equation for the motion of a single spike as well as the distribution of spike location from the associated Fokker-Plank equation. With sufficiently large amplitude noise, it is possible for the spike to reach the boundary of the domain and become effectively trapped for some time. In this case, we calculate the expected time to capture.

The Phase-Field-Crystal (PFC) model is a simple yet surprisingly useful model for successfully capturing the phenomenology of grain growth in polycrystalline materials. PFC models are variational with a free energy functional which is very similar (in some cases, identical) to the well-known Swift-Hohenberg free energy. In this talk, we will discuss the simplest PFC functional and its gradient flow.

The first part of the talk will focus on large scales and address the model’s uncanny ability to o capture certain features of grain growth. We introduce a novel atom-based grain extraction and measurement method, and then use it to provide a comparison of multiple statistical grain metrics between (i) PFC simulations, (ii) experimental thin film data for aluminum, and (iii) simulations from the Mullins model.

In the second part of the talk, we investigate the PFC energy landscape at small scales (the local arrangement of atoms). We address patterns which are numerically observed as steady states via the framework of the modern theory of rigorous computations. In doing so, we make rigorous conclusions on the existence of similar states. In particular, we show that localized patterns and grain boundaries are critical and not simply metastable states. Finally, we present preliminary work on connections and parameter continuation in the PFC system. This talk consists of work from the PhD thesis of Gabriel Martine La Boissoniere at McGill. Parts of the talk also involve joint work with S. Esedoglu (Michigan), K. Barmak (Columbia) and J.-P. Lessard (McGill).

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).