Applied Mathematics Seminar
University of Leeds
Welcome to the Leeds Applied Mathematics Seminar
Autumn 2019
Wednesday 2 October 2019 at 12noon
Uwe Täuber (Virginia Tech)
Stochastic spatial predator-prey models
Stochastic, spatially extended models for predator-prey interaction display striking spatio-temporal structures. These spreading activity fronts induce persistent correlations between predators and prey that can be studied through field-theoretic methods. Introducing local restrictions on the prey population induces predator extinction. The critical dynamics at this continuous absorbing state transition are governed by the scaling exponents of directed percolation. I will also address the influence of spatially varying reaction rates: Fluctuations in rare favorable regions cause a remarkable increase in both predator and prey populations. Intriguing novel features are found when variable interaction rates are affixed to individual particles rather than lattice sites. The ensuing stochastic dynamics combined with inheritance rules causes rapid time evolution for the rate distributions, with however overall neutral effect on stationary population densites. I will finally discuss noise-induced spontaneous pattern formation in systems with three cyclically competing species akin to spatial rock-paper-scissors games.
Monday 14 October 2019
Michael McIntyre (University of Cambridge)
Wave-vortex interactions, remote recoil, the Aharonov–Bohm effect and the Craik–Leibovich equation
Three examples of non-dissipative yet cumulative interaction between a wavetrain and a vortex are analysed, with a focus on effective recoil forces, local and remote. Local recoil occurs when the wavetrain overlaps the vortex core. All three examples comply with the pseudomomentum rule. The first two examples are two-dimensional and non-rotating (shallow water or gas dynamical), and the third is rotating, with deep-water gravity waves inducing an Ursell `anti-Stokes flow'. The Froude or Mach number, and the Rossby number in the third example, are assumed small. Remote recoil is all or part of the interaction in all three examples, except in one special limiting case. That case is found only within a severely restricted parameter regime and is the only case in which, exceptionally, the effective recoil force can be regarded as purely local and identifiable with the celebrated Craik–Leibovich vortex force—which corresponds, in the quantum fluids literature, to the Iordanskii force due to a phonon current incident on a vortex. Another peculiarity of that exceptional case is that the only significant wave refraction effect is the Aharonov–Bohm topological phase jump. (Preprint at https://arxiv.org/abs/1901.11525).
Monday 28 October 2019
Carola-Bibiane Schönlieb (University of Cambridge)
From differential equations to deep learning for image processing
How mathematics can help to conserve paintings and trees
Images are a rich source of beautiful mathematical formalism and analysis. Associated mathematical problems arise in functional and non-smooth analysis, the theory and numerical analysis of partial differential equations, harmonic, stochastic and statistical analysis, and optimisation. Starting with a discussion on the intrinsic structure of images and their mathematical representation, in this talk we will learn about some of these mathematical problems, about variational models for image analysis and their connection to partial differential equations and deep learning. The talk is furnished with applications to art restoration, forest conservation and cancer research.
Monday 11 November 2019
Lucas Goehring (Nottingham Trent University)
Salt polygons and groundwater convection
From fairy circles to patterned ground and columnar joints, natural patterns spontaneously appear in many complex geophysical settings. Here, we explain the origins of polygonally patterned crusts of salt playa and salt pans. These beautifully regular features, approximately a meter in diameter, are found worldwide and are fundamentally important to the transport of salt and dust in arid regions, yet there has been no convincing mechanism known for their formation. We show that they are the surface expression of buoyancy-driven convection in the porous soil beneath a salt crust. By combining consistent results from direct field observations, analogue experiments, linear stability theory, and numerical simulations, we further determine the conditions under which salt polygons will form, as well as how their characteristic size emerges.
Monday 16 December 2019
Tomos David (University of Oxford)
An ensemble modelling method for exploring palaeo-tides: middle Devonian tides and their influence on early tetrapods
It has been hypothesized that the near equal angular radii of the Sun and Moon influenced the evolution of the early tetrapods via the creation of upper inter-tidal zones. In order to test this hypothesis it is necessary to use hydro-dynamical modelling to infer what the tides looked like during the middle Devonian period. Due the large and unconstrained uncertainties in bathymetric reconstructions for that time period a statistical approach is taken. Despite the tides being some of the most predicable components of present day ocean dynamics, on large, geological, time scales we must think statistically. In this study we present an ensemble modelling framework for paleo-tidal computations. We compute the tides for an ensemble of $N$ randomly perturbed Devonian bathymetric reconstructions and examine their statistics. This preliminary study indicates the presence of robust features of the Devonian tides, supporting the suggestion of their importance to the evolution of early Tetrapods.