L3

Most of SDE models in epidemics, ecology, biology, finance etc. are highly nonlinear and do not have explicit solutions. Monte Carlo simulations have played a more and more important role. This talk will point out several well-known numerical schemes may fail to preserve the positivity or moment of the solutions to SDE models. Reliable numerical schemes are therefore required to be designed so that the corresponding Monte Carlo simulations can be trusted. The talk will then concentrate on new numerical schemes for the well-known stochastic Lotka--Volterra model for interacting multi-species. This model has some typical features: highly nonlinear, positive solution and multi-dimensional. The known numerical methods including the tamed/truncated Euler-Maruyama (EM) applied to it do not preserve its positivity. The aim of this talk is to modify the truncated EM to establish a new positive preserving truncated EM (PPTEM).

Distribution dependent equations (or McKean—Vlasov equations) have found many applications to problems in physics, biology, economics, finance and computer science. Historically, equations with either Brownian noise or zero noise have received the most attention; many well known results can be found in the monographs by A. Sznitman and F. Golse. More recently, attention has been paid to distribution dependent equations driven by random continuous noise, in particular the recent works by M. Coghi, J-D. Deuschel, P. Friz & M. Maurelli, with applications to battery modelling. Furthermore, the phenomenon of regularisation by noise has received new attention following the works of D. Davie and M. Gubinelli & R. Catellier using techniques of averaging along rough trajectories. Building on these ideas I will present recent joint work with L. Galeati and F. Harang concerning well-posedness and stability results for distribution dependent equations driven first by merely continuous noise and secondly driven by fractional Brownian motion.

Climate- and weather prediction centres such as the Met Office rely on efficient numerical methods for simulating large scale atmospheric flow. One computational bottleneck in many models is the repeated solution of a large sparse system of linear equations. Preconditioning this system is particularly challenging for state-of-the-art discretisations, such as (mimetic) finite elements or Discontinuous Galerkin (DG) methods. In this talk I will present recent work on developing efficient multigrid preconditioners for practically relevant modelling codes. As reported in a REF2021 Industrial Impact Case Study, multigrid has already led to runtime savings of around 10%-15% for operational global forecasts with the Unified Model. Multigrid also shows superior performance in the Met Office next-generation LFRic model, which is based on a non-trivial finite element discretisation.

As biological data has become more accessible and available in biology and healthcare, we find increasing opportunities to leverage artificial intelligence to help with data integration and picking out patterns in complex data. In this short talk, we will provide glimpses of what we do at the Novo Nordisk Research Centre Oxford towards understanding patient journeys, and in the use of knowledge graphs to draw insights from diverse biomedical data streams

Fibonacci numbers in plants, such as in sunflower spiral counts, have long fascinated mathematicians. For the last thirty years, most analyses have been variants of a Standard Model in which plant organs are treated as point nodes successively placed on a cylinder according to a given function of the previous node positions, not too close or too far away from the existing nodes. These models usually lead to lattice solutions. As a parameter of the model, like the diameter of the cylinder, is changed, the lattice can transition to another, more complex lattice, with a different spiral count. It can typically be proved that these transitions move lattice counts to higher Fibonacci numbers. While mathematically compelling, empirical validation of this Standard Model is as yet weak, even though the underlying molecular mechanisms are increasingly well characterised. 

In this talk I'll show a gallery of Fibonacci patterning and give a brief history of mathematical approaches, including a partially successful attempt by Alan Turing. I'll describe how the classification of lattices on cylinders connects both to a representation of $SL(2,Z)$ and to applications through defining the constraint that any model must satisfy to show Fibonacci structure. I'll discuss a range of such models, how they might be used to make testable predictions, and why this matters.

From 2011 to 2017 Jonathan Swinton  was a visiting professor to MPLS in Oxford in Computational Systems Biology. His new textbook Mathematical Phyllotaxis will be published  soon, and his Alan Turing's Manchester will be republished by The History Press in May 2022. 

 In addition, another class of cell-cell communication is by long, thin cellular protrusions that are ~100 microns (many cell-lengths) in length and ~100 nanometers (below traditional microscope resolution) in width. These protrusions have been recently discovered in many organisms, including nanotubes humans and airinemes in zebrafish. But, before establishing communication, these protrusions must find their target cell. Here we demonstrate airinemes in zebrafish are consistent with a finite persistent random walk model. We study this model by stochastic simulation, and by numerically solving the survival probability equation using Strang splitting. The probability of contacting the target cell is maximized for a balance between ballistic search (straight) and diffusive (highly curved, random) search. We find that the curvature of airinemes in zebrafish, extracted from live cell microscopy, is approximately the same value as the optimum in the simple persistent random walk model. We also explore the ability of the target cell to infer direction of the airineme’s source, finding the experimentally observed parameters to be at a Pareto optimum balancing directional sensing with contact initiation.

In collective navigation a population travels as a group from an origin to a destination. Famous examples include the migrations of birds and whales, between winter and summer grounds, but collective movements also extend down to microorganisms and cell populations. Collective navigation is believed to improve the efficiency of migration, for example through the presence of more knowledgeable individuals that guide naive members ("leader-follower behaviour") or through the averaging out of individual uncertainty ("many wrongs"). In this talk I will describe both individual and continuous approaches for modelling collective navigation. We investigate the point at which group information becomes beneficial to migration and how it can help a population navigate through areas with poor guidance information. We also explore the effectiveness of different modes through which a leader can herd a group of naïve followers. As an application we will consider the impact of noise pollution on the migration of whales through the North Sea.

Gastrulation is a critical event in vertebrate morphogenesis, characterized by coordinated large-scale multi-cellular movements. One grand challenge in modern biology is understanding how spatio-temporal morphological structures emerge from cellular processes in a developing organism and vary across vertebrates. We derive a theoretical framework that couples tissue flows, stress-dependent myosin activity, and actomyosin cable orientation. Our model, consisting of a set of nonlinear coupled PDEs, predicts the onset and development of observed experimental patterns of wild-type and perturbations of chick gastrulation as a spontaneous instability of a uniform state. We use analysis and numerics to show how our model recapitulates the phase space of gastrulation morphologies seen across vertebrates, consistent with experiments. Altogether, this suggests that early embryonic self-organization follows from a minimal predictive theory of active mechano-sensitive flows. 

 https://www.biorxiv.org/content/10.1101/2021.10.03.462928v2 

We will report on recent progress regarding the near-critical behavior of certain statistical physics models in dimension 3. Our results deal with the second-order phase transition associated to two percolation problems involving the Gaussian free field in 3D. In one case, they determine a unique ``fixed point'' corresponding to the transition, which is proved to obey one of several scaling relations. Such laws are classically conjectured to hold by physicists on the grounds of a corresponding scaling ansatz.