Collective Motion Under Non-Reciprocal Pairwise Interactions

This project, to investigate the effect on non-reciprocal inter-particle forces on long-term collective motion, is motivated by collaboration with experimental behavioural ecologists in Exeter. Their particles of interest are Trinidadian Guppies (Poecilia reticulata), and the non-reciprocal forces arise as a result of sexual conflict between males and females of the species. In the long-term, their behaviour also results in key ecological and evolutionary processes such as population dispersal and the invasion of alien species. Pilot experimental work and some preliminary mathematical modelling suggest that this dynamic, driven by sexual conflict, can give rise to anomalously fast diffusion of pairs of guppies. To combine the biology of pairwise interactions of Trinidadian Guppy fish through to population level consequences throughout the system will require mathematical models of processes at different time, spatial and social scales, with assumptions founded in empirical evidence from the experimental data.

The aim of this project is to develop a general mathematical framework to interpolate between and extrapolate from one social, spatial and temporal scale to the next, modelling the effect of non-reciprocal (and reciprocal) social forces on large-scale population processes, movement and structure. In biological terms this will sit in the intersection of movement ecology and collective behaviour, and motivates the need to address the 'problem of scale' - understanding how individual level interactions and decisions, feed into and drive patterns and processes at the population level over time. As the movement of guppies is in part stochastic, and they live in 'fission-fusion' societies - with loose groups subject to rapid coagulation and fragmentation, a wide range of mathematical tools and techniques will be needed to generate a suite of models.

Likely approaches include but are certainly not limited to: a data-integrative agent-based model to develop and test the effects of measured and modelled local interactions on larger-scale outcomes. Changes in position and state will occur in response to the environment and other nearby individuals, with noise added to account for uncertainty; coagulation-fragmentation model of group dynamics, in which spatial coarse graining would map from individual dynamics to a stochastic PDE describing population density within small groups. Feedback between the group configuration and the motion of individuals within it means this equation will be of the McKean-Vlasov type. This model should yield effective rates of group formation and dissolution to inform a description of the population as an exchangeable fragmentation coagulation process (EFCP); and population level models, obtained by taking large scaling limits either directly from the McKean-Vlasov SPDE description, or from the EFCP (agreement between these results will be an important consistency check for our modelling). In either case, emergent population level properties can be predicted. This includes quantities such as the global speed of dispersal in populations with different concentrations of reciprocal and non-reciprocal dynamics, or possible emergent phenotypic gradients in expanding populations.

Combining specialised modelling techniques with complex data analysis in order to deliver prediction with quantified uncertainties lies at the heart of many of the major challenges facing UK industry and society over the next decades. Indeed, the recent Government Office for Science report "Computational Modelling, Technological Futures, 2018" specifies putting the UK at the forefront of the data revolution as one of their Grand Challenges.

The beneficiaries of our research portfolio will include a wide range of UK industrial sectors such as the pharmaceutical industry, risk consultancy, telecommunications and advanced materials, as well as government bodies, including the NHS, the Met Office and the Environment Agency.

Examples of current impactful projects pursued by students and in collaboration with stake-holders include:

- Using machine learning techniques to develop automated assessment of psoriatic arthritis from hand X-Rays, freeing up consultants' time (with the NHS).

- Uncertainty quantification for the Neutron Transport Equation improving nuclear reactor safety (co-funded by Wood).

- Optimising the resilience and self-configuration of communication networks with the help of random graph colouring problems (co-funded by BT).

- Risk quantification of failure cascades on oil platforms by using Bayesian networks to improve safety assessment for certification (co-funded by DNV-GL).

- Krylov regularisation in a Bayesian framework for low-resolution Nuclear Magnetic Resonance to assess properties of porous media for real-time exploration (co-funded by Schlumberger).

- Machine learning methods to untangle oceanographic sound data for a variety of goals in including the protection of wildlife in shipping lanes (with the Department of Physics).

Future committed partners for SAMBa 2.0 are: BT, Syngenta, Schlumberger, DNV GL, Wood, ONS, AstraZeneca, Roche, Diamond Light Source, GKN, NHS, NPL, Environment Agency, Novartis, Cytel, Mango, Moogsoft, Willis Towers Watson.

SAMBa's core mission is to train the next generation of academic and industrial researchers with the breadth and depth of skills necessary to address these challenges. SAMBa's most sustained impact will be through the contributions these researchers make over the longer term of their careers. To set the students up with the skills needed to maximise this impact, SAMBa has developed a bespoke training experience in collaboration with industry, at the heart of its activities. Integrative Think Tanks (ITTs) are week-long workshops in which industrial partners present high-level research challenges to students and academics. All participants work collaboratively to formulate mathematical
models and questions that address the challenges. These outputs are meaningful both to the non-academic partner, and as a mechanism for identifying mathematical topics which are suitable for PhD research. Through the co-ownership of collaboratively developed projects, SAMBa has the capacity to lead industry in capitalising on recent advances in mathematics. ITTs occur twice a year and excel in the process of problem distillation and formulation, resulting in an exemplary environment for developing impactful projects.

SAMBa's impact on the student experience will be profound, with training in a broad range of mathematical areas, in team working, in academic-industrial collaborations, and in developing skills in communicating with specialist and generalist audiences about their research. Experience with current SAMBa students has proven that these skills are highly prized: "The SAMBa approach was a great template for setting up a productive, creative and collaborative atmosphere. The commitment of the students in getting involved with unfamiliar areas of research and applying their experience towards producing solutions was very impressive." - Dr Mike Marsh, Space weather researcher, Met Office.