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.