BB12: Understanding collective behaviour of systems of interacting particles

Background

We need only look out of the window to see collective behaviour – for example, in a flock of birds. In this project, we investigate systems of many interacting particles (individuals) with the aim to better understand their collective behaviour. We attempt to answer: How do local interactions between components of a system determine the behaviour of the whole system? What changes can be made in the individual-level model to achieve a desired change in the collective behaviour of the system?

Techniques and Challenges

Our research focuses on specific examples – bacterial cells that interact by releasing extracellular chemical signals, and an ion channel occupied by ions that interact through electrical forces – with the intent to explore applications of the developed mathematical approaches to other relevant systems. The mathematical techniques used include partial differential equations (PDEs), perturbation theory, numerical methods and individual-based computer simulations.

Results

First, we concentrated on hybrid models that combined agent-based stochastic processes with PDEs. We studied hybrid models of chemotaxis, where bacteria are modelled as particles and extracellular molecules are described by PDEs and found a qualitative difference in behaviour between the hybrid model and the corresponding continuum model [11/20]. To gain a deeper understanding of the difference, we then studied pattern formation in simplified models of chemotaxis [11/04]. We also analysed the possibility of travelling wave solutions in systems with a non-singular chemotactic sensitivity [10/15]. We have begun extending a simulation algorithm that allows the modelling of one species through different methods in different parts of a computational domain [11/15].

The Future

We hope to further increase our understanding of hybrid models in chemotaxis; specifically we want to investigate the effects that the introduction of a second length-scale has on the system. We also aim to generalise the hybrid simulation algorithm for reaction-diffusion systems in higher dimensions and for higher-order reactions, and to eventually introduce charged particles to use this type of model for an ion-channel system.

References

[11/20] Franz B., Erban R.: Hybrid modelling of individual movement and collective behaviour: Dispersal, individual movement and spatial ecology: A mathematical perspective (book chapter)

[11/15] Flegg M.B., Chapman S.J., Erban R.: The Two Regime method for optimizing stochastic reaction-diffusion simulations

[11/04] Erban R., Haskovec, J.: From individual to collective behaviour of coupled velocity jump processes - a locust example

[10/15] Xue C., Hwang H.J., Painter K.J., Erban R.: Travelling waves in hyperbolic chemotaxis equations, Bulletin of Mathematical Biology

[09/46] Yates C.A., Baker R.E., Erban R., Maini P.K.: Refining self-propelled particle models for collective behaviour, Canadian Applied Mathematics Quarterly

[09/06] Yates C. A., Erban R., Escudero C., Couzin I.D., Buhl J., Kevrekidis I.G., Maini P.K., Sumpter D.J.T.: Inherent noise can facilitate coherence in collective swarm motion, PNAS (Proceedings of the National Academy of Sciences of the United States of America)

Erban R., Othmer H.: From individual to collective behavior in bacterial chemotaxis, SIAM Journal on Applied Mathematics, Volume 65, Number 2, pp. 361-391, 2004

This project is funded by the European Research Council Starting Independent Researcher Grant awarded to Dr Erban.