1 Introduction

The physical world is filled with systems of interacting agents, from the nanoscale of atomic interactions to the terascale of galaxies filled with stars all exerting a force on one another. Such systems are not limited to physics; biology also provides many examples of complex emergent behaviour such as the slime mould, Dictyostelium. These amoebae coalesce to form slugs consisting of thousands of individuals to move towards an ideal site for the formation of a “fruiting body” as part of their life cycle [2]. Beyond the microsopic scale, “safety in numbers” rules the seas and the skies, with starlings flocking to avoid predation, bees swarming to find new nest sites and fish shoaling to spawn [3]. Even human behaviour can be seen through this lens: traffic jams, opinions and segregation can all be thought of as interacting particle systems [4, 5]. Although in the physical world the laws governing these systems are generally well understood, the same cannot be said of the biological world where common events like animal migration are still yet to be fully explained. By viewing these phenomena as interacting particle systems, we aim to model these interactions and recreate the emergent behaviours in silico. This, in combination with an analytical approach, will lead to better understanding of these complex events.

Many attempts have been made to explain the mechanics of these systems, and they broadly fall in to two categories: kinetic models and particle models. Particle models, also known as agent-based models, are favoured by many as they are simple to create and simulate. One can easily develop a set of rules to be followed by individuals and then simulate the behaviour of a group. This was done to great effect by Ballerini et al. who were able to recreate the behaviour of a flock of starlings using very few rules [6]. Kinetic models on the other hand, are obtained in the limit as the number of particles tends to infinity and they describe the evolution of the particles’ density rather than the behaviour of each single agent in the system. It is this aggregation across particles that has seen this method be of great use in statistical mechanics. However, when the interaction between particles becomes more complex, so too does the corresponding PDE which often becomes nonlocal and nonlinear.

The methods are thus inextricably linked, however the behaviour of the two can be markedly different, as we will show in this report. We will begin to look at the differences that arise in these two approaches by focussing on one such model: that of Buttà et al. [7]. We are particularly interested in the longtime dynamics of this model. The report is structured as follows:

In Section 2 we introduce the main models of our study and present some analytic results. We also set out the main aims of our work and this report.

In Section 3, we numerically simulate the models introduced to demonstrate their behaviour and aim to answer questions raised by the analysis.

In Section 4, we briefly discuss some related models from the literature and suggest directions for our future research.

References

[2] Bonner, J. T. (2009). The social amoebae: The biology of cellular slime molds. Princeton University Press.

[3] Sumpter, D. J. T. (2010). Collective animal behavior. Princeton University Press.

[4] Schelling, T. C. (1971). Dynamic models of segregation. The Journal of Mathematical Sociology 1 143–86.

[5] Hegselmann, R. and Krause, U. (2002). Opinion dynamics and bounded confidence: Models, analysis and simulation. Journal of Artificial Societies and Social Simulation 5 1–24.

[6] Ballerini, M., Cabibbo, N., Candelier, R., Cavagna, A., Cisbani, E., Giardina, I., Orlandi, A., Parisi, G., Procaccini, A., Viale, M. and Zdravkovic, V. (2008). Empirical investigation of starling flocks: A benchmark study in collective animal behaviour. Animal Behaviour 76 201–15.

[7] Buttà, P., Flandoli, F., Ottobre, M. and Zegarlinski, B. (2019). A non-linear kinetic model of self-propelled particles with multiple equilibria. Kinetic & Related Models 12(4) 791–827.