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Refining Self-Propelled Particle Models for Collective Behaviour

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Refining Self-Propelled Particle Models for Collective Behaviour CANADIAN APPLIED MATHEMATICS QUARTERLY Volume 18, Number 3, Fall 2010 REFINING SELF-PROPELLED PARTICLE MODELS FOR COLLECTIVE BEHAVIOUR CHRISTIAN A. YATES, RUTH E. BAKER, RADEK ERBAN AND PHILIP K. MAINI ABSTRACT. Swarming, schooling, flocking and herding are all names given to the wide variety of collective behaviours exhibited by groups of animals, bacteria and even individual cells. More generally, the term swarming describes the be- haviour of an aggregate of agents (not necessarily biological) of similar size and shape which exhibit some emergent property such as directed migration or group cohesion. In this paper we review various individual-based models of collective behaviour and discuss their merits and drawbacks. We further analyse some one-dimensional models in the context of locust swarm- ing. In specific models, in both one and two dimensions, we demonstrate how varying parameters relating to how much at- tention individuals pay to their neighbours can dramatically change the behaviour of the group. We also introduce leader individuals to these models. Leader individuals have the ability to guide the swarm to a greater or lesser degree as we vary the parameters of the model. Finally, we consider evolutionary sce- narios for models with leaders in which individuals are allowed to evolve the degree of influence neighbouring individuals have on their subsequent motion. 1 Introduction Often we need only look out of the window to see the principles of collective behaviour at work; in a flock of birds for example. Other collective behaviours which capture our imaginations are those of fish schools, locust plagues and bee swarms. Sometimes we almost forget that, as humans in crowded environments, we exhibit some of the most interesting collective behaviours of any species [14]. As well as the aesthetically pleasing aspect of watching a swarm in motion, studying collective behaviour has practical applications: understand- ing fish schooling can lead to more well developed fishing strategies; a knowledge of the way locusts interact and stay together in devastat- ingly large, coherent groups may shed light on potential strategies which Copyright c Applied Mathematics Institute, University of Alberta. 299 300 C. A. YATES ET AL. may be used to disrupt these groups and halt the swarm's destructive progress [8, 72]; studying how humans interact in crowds can be key in devising and implementing efficient and safe crowd control policies, which will help to avoid fatalities in possibly overcrowded situations such as football grounds, music concerts and shopping centres [14]. Over the course of the last twenty-five years several so called `self- propelled particle' (SPP) models (a specific class of individual-based model (IBM)) have been introduced and analysed, in one dimension [22], in higher dimensions [15, 19, 69], both without and with leaders [16, 41, 61], with and without internally or externally generated noise and incorporating more biologically realistic interaction rules [2, 15, 31, 39, 49, 60]. These models attempt to replicate naturally observed phenomena from not only animal groups but other self-propelled inter- acting groups of individuals such as robots. Several attempts have been made to evolve the characteristics of such models [63] and to compare the evolved characteristics with those of actual animal swarms [71] in order to understand better the possible mechanisms by which these char- acteristics may have evolved. Many of the models have been found, at least qualitatively, to replicate some of the complex emergent behaviours exhibited by these groups. In this paper we aim to give a brief overview of some of the above classes of SPP models or IBM. We will discuss a paradigmatic example of each in more detail, illustrating the highlights and pitfalls. For some of the most important and seminal models [22, 69] we derive novel re- sults pertaining to phase transitions as parameters of the model vary (see Sections 2 and 3). We also consider the effects of evolving these SPP models in the presence of an `informed individual' allowing param- eters to change in subsequent generations of particles in order to find stable parameter values in a manner analogous to [71]. The overview we give is by no means an exhaustive review of SPP models or IBMs. We have selected a variety of archetypal models which cover a wide range of biological situations and modelling regimes but there are, of course, several areas which we will not have space to discuss. These include traffic [36, 37] and human crowd modelling [38, 68] to name but two. The remainder of this paper is structured as follows. In Section 2 we discuss the most basic SPP model, formulated in one dimension by Czir´ok et al. [22]. After discussing alterations to the model we go on to consider suitable application areas in the specific context of locust swarming. We present higher dimensional models in Section 3. We be- gin by discussing the simple model proposed by Vicsek et al. [69] and show that complex emergent behaviours are possible given even the sim- REFINING SELF-PROPELLED PARTICLE MODELS 301 plest of interaction rules in two dimensions. We consider alterations to the model and summarise the analysis of the potential convergence of the individuals in the model to a common direction. We also consider the archetypal model of [19] in both a discrete and continuous formulation and the biologically motivated model of [15]. In Section 4 we introduce the concept of leadership in SPP models. We analyse three different types of leadership in the three paradigmatic models previously intro- duced and discuss possible effects these leader particles can have upon the behaviour of the group as a whole. Finally we discuss the concept of evolving SPP models in order to test theories about the selective pres- sures acting on individuals which give rise to specific behavioural types. We combine the work of previous sections by introducing evolvable pa- rameters to SPP models in one and two dimensions, studying how the parameters vary as a result of each individual's ability to follow an in- formed leader. We conclude with a short discussion of the current status of SPP modelling and possible avenues for further research. 2 One-dimensional models The simplest case of a SPP model is the Czir´ok model in one dimension. The system studied by Czir´ok et al. [22], for example, consists of N autonomous agents on a line of length L with periodic boundary conditions. The ith particle, with lateral po- sition xi moves in one direction or the other along the line with variable speed ui. The velocity of each agent is updated by considering an av- erage of its own velocity and those of its `neighbours'. The neighbours of agent i are those particles which lie within a distance r of agent i's position, xi. The interaction radius, r, is a predefined constant which, in the original model, was chosen to be the same for each particle in the r simulation. We use i to denote the set of neighbours of particle i, ex- J r cluding itself, and ni(t) = is the number of these neighbours. The jJi j particles are initialised with a random initial position, xi [0; L), and 2 a random initial velocity, ui 1; 1 . Their positions and nondimen- sionalised velocities are evolv2ed{−usinggthe following two rules, identical for each particle, i = 1; 2; : : : ; N, (2.1) xi(t + 1) = xi(t) + v0ui(t); (2.2) ui(t + 1) = G( u(t) i) + ∆Qi; f h i g 302 C. A. YATES ET AL. and 1 (2.3) u i = ui + uj : h i ni + 1 j2J r Xi u i is the mean of the nondimensionalised velocity of the particles in the h i interaction radius of particle i and v0 is a velocity scaling necessitated by choosing a specific time-step. The role of the function G is to scale the average nondimensional velocity measured by each particle towards unity. A generalisation of the odd function, G, representing this velocity adjustment, is as follows: z + β sign(z) (2.4) G(z) = ; 1 + β where 1; for z > 0; sign(z) = 8 0; for z = 0; <> 1; for z < 0; − and in [22] β is taken to be 1.:> ∆Qi represents noise drawn from a uniform distribution with support [ η=2; η=2] where η will be referred to as the amplitude of the noise. In this,− original, statement of the Czir´ok model (or adapted versions thereof) the time-step is simply taken to be unity, which avoids the necessity of scaling the noise with the time- step, but leaves the model with one fewer degree of freedom. When recapitulating the models we prefer to state them with generality and consistency, which necessitates, in some cases, the introduction of an arbitrary time-step, ∆t: (2.5) xi(t + ∆t) = xi(t) + ∆tvi(t) (2.6) vi(t + ∆t) = vi(t) + ∆t G( v(t) i) vi(t) + ∆Qi: f h i − g Note that vi now represents the dimensional velocity of particle i. The noise parameter η should scale with the square root of the time-step, ∆t. We note that the choice of distribution from which the noise is drawn is perhaps less than optimal. Mathematically, at least, it would make more sense to consider normal distributions, especially when con- sidering ensemble properties of the group, such as the average velocity. In order to obtain an update equation for the average velocity of the group, we must sum equation (2.6) over all individuals. Normal noise REFINING SELF-PROPELLED PARTICLE MODELS 303 has the convenient property that the sum of two (or more) normal dis- tributions is still normal, whereas the sum of two uniform distributions gives the nonstandard, nondifferentiable triangle distribution.
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