A numerical study of attraction/repulsion collective behavior models: 3D particle analyses and 1D kinetic simulations
Highlights
► Numerical study of an adimensionalized recent model describing collective behavior. ► Systematic 3D particle simulations (system of ODEs) in an unbounded domain. ► Systematic 1D periodic kinetic simulations (PDE). ► Similarities and differences with the 2D studies pointed out.
Introduction
Collective behaviors arise when a set of individuals organize into macroscopically observable patterns without the active role of a leader, rather by self-organization. As examples we can give the diffusion of languages in primitive societies [1], the averaging of prices in a stock exchange, or in biology the movement of bird flocks [2], [3], fish schools [4], [5], [6], insect swarms [7], sheep herds and even some micro-organisms [8]. Application of collective behavior theory includes various fields: for instance, in engineering, it can be used to coordinate robots or autonomous vehicles for unmanned operations, see [9], [10], [3], [11], [12], [13] and included references. Another example is their use in sociology to predict criminal behavior, see [14] for instance.
Depending on the number of agents taken into account, microscopic [15], [16], [17], [18] or macroscopic [19], [20], [21], [22] models can be used: the first ones go by the name of discrete, or particle, or individual-based models, and in them the agents are numbered and identified by their position and velocity, plus any other feature which might be considered useful for the proposed goal, for instance the size of cells or the age of human beings; moreover, the models must include the dynamics, the rules describing the behavior of the agents, like the tendency animals have to group together, nonetheless repelling each other when they are too close. When the number of agents is very large, for example when studying the migration of fish schools involving millions of individuals, numerical simulations become unaffordable, therefore macroscopic models have to be used, divided essentially into two categories: kinetic and hydrodynamic. The macroscopic models might be written directly following phenomenological rules [2], [20], [19], [23]. Otherwise, kinetic models can be derived from individual-based models, as in [24], [25], [26], [27], [28] where a formal derivation from the microscopic model is obtained, and qualitative properties are studied. In [29], the authors are able to reduce the dimensionality of the mesoscopic model by constraining the velocities on a sphere , in the spirit of the Vicsek model [30]. Hydrodynamic models, also derived from individual-based models, are used to reduce even more the dimensionality of the system; see [31], [27], [26], [32], [33], [34], [35].
The goal of this work is to investigate numerically a model reflecting self-propulsion, friction and the attraction/repulsion phenomena. The particles we are considering can model a variety of physical and biological situations, such as fish, polymers, and so on. One can find a review in [34]. Let be the time variable and the dimension of spaces for position and velocity. Assuming that there are particles of masses , Newton’s second law reads, for all , where the potential is defined as: with , . As in the gravitational or Coulombian models of interaction, we adopt a quadratic dependency on the mass for the attraction/repulsion force. From now on, we assume that the particles are indiscernible (), so that the total mass is the only mass parameter of the system; furthermore, we shall normalize the system so that in our simulations. The self-propulsion and friction constants per mass unit and , the characteristic lengths of attraction and repulsion and and the amplitude constants and are given data.
We are interested in comparing the numerical results obtained in this microscopic description with simulations derived at a mesoscopic level. In other words, based on a kinetic equation, we will describe the evolution of the particles from a statistical point of view as a function of the time , the position and the velocity . Let be the distribution function of the particles, that is, at time , in a volume , there are particles. Let be the macroscopic concentration of particles at time and position . The kinetic equation, obtained from (1) as a mean-field limit [36], reads with . We assume that , that is, a probability measure with finite first moment, and we moreover assume that it has a compact support. It was proved by Cañizo, Carrillo and Rosado [32] that, at least in the case , the solution lies in . The model was originally studied for the Morse potential () in [17] for particles and later on, in [36] at a kinetic level.
The effect of the self-propulsion/friction part will be to fix the modulus of the velocities to , and that of the Morse potential to make the particles repel each other when they are too close, and attract when they are far, roughly fixing relative distances corresponding to its minimum and thus favoring the formation of crystalline structures. According to [17], the dimensionless quantities of interest are and , that lead to the formation of different patterns. We study how the hierarchy of three other quantities, namely the characteristic times associated to the transport term , to the self-propulsion/friction term and to the Morse term , is also relevant in order to predict the behavior. Of course, in order to present a readable approach of this multiparameter model, we focused our attention on a restricted number of cases.
This paper is organized as follows: in Section 2 our reference model at particle level is introduced, adimensionalized and a numerical study on its properties is performed in the 3D setting paying attention to the fact that the dimension is higher than what can be found in the literature and the potential is rescaled; in Section 3 we detail the kinetic model corresponding to the discrete one as the mean-field limit , we propose a numerical scheme to solve it and perform numerical tests in the 1D periodic case; finally, in Section 4 we present our conclusions and plans for the future.
Section snippets
The particle model
In this section we describe in detail the particle model and perform a numerical analysis to have an intuition on the behavior in the limit for different regimes; this helps us understand whether the continuum model is meaningful or not. We set ourselves in the 3D space .
General guide rules
In this section we want to study numerically the solutions to the 1D kinetic version of the model, with periodic boundary conditions, modeling the behavior of the school on a ring of fixed radius ( should be thought of as an arc length).
Indeed, it was shown in the particle simulations that, for example for MRK-VII-2, the particles tend to turn on a donut shape. It therefore seems adequate to study the long-time behavior of the
Conclusions and future plans
In this paper, we have implemented a 3D solver at particle level and a 1D solver on a ring (i.e. with periodic boundaries) at kinetic level. We have given a key to interpret the qualitative behavior of the system based on ranking the typical times for the free motion, the self-propulsion/friction term and the attractive/repulsive term. We have performed a numerical analysis on the 3D particle system, to show the emergence of several different patterns (clumps, rings, mills, rigid-body
Acknowledgments
The authors would like to thank Professor José Antonio Carrillo for encouraging them and for many fruitful and interesting discussions and Professor Andrea L. Bertozzi for her enlightening comments. PL was partly supported by the University of Lille 1, the EPI SIMPAF, INRIA Lille Nord Europe and the MAS Lab. at Ecole Centrale Paris. JR was partially supported by the FSMP, CBDif, ARO MURI grant W911NF-11-1-0332, NSF grant EFRI-1024765 and NSF grant DMS-0907931. FV acknowledges the MINECO project
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