Narrow Results

In this work, we consider the phase transition from ordered to disordered states that occur in the Vicsek model of self-propelled particles. This model was proposed to describe the emergence of collective order in swarming systems. When noise is added to the motion of the particles, the onset of collective order occurs through a dynamical phase transition. Based on their numerical results, Vicsek and his colleagues originally concluded that this phase transition was of second order (continuous). However, recent numerical evidence seems to indicate that the phase transition might be of first order (discontinuous), thus challenging Vicsek's original results. In this work, we review the evidence supporting both aspects of this debate. We also show new numerical results indicating that the apparent discontinuity of the phase transition may in fact be a numerical artifact produced by the artificial periodicity of the boundary conditions.

In this paper, an adaptive attractive/repulsive (A/R) swarming model is proposed to explore the role of self-organized formation in swarming systems. By defining the adjustable A/R range γ_i, which is affected by the localized steady state of agents, the standard collective crystal-like swarming formations are straightforwardly unfolded in different scale. Meanwhile, with numerical simulations and analyses, the results show that the adaptive A/R swarming model provides an effective solution to the current existing dilemma of the collective liquid-like formation with unexpected neighbor distances and the split crystal-like formation. The actual neighbor distance of the adaptive A/R model could converge to the expected neighbor distance as planned, based on the different settings of the expected neighbor distance and the A/R range. Moreover, such adjustable A/R swarming formations may find their potential applications such as the formation of self-organized multi-robots and unmanned aerial vehicles, the automatic networking of sensors, etc.

We present existence, uniqueness and continuous dependence results for some kinetic equations motivated by models for the collective behavior of large groups of individuals. Models of this kind have been recently proposed to study the behavior of large groups of animals, such as flocks of birds, swarms, or schools of fish. Our aim is to give a well-posedness theory for general models which possibly include a variety of effects: an interaction through a potential, such as a short-range repulsion and long-range attraction; a velocity-averaging effect where individuals try to adapt their own velocity to that of other individuals in their surroundings; and self-propulsion effects, which take into account effects on one individual that are independent of the others. We develop our theory in a space of measures, using mass transportation distances. As consequences of our theory, we show also the convergence of particle systems to their corresponding kinetic equations, and the local-in-time convergence to the hydrodynamic limit for one of the models.

This brief note presents the papers published in a special issue devoted to complex systems in life sciences. Out of the set of papers some perspective ideas on conceivable future researches are extracted and brought to the attention of the readers. The final ambitious aim is to contribute to the development of a mathematical theory for complex living systems.

This paper deals with the modeling and simulation of swarms viewed as a living, hence complex, system. The approach is based on methods of kinetic theory and statistical mechanics, where interactions at the microscopic scale are nonlinearly additive and modeled by stochastic games.

This brief note is an introduction to the papers published in this special issue devoted to complex systems in life sciences. Out of this presentation some perspective ideas on conceivable future research objectives are extracted and brought to the reader's attention. The final (ambitious) aim is to develop a mathematical theory for complex living systems.

We perform an asymptotic analysis of general particle systems arising in collective behavior in the limit of large self-propulsion and friction forces. These asymptotics impose a fixed speed in the limit, and thus a reduction of the dynamics to a sphere in the velocity variables. The limit models are obtained by averaging with respect to the fast dynamics. We can include all typical effects in the applications: short-range repulsion, long-range attraction, and alignment. For instance, we can rigorously show that the Cucker–Smale model is reduced to a Vicsek-like model without noise in this asymptotic limit. Finally, a formal expansion based on the reduced dynamics allows us to treat the case of diffusion reducing the Cucker–Smale model with diffusion to the non-normalized Vicsek model as in Ref. 29. This technique follows closely the gyroaverage method used when studying the magnetic confinement of charged particles. The main new mathematical difficulty is to deal with measure solutions in this expansion procedure.

In the present paper we study the macroscopic limits of a kinetic model for interacting entities (individuals, organisms, cells). The kinetic model is one-dimensional and the entities are characterized by their position and orientation (+/-) with swarming interaction controlled by a sensitivity parameter. The macroscopic limits of the model are considered for solutions close either to the diffusive (isotropic) or to the aligned (swarming) equilibrium states for various values of that parameter. In the former case the classical linear diffusion equation results whereas in the latter a traveling wave solution does both in the zeroth ("Euler") and first ("Navier–Stokes") order of approximation.

The hydrodynamic limit of a kinetic Cucker–Smale flocking model is investigated. The starting point is the model considered in [Existence of weak solutions to kinetic flocking models, SIAM Math. Anal.45 (2013) 215–243], which in addition to free transport of individuals and a standard Cucker–Smale alignment operator, includes Brownian noise and strong local alignment. The latter was derived in [On strong local alignment in the kinetic Cucker–Smale equation, in Hyperbolic Conservation Laws and Related Analysis with Applications (Springer, 2013), pp. 227–242] as the singular limit of an alignment operator first introduced by Motsch and Tadmor in [A new model for self-organized dynamics and its flocking behavior, J. Stat. Phys.141 (2011) 923–947]. The objective of this work is the rigorous investigation of the singular limit corresponding to strong noise and strong local alignment. The proof relies on a relative entropy method. The asymptotic dynamics is described by an Euler-type flocking system.

The asymptotic analysis of kinetic models describing the behavior of particles interacting through alignment is performed. We will analyze the asymptotic regime corresponding to large alignment frequency where the alignment effects are dominated by the self-propulsion and friction forces. The former hypothesis leads to a macroscopic fluid model due to the fast averaging in velocity, while the second one imposes a fixed speed in the limit, and thus a reduction of the dynamics to a sphere in the velocity space. The analysis relies on averaging techniques successfully used in the magnetic confinement of charged particles. The limiting particle distribution is supported on a sphere, and therefore we are forced to work with measures in velocity. As for the Euler-type equations, the fluid model comes by integrating the kinetic equation against the collision invariants and its generalizations in the velocity space. The main difficulty is their identification for the averaged alignment kernel in our functional setting of measures in velocity.

This paper addresses some preliminary steps toward the modeling and qualitative analysis of swarms viewed as living complex systems. The approach is based on the methods of kinetic theory and statistical mechanics, where interactions at the microscopic scale are nonlocal, nonlinearly additive and modeled by theoretical tools of stochastic game theory. Collective learning theory can play an important role in the modeling approach. We present a kinetic equation incorporating the Cucker–Smale flocking force and stochastic game theoretic interactions in collision operators. We also present a sufficient framework leading to the asymptotic velocity alignment and global existence of smooth solutions for the proposed kinetic model with a special kernel. Analytic results on the global existence and flocking dynamics are presented, while the last part of the paper looks ahead to research perspectives.

The aim of this paper is first to provide a presentation of the papers published in a special issue devoted to modeling, qualitative analysis, control and simulations of large systems of living entities, viewed as active particles, related to real-life applications, and subsequently to present some prospective ideas on possible future research programs for the development of mathematical tools to model living systems.

We study emergent dynamics of the swarmalator model [K. P. O’Keeffe, H. Hong and S. H. Strogatz, Oscillators that sync and swarm, Nature Commun.8 (2017) 1504] describing the dynamic interplay of aggregation and synchronization dynamics for interacting many-particle systems. For the particle aggregation, we employ the nonlinear aggregation system with singular attractive–repulsive couplings depending on the phase differences, while we use the Kuramoto-type model with the singular coupling strength depending on the spatial distances for the dynamics of phases. We show how collective behaviors emerge from the dynamic interplay between position aggregation and phase synchronization. We introduce some sufficient framework leading to the positive minimal relative distances between particles and its uniform upper bound. We also show the convergence of position under some sufficient conditions.

We concentrate on kinetic models for swarming with individuals interacting through self-propelling and friction forces, alignment and noise. We assume that the velocity of each individual relaxes to the mean velocity. In our present case, the equilibria depend on the density and the orientation of the mean velocity, whereas the mean speed is not anymore a free parameter and a phase transition occurs in the homogeneous kinetic equation. We analyze the profile of equilibria for general potentials identifying a family of potentials leading to phase transitions. Finally, we derive the fluid equations when the interaction frequency becomes very large.

The concept of swarm comes from the biological world. Drones gather in groups of 100 or even 1000 to fly like a flock of birds, called swarms. Swarm systems satisfy several assumptions such as decentralized controls, local information and simple platforms. Swarm systems have attractive properties such as resilience, scalability, and ease of development and implementation. Swarm techniques can perform simple tasks such as moving in a coordinated direction. The flocking behavior of a group of animals that converge by local interactions toward the same heading is an example of simple consensus for decentralized dynamic systems. However, the notion of decentralized control based on local information suffers from taking into account the overall behavior of the group. For example, in a complex environment, a swarm will adapt to the presence of obstacles and congestion reactively, whereas we would like more anticipatory control. The objective of this paper is to propose a solution based on Mean Field Game (MFG) concepts to integrate macro-level knowledge at the micro-level in decentralized flocking.