Narrow Results

A model for the displacement of self-driven organisms is studied by means of extensive computer simulations. Local interactions influenced by noisy communications among organisms, leads to the onset of collective motion at low noise levels. When the noise is increased the system undergoes first-order transitions into disordered states of motion. We have also studied the relaxation process between these states. By fitting the time dependence of the order parameter when the system is annealed from a state below coexistence to another above it, we conclude that the relaxation can be well described by means of a stretched exponential. In this way the characteristic relaxation times are obtained.

This paper focuses on the problem of finite-time flocking in the multi-agent nonholonomic system with proximity graph. With the aid of singular communication function and results from graph theory, a novel distributed control protocol is proposed, where each agent merely obtains state information of its neighbors. Based on the theory of finite-time stability, the proposed protocol can achieve flocking within a finite-time and an upper bound on the setting time can be derived. Furthermore, we can deduce several sufficient conditions to the problem under the assumption that the positions and relative distances of agents are unknown. These sufficient conditions show that there are always suitable gains that enable our proposed protocol to perform finite-time flocking control and maintain the connectivity of the graph for any given initial connected graphs. Finally, the theoretical results are validated by numerical simulations.

Very recently, a model for flocking was introduced by Cucker and Smale together with a proof of convergence. This proof established unconditional convergence to a common velocity provided the interaction between agents was strong enough and conditional convergence otherwise. The strength of the interaction is measured by a parameter β ≥ 0 and the critical value at which unconditional convergence stops holding is β = 1/2. This model was extended by Shen to allow for a hierarchical leadership structure among the agents and similar convergence results were proved. But, for discrete time, unconditional convergence was proved only for (k being the number of agents). In this note we improve on this result showing that unconditional convergence holds indeed for β < 1/2.

In this paper, we provide the O(ε) corrections to the hydrodynamic model derived by Degond and Motsch from a kinetic version of the model by Vicsek and co-authors describing flocking biological agents. The parameter ε stands for the ratio of the microscopic to the macroscopic scales. The O(ε) corrected model involves diffusion terms in both the mass and velocity equations as well as terms which are quadratic functions of the first-order derivatives of the density and velocity. The derivation method is based on the standard Chapman–Enskog theory, but is significantly more complex than usual due to both the non-isotropy of the fluid and the lack of momentum conservation.

Animal groups represent magnificent archetypes of self-organized collective behavior. As such, they have attracted enormous interdisciplinary interest in the last years. From a mechanistic point of view, animal aggregations remind physical systems of particles or spins, where the individual constituents interact locally, giving rise to ordering at the global scale. This analogy has fostered important research, where numerical and theoretical approaches from physics have been applied to models of self-organized motion. In this paper, we discuss how the physics methodology may provide precious conceptual and technical instruments in empirical studies of collective animal behavior. We focus on three-dimensional groups, for which empirical data have been extremely scarce until recently, and describe novel experimental protocols that allow reconstructing aggregations of thousands of individuals. We show how an appropriate statistical analysis of these large-scale data allows inferring important information on the interactions between individuals in a group, a key issue in behavioral studies and a basic ingredient of theoretical models. To this aim, we revisit the approach we recently used on starling flocks, and apply it to a much larger data set, never analyzed before. The results confirm our previous findings and indicate that interactions between birds have a topological rather than metric nature, each individual interacting with a fixed number of neighbors irrespective of their distances.

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.

We present a class of extended Kuramoto models describing a flocking motion of particles on the infinite cylinder and provide sufficient conditions for the asymptotic formation of locked solutions where the distance between particles remains constant. Our proposed model includes the complex Kuramoto model for synchronization. We also provide several numerical simulation results and compare them with analytical results.

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.

In this study, we present a new stochastic volatility model incorporating a flocking mechanism between individual volatilities of assets. Collective phenomena of asset pricing and volatilities in financial markets are often observed; these phenomena are more apparent when the market is in critical situations (market crashes). In the classical Heston model, the constant theoretical mean of the square of the volatility was employed, which can be assumed a priori. Our proposed model does not assume this mean value a priori, we instead use the flocking effect to continuously update the theoretical mean value using the local weighted average of individual volatility values. To perform this function, we use the Cucker–Smale flocking mechanism to calculate the local mean. For some classes of interaction weights such as all-to-all and symmetric coupling with a positive lower bound, we show that the fluctuations of the square process of volatility are uniformly bounded, such that the overall dynamics are mainly dictated by the averaged process. We also provide several numerical examples showing the dynamics of volatility.

This brief note is an introduction to the special issue devoted to modeling, qualitative analysis and simulations of vehicular traffic, crowds and swarms, viewed as living, hence complex, systems. The first part focuses on the conceptual difficulties to be tackled when dealing with the most challenging topics of this subject, mainly on interactions and complexity features. The second part provides a brief overview and critical analysis of the contents. Finally, some perspective ideas on conceivable future research objectives are brought to the reader's attention.

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.

We present a mathematical model for stock market volatility flocking. Our proposed model consists of geometric Brownian motions with time-varying volatilities coupled with Cucker–Smale (C–S) flocking and regime switching mechanisms. For all-to-all interactions, we assume that all assets' volatilities are coupled to each other with a constant interaction weight, and we show that the common volatility emerges asymptotically and discuss its financial applications. We also provide several numerical simulations and compare them to existing analytical results.

This issue is devoted to complex systems in life sciences. Some perspective ideas on possible objectives of future research are extracted from the contents of this issue and brought to the reader’s attention. The final ambitious aim is the development of a mathematical theory for complex living systems.

Collective migration of animals in a cohesive group is rendered possible by a strategic distribution of tasks among members: some track the travel route, which is time and energy-consuming, while the others follow the group by interacting among themselves. In this paper, we study a social dynamics system modeling collective migration. We consider a group of agents able to align their velocities to a global target velocity, or to follow the group via interaction with the other agents. The balance between these two attractive forces is our control for each agent, as we aim to drive the group to consensus at the target velocity. We show that the optimal control strategies in the case of final and integral costs consist of controlling the agents whose velocities are the furthest from the target one: these agents sense only the target velocity and become leaders, while the uncontrolled ones sense only the group, and become followers. Moreover, in the case of final cost, we prove an “Inactivation” principle: there exist initial conditions such that the optimal control strategy consists of letting the system evolve freely for an initial period of time, before acting with full control on the agent furthest from the target velocity.

We study the critical thresholds for the compressible pressureless Euler equations with pairwise attractive or repulsive interaction forces and non-local alignment forces in velocity in one dimension. We provide a complete description for the critical threshold to the system without interaction forces leading to a sharp dichotomy condition between global-in-time existence or finite-time blowup of strong solutions. When the interaction forces are considered, we also give a classification of the critical thresholds according to the different type of interaction forces. We also remark on global-in-time existence when the repulsion is modeled by the isothermal pressure law.

We analyze the one-dimensional pressureless Euler–Poisson equations with linear damping and nonlocal interaction forces. These equations are relevant for modeling collective behavior in mathematical biology. We provide a sharp threshold between the supercritical region with finite-time breakdown and the subcritical region with global-in-time existence of the classical solution. We derive an explicit form of solution in Lagrangian coordinates which enables us to study the time-asymptotic behavior of classical solutions with the initial data in the subcritical region.

We prove, both for continuous time and discrete time, results establishing convergence to flocking in a model in which each agent is influenced by only a few closest neighbors. We show unconditional convergence to flocking when this number of closest neighbors is at least half of the population and, otherwise, conditional convergence. That is, convergence to flocking provided the initial positions and velocities satisfy an explicit constraint. We also show that the proportion of neighbors necessary for unconditional convergence, one-half, is sharp. Nothing less will do.

We study kinetic representations of flocking models. They arise from agent-based models for self-organized dynamics, such as Cucker–Smale [Emergent behaviors in flocks, IEEE Trans. Autom. Control.52 (2007) 852–862] and Motsch–Tadmor [A new model for self-organized dynamics and its flocking behavior, J. Statist. Phys.144 (2011) 923–947] models. We first establish a well-posedness theory and large-time flocking behavior for the kinetic systems, which indicates a concentration in velocity variable in infinite time. We then apply a discontinuous Galerkin method to treat the asymptotic $\delta$-singularity, and construct high-order positive-preserving schemes to solve kinetic flocking systems.

In this paper, we revisit an interaction problem of two homogeneous Cucker–Smale (in short CS) ensembles with attractive–repulsive couplings, possibly under the effect of Rayleigh friction, and study three sufficient frameworks leading to bi-cluster flocking in which two sub-ensembles evolve to two clusters departing from each other. In the previous literature, the interaction problem has been studied in the context of attractive couplings. In our interaction problem, inter-ensemble and intra-ensemble couplings are assumed to be repulsive and attractive, respectively. When the Rayleigh frictional forces are turned on, we show that the total kinetic energy is uniformly bounded so that spatially mixed initial configurations evolve toward the bi-cluster configuration asymptotically fast under some suitable conditions on system parameters, communication weight functions and initial configurations. In contrast, when Rayleigh frictional forces are turned off, the flocking analysis is more delicate mainly due to the possibility of an exponential growth of the kinetic energy. In this case, we employ two mutually disjoint frameworks with constant inter-ensemble communication function and exponentially localized inter-ensemble communication functions, respectively, and prove the bi-clustering phenomenon in both cases. This work extends the previous work on the interaction problem of CS ensembles. We also conduct several numerical experiments and compare them with our theoretical results.