Flocking Model Research Articles

Self-organization in brain tumors: How cell morphology and cell density influence glioma pattern formation.

Modeling cancer cells is essential to better understand the dynamic nature of brain tumors and glioma cells, including their invasion of normal brain. Our goal is to study how the morphology of the glioma cell influences the formation of patterns of collective behavior such as flocks (cells moving in the same direction) or streams (cells moving in opposite direction) referred to as oncostream. We have observed experimentally that the presence of oncostreams correlates with tumor progression. We propose an original agent-based model that considers each cell as an ellipsoid. We show that stretching cells from round to ellipsoid increases stream formation. A systematic numerical investigation of the model was implemented in [Formula: see text]. We deduce a phase diagram identifying key regimes for the dynamics (e.g. formation of flocks, streams, scattering). Moreover, we study the effect of cellular density and show that, in contrast to classical models of flocking, increasing cellular density reduces the formation of flocks. We observe similar patterns in [Formula: see text] with the noticeable difference that stream formation is more ubiquitous compared to flock formation.

Pricing through ambiguity: a flocking model of the inter-dynamics between pricing practices and market uncertainties

ABSTRACT Pricing practices of firms are an important yet the least studied aspect of the price phenomenon in sociology. This paper answers the question: why do firms, even in the same market, tend to use different pricing practices – value-informed, competition-informed, or cost-informed pricing – to set prices? To that end, this study constructs a formal dynamic flocking model to investigate the inter-dynamics between market uncertainties and the viability of the three pricing practices. The model is a substantial revision and extension of Harrison White’s static W(y) market model by reformulating the latter into a dynamic one and by explicitly incorporating different market uncertainties into the model as variables. The study shows that each kind of pricing practice is only viable under certain distributions of market uncertainties. The theory is then used to explain the distribution of pricing practices among firms in the Burgundy wine market. Theoretical and methodological innovations and the implications for firms and for sociological research on markets and uncertainties are also discussed.

Coarsening dynamics in the Vicsek model of active matter.

We study the flocking model introduced by Vicsek et al. (Phys. Rev. Lett. 75, 1226 (1995)) in the "coarsening" regime. At standard self-propulsion speeds, we find two distinct growth laws for the coupled density and velocity fields. The characteristic length scale of the density domains grows as [Formula: see text] (with [Formula: see text] , while the velocity length scale grows much faster, viz., [Formula: see text] (with [Formula: see text] . The spatial fluctuations in the density and velocity fields are studied by calculating the two-point correlation function and the structure factor, which show deviations from the well-known Porod's law. This is a natural consequence of scattering from irregular morphologies that dynamically arise in the system. At large values of the scaled wave vector, the scaled structure factors for the density and velocity fields decay with powers -2.6 and -1.52 , respectively.

Emergent dynamics of an orientation flocking model for multi-agent system

We study the orientation flocking for the deterministic counterpart of a stochastic agent-based model introduced by Degond, Frouvelle and Merino-Aceituno in 2017, where the orientation is defined as a \begin{document}$ {\rm SO}(3) $\end{document} matrix. Their proposed model can be reduced to the other collective dynamics models such as the Lohe matrix model and the Viscek-type model as special cases. In this work, we study the emergent dynamics of the orientation flocking model in two frameworks. First, we present sufficient conditions leading to the orientation flocking when the natural frequency matrices are identical. To be precise, we prove that all orientation matrices asymptotically converge to the common one, and the spatial position diameter remains uniformly bounded. Second, we show the emergence of orientation-locked states for non-identical natural frequency matrices, that is, the difference of any two orientation matrices tends to the definite constant matrix. On the other hand, we establish the finite-in-time stability with respect to initial data of the proposed orientation flocking model. We also present the numerical results consistent with our rigorous analysis. Our work remains valid even for dimensions greater than three.

Multi-Robot Flocking Control Based on Deep Reinforcement Learning

In this paper, we apply deep reinforcement learning (DRL) to solve the flocking control problem of multi-robot systems in complex environments with dynamic obstacles. Starting from the traditional flocking model, we propose a DRL framework for implementing multi-robot flocking control, eliminating the tedious work of modeling and control designing. We adopt the multi-agent deep deterministic policy gradient (MADDPG) algorithm, which additionally uses the information of multiple robots in the learning process to better predict the actions that robots will take. To address the problems such as low learning efficiency and slow convergence speed of the MADDPG algorithm, this paper studies a prioritized experience replay (PER) mechanism and proposes the Prioritized Experience Replay-MADDPG (PER-MADDPG) algorithm. Based on the temporal difference (TD) error, a priority evaluation function is designed to determine which experiences are sampled preferentially from the replay buffer. In the end, the simulation results verify the effectiveness of the proposed algorithm. It has a faster convergence speed and enables the robot group to complete the flocking task in the environment with obstacles.

FLOCKING AND COLLISION AVOIDANCE OF A CUCKER-SMALE TYPE SYSTEM WITH SINGULAR WEIGHTS

The dynamical behavior of a flock model with a singular communication rate and extra interaction terms is investigated in this paper. A rigorous theoretical proof of collision avoidance between any two agents is obtained which guarantees the existence of global solutions. Moreover, a sufficient condition for the existence of time-asymptotic flocking is also acquired and numerical simulations verified these results which show that a compact equilibrium configuration may emerge.

Uniform-in-time bound for kinetic flocking models

In this paper, we provide a uniform-in-time bound of the spatial density of kinetic flocking models, in particular, kinetic Cucker–Smale type equations, in all dimensions. By a careful analysis of the structure of the main equations, under certain assumptions on the initial data and the communication weight function, we provide the uniform-in-time bound of the spatial density.

Flocking with General Local Interaction and Large Population

This paper studies a flocking model in which the interaction between agents is described by a general local nonlinear function depending on the distance between agents.

Three-body interactions drive the transition to polar order in a simple flocking model.

A large class of mesoscopic or macroscopic flocking theories are coarse-grained from microscopic models that feature binary interactions as the chief aligning mechanism. However, while such theories seemingly predict the existence of polar order with just binary interactions, actomyosin motility assay experiments show that binary interactions are insufficient to obtain polar order, especially at high densities. To resolve this paradox, here we introduce a solvable one-dimensional flocking model and derive its stochastic hydrodynamics. We show that two-body interactions are insufficient to generate polar order unless the noise is non-Gaussian. We show that noisy three-body interactions in the microscopic theory allow us to capture all essential dynamical features of the flocking transition, in systems that achieve orientational order above a critical density.

Multistate voter model with imperfect copying.

The voter model with multiple states has found applications in areas as diverse as population genetics, opinion formation, species competition, and language dynamics, among others. In a single step of the dynamics, an individual chosen at random copies the state of a random neighbor in the population. In this basic formulation, it is assumed that the copying is perfect, and thus an exact copy of an individual is generated at each time step. Here, we introduce and study a variant of the multistate voter model in mean field that incorporates a degree of imperfection or error in the copying process, which leaves the states of the two interacting individuals similar but not exactly equal. This dynamics can also be interpreted as a perfect copying with the addition of noise: a minimalistic model for flocking. We found that the ordering properties of this multistate noisy voter model, measured by a parameter ψ in [0,1], depend on the amplitude η of the copying error or noise and the population size N. In the case of perfect copying η=0, the system reaches an absorbing configuration with complete order (ψ=1) for all values of N. However, for any degree of imperfection η>0, we show that the average value of ψ at the stationary state decreases with N as 〈ψ〉≃6/(π^{2}η^{2}N) for η≪1 and η^{2}N≳1, and thus the system becomes totally disordered in the thermodynamic limit N→∞. We also show that 〈ψ〉≃1-π^{2}/6η^{2}N in the vanishing small error limit η→0, which implies that complete order is never achieved for η>0. These results are supported by Monte Carlo simulations of the model, which allow to study other scenarios as well.

Systemic risk in market microstructure of crude oil and gasoline futures prices: A Hawkes flocking model approach

AbstractWe propose the Hawkes flocking model that assesses systemic risk in high‐frequency processes at the two perspectives—endogeneity and interactivity. We examine the futures markets of West Texas Intermediate (WTI) crude oil and gasoline for the past decade, and perform a comparative analysis with conditional value‐at‐risk as a benchmark measure. In terms of high‐frequency structure, we derive the empirical findings. The endogenous systemic risk in WTI was significantly higher than that in gasoline, and the level at which gasoline affects WTI was constantly higher than that in the opposite case. Moreover, although the relative influence's degree was asymmetric, its difference has gradually reduced.

A flocking model base on selective attention mechanism

Motivated by the capability to compress information efficiently of selective attention mechanism, we proposed a selective attention based flocking model. In this model, individuals sense other members by fixed number of neighbors guided by line of sight method and calculate saliency of these neighbors according to their contribution to local order. Then individuals decide whether to activate themselves to respond to stimuli determined by threshold model. We found a well-defined hierarchical network concerning stimulus transfer among the flock. Individuals switch between hierarchical and egalitarian strategies according to their status of activation. Results reveal that the selective attention can enhance the transfer of social cues and make flock members able to respond to social cues rapidly. The decision to turn caused by external stimuli can propagate across the group quickly using this model, which is consistent with natural flocks.

Some weak flocking models and its application to target tracking

In this paper, we introduce a new concept named the weak flocking behavior. That is, if the complex system has the weak flocking property, the agents do not need have the same velocities to keep together when t→∞. This shows the biggest difference between weak flocking behavior and the flocking behavior. The advantage of this concept is that its practical significance—the agents don't need always keep in step in lots of the complex systems but they still keep together. Then, we propose a weak flocking model with two agents and study two different mechanisms for this model. Moreover, we also consider a system with N agents. Under the weak link mechanism, the system with N agents can keep weak flocking.

Bacteria Floc, but Do They Flock? Insights from Population Interaction Models of Quorum Sensing.

Quorum sensing (QS) enables coordinated, population-wide behavior. QS-active bacteria "communicate" their number density using autoinducers which they synthesize, collect, and interpret. Tangentially, chemotactic bacteria migrate, seeking out nutrients and other molecules. It has long been hypothesized that bacterial behaviors, such as chemotaxis, were the primordial progenitors of complex behaviors of higher-order organisms. Recently, QS was linked to chemotaxis, yet the notion that these behaviors can together contribute to higher-order behaviors has not been shown. Here, we mathematically link flocking behavior, commonly observed in fish and birds, to bacterial chemotaxis and QS by constructing a phenomenological model of population-scale QS-mediated phenomena. Specifically, we recast a previously developed mathematical model of flocking and found that simulated bacterial behaviors aligned well with well-known QS behaviors. This relatively simple system of ordinary differential equations affords analytical analysis of asymptotic behavior and describes cell position and velocity, QS-mediated protein expression, and the surrounding concentrations of an autoinducer. Further, heuristic explorations of the model revealed that the emergence of "migratory" subpopulations occurs only when chemotaxis is directly linked to QS. That is, behaviors were simulated when chemotaxis was coupled to QS and when not. When coupled, the bacterial flocking model predicts the formation of two distinct groups of cells migrating at different speeds in their journey toward an attractant. This is qualitatively similar to phenomena spotted in our Escherichiacoli chemotaxis experiments as well as in analogous work observed over 50 years ago.IMPORTANCE Our modeling efforts show how cell density can affect chemotaxis; they help to explain the roots of subgroup formation in bacterial populations. Our work also reinforces the notion that bacterial mechanisms are at times exhibited in higher-order organisms.

New Flocking Models Apply for UAV Formation

In this paper some new flocking models are introduced. The contact mechanism between agents is from the famous Cooker-Smile model. In these models, the information exchange in the system is fully connected and the agents can adjust their velocities to form a flock. The important innovation of this paper is the flocking agents will automatic slow their velocities to avoid collision when they detects their relatively velocities so quickly and they are too closed with other agents. Using these models, the information interaction between the agents is completely automated which means they do not need human help to finish the flock. We only need to give the initial velocity for each agents they will complete the flocking and not run into each other. Also, we give the final velocity of the system and from the expression it show this velocity is only depended on their agent’s initial velocity.

Flocking-Inspired Transmission Power Control for Fair Resource Allocation in Vehicle-Mounted Mobile Relay Networks

The Cucker–Smale flocking model is designed to analyze the velocity and position-control mechanisms in a group of living individuals, such as a flock of birds. In this model, each agent (a bird or fish) adjusts its velocity according to its own velocity and the weighted mean of the relative velocities of the other members of the flock. We herein focus on the simple, distributed, and autonomous nature of the flocking model and apply its operating principles to fair resource allocation for vehicle-mounted mobile relays (VMRs) by considering their need for distributed fair resource allocation in various network topologies. By adaptively adjusting the VMR transmission power, the proposed method effectively alleviates the interference between users in vehicles and users directly connected to the base station. We evaluate the performance of the proposed method in the context of the IEEE 802.16j based system. The achievable bandwidth performance of the proposed method is evaluated relative to that of both the fixed power allocation (FPA) method and the signal-to-interference plus noise ratio (SINR)-based transmit power control method. The result shows that the proposed method adjusts to various network topologies and provides better achievable bandwidth than both the SINR-based method and the FPA method.

Emergent behavior of Cucker-Smale model with normalized weights and distributed time delays

We study a Cucker-Smale-type flocking model with distributed time delay where individuals interact with each other through normalized communication weights. Based on a Lyapunov functional approach, we provide sufficient conditions for the velocity alignment behavior. We then show that as the number of individuals \begin{document}$ N $\end{document} tends to infinity, the \begin{document}$ N $\end{document} -particle system can be well approximated by a delayed Vlasov alignment equation. Furthermore, we also establish the global existence of measure-valued solutions for the delayed Vlasov alignment equation and its large-time asymptotic behavior.

Hydrodynamic limits for kinetic flocking models of Cucker-Smale type.

We analyse the asymptotic behavior for kinetic models describing the collective behavior of animal populations. We focus on models for self-propelled individuals, whose velocity relaxes toward the mean orientation of the neighbors. The self-propelling and friction forces together with the alignment and the noise are interpreted as a collision/interaction mechanism acting with equal strength. We show that the set of generalized collision invariants, introduced in [1], is equivalent in our setting to the more classical notion of collision invariants, i.e., the kernel of a suitably linearized collision operator. After identifying these collision invariants, we derive the fluid model, by appealing to the balances for the particle concentration and orientation. We investigate the main properties of the macroscopic model for a general potential with radial symmetry.

A rigorous derivation from the kinetic Cucker–Smale model to the pressureless Euler system with nonlocal alignment

We consider the kinetic Cucker–Smale model with local alignment as a mesoscopic description for the flocking dynamics. The local alignment was first proposed by Karper, Mellet and Trivisa (2014), as a singular limit of a normalized nonsymmetric alignment introduced by Motsch and Tadmor (2011). The existence of weak solutions to this model was obtained by Karper, Mellet and Trivisa (2014), and in the same paper they showed the time-asymptotic flocking behavior. Our main contribution is to provide a rigorous derivation from a mesoscopic to a macroscopic description for the Cucker–Smale flocking models. More precisely, we prove the hydrodynamic limit of the kinetic Cucker–Smale model with local alignment towards the pressureless Euler system with nonlocal alignment, under a regime of strong local alignment. Based on the relative entropy method, a main difficulty in our analysis comes from the fact that the entropy of the limit system has no strict convexity in terms of density variable. To overcome this, we combine relative entropy quantities with the 2-Wasserstein distance.

Convergence Analysis of Classes of Asymmetric Networks of Cucker–Smale Type With Deterministic Perturbations

We discuss two extensions of the Cucker-Smale flocking model with asymmetric coupling weights. The first model assumes a finite collection of autonomous agents aiming to perform a consensus process in the presence of identical internal dynamics. The second model describes a similar population of agents that perform velocity alignment with the restriction of collision-free orbits. Although qualitatively different, we explain how these models can be analyzed under a common framework. Rigorous analysis is conducted toward establishing sufficient conditions for asymptotic flocking to a synchronized motion. Applications of our results are compared with simulations to illustrate the effectiveness of our theoretical estimates.