Swarming Patterns in a Two-Dimensional Kinematic Model for Biological Groups

By Chad M. Topaz, Andrea L. Bertozzi. SIAM Journal on Applied Mathematics, 2004

503 citations recorded by the source corpus.

We construct a continuum model for the motion of biological organisms experiencing social interactions and study its pattern-forming behavior. The model takes the form of a conservation law in two spatial dimensions. The social interactions are modeled in the velocity term, which is nonlocal in the population density and includes a parameter that controls the interaction length scale. The dynamics of the resulting partial integrodifferential equation may be uniquely decomposed into incompressible motion and potential motion. For the purely incompressible case, the model resembles one for fluid dynamical vortex patches. There exist solutions which have constant population density and compact support for all time. Numerical simulations produce rotating structures which have circular cores and spiral arms and are reminiscent of naturally observed phenomena such as ant mills. The sign of the social interaction term determines the direction of the rotation, and the interaction length scale affects the degree o...

Subfields: Computer Networks and Communications, Public Health, Environmental and Occupational Health, Sociology and Political Science

View Paper: https://www.semanticscholar.org/paper/c4cf58fa0e569eb4a0e74b7b9e6dde0566100ee3

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Computer Networks and Communications

In Public Health, Environmental and Occupational Health

In Sociology and Political Science

Citations in the corpus

Flocking for multi-agent dynamic systems: algorithms and theory (2006, background): “Other continuum models of swarms were proposed by Mogilner and Eldstein-Keshet [18], [9] and Topaz and Bertozzi [ 19 ].”
Finite-Time Singularities of an Aggregation Equation in $${\mathbb {R}^n}$$ with Fractional Dissipation (2009, background): “2), Topaz and Bertozzi [43] constructed a kinematic two-dimensional swarming model which takes the form ut +∇ · (u (G ∗ u)) = 0, (1.”
Emergence of time-asymptotic flocking in a stochastic Cucker-Smale system (2009, background): “Flocking phenomena are ubiquitous in many biological and physical systems such as bacteria, bees, flies, birds, fish, humans, and interacting oscillators, to name a few [1, 4, 5, 6, 9, 11, 12, 13, 14, 15, 17, 18, 19].”
Ultrafast consensus in small-world networks (2005, background): “Some examples include power networks, metabolic and gene networks [17], coauthorship network of scientists [27], biological network of oscillators [44], [19], [18], [24], [39], economic networks [15], sensor networks [10], [30], [7], [38], swarms of networked unmanned autonomous vehicles (UAVs) [32]”
From Swarm Intelligence to Swarm Robotics (2004, methodology): “For example, “swarming” is used to describe mathematical solutions to some high order PDE [14].”
Stationary states of quadratic diffusion equations with long-range attraction (2011, background): “ion acts at a smaller scale with respect to nonlocal attractive forces in the large particle limit, then a nonlocal repulsion term can be replaced by a local term with nonlinear diusion, we refer to [42, 11, 51, 52, 20, 22, 31, 49, 10, 36] for several examples. A prototype model, which we shall also”
Finite-Time Blow-up of Solutions of an Aggregation Equation in Rn (2007, background): “This bound was originally derived in [27] and global existence of smooth solutions with such kernels was established in [20].”
Differential inclusions in Wasserstein spaces: The Cauchy-Lipschitz framework (2020, background): “n several mathematical communities. Multi-agent systems are ubiquitous in a wide number of applications, ranging from the understanding of patterns formation in several branches of the animal kingdom [14, 19, 33] to the analysis of pedestrian dynamics [2, 32], ensembles of autonomous vehicles [15, 2”
On collision-avoiding initial configurations to Cucker-Smale type flocking models (2012, background): “Introduction Flocking phenomena [3, 11, 12, 13, 14, 32, 33, 34] such as flocking of birds, schooling of fishes, and swarming of bacteria are often observed in complex biological systems, and are some of the collective self-organizing phenomena in biological, mechanical, and social groups of self-pro”
From particle to kinetic and hydrodynamic descriptions of flocking (2008, background): “, biological organism [3, 8, 9, 10, 18, 19, 22, 25, 26], mobile network [2, 5, 11, 12] appears in many contexts of biological system, mobile and human network [6, 7].”
Distributed Intelligence: Overview of the Field and Its Application in Multi-Robot Systems (2008, background): “This form of interaction also has clear relevance for many military applications.”
State Transitions and the Continuum Limit for a 2D Interacting, Self-Propelled Particle System (2006, background): “tion of Eqs.(1)-(3). 3 Continuum model The continuum approach is widely adopted for modeling swarming systems, especially on the ecological scale where massive movements of populations are considered [2,3,22,23,25,26]. It is also more suitable for theoretical analysis especially in large N limit. Du”
Global-in-time weak measure solutions and finite-time aggregation for nonlocal interaction equations (2011, methodology): “Mathematical modeling of the collective behavior of individuals, such as swarming, has also been treated by continuum models steaming from discrete particle models [33, 14, 41, 34, 42, 15, 36, 13, 21, 23, 16, 17].”
Double milling in self-propelled swarms from kinetic theory (2009, background): “The ubiquity of the self-organizing phenomenon has lead to the development of several minimal models to describe a collection of interacting agents, both as discrete particles [35, 18, 10] or as a continuous density [32, 33, 34].”