A primer of swarm equilibria

We study equilibrium configurations of swarming biological organisms subject to exogenous and pairwise endogenous forces. Beginning with a discrete dynamical model, we derive a variational description of the corresponding continuum population density. Equilibrium solutions are extrema of an energy functional, and satisfy a Fredholm integral equation. We find conditions for the extrema to be local minimizers, global minimizers, and minimizers with respect to infinitesimal Lagrangian displacements of mass. In one spatial dimension, for a variety of exogenous forces, endogenous forces, and domain configurations, we find exact analytical expressions for the equilibria. These agree closely with numerical simulations of the underlying discrete model.The exact solutions provide a sampling of the wide variety of equilibrium configurations possible within our general swarm modeling framework. The equilibria typically are compactly supported and may contain $\delta$-concentrations or jump discontinuities at the edge of the support. We apply our methods to a model of locust swarms, which are observed in nature to consist of a concentrated population on the ground separated from an airborne group. Our model can reproduce this configuration; quasi-two-dimensionality of the model plays a critical role.Comment: 38 pages, submitted to SIAM J. Appl. Dyn. Sy

Finite Amplitude Convection Between Stress-Free Boundaries; Ginzburg-Landau Equations and Modulation Theory

The stability theory for rolls in stress-free convection at finite Prandtl number is affected by coupling with low wavenumber two-dimensional mean-flow modes. In this work, a set of modified Ginzburg-Landau equations describing the onset of convection is derived which accounts for these additional modes. These equations can be used to extend the modulation equations of Zippelius & Siggia describing the breakup of rolls, bringing their stability theory into agreement with the results of Busse & Bolton

Biological Aggregation Driven By Social and Environmental Factors: A Nonlocal Model and Its Degenerate Cahn-Hilliard Approximation

Biological aggregations such as insect swarms and bird flocks may arise from a combination of social interactions and environmental cues. We focus on nonlocal continuum equations, which are often used to model aggregations, and yet which pose significant analytical and computational challenges. Beginning with a particular nonlocal aggregation model [Topaz et al., Bull. Math. Bio., 2006], we derive the minimal well-posed long-wave approximation, which is a degenerate Cahn-Hilliard equation. Energy minimizers of this reduced, local model retain many salient features of those of the nonlocal model, especially for large populations and away from an aggregation\u27s boundaries. Using the Cahn-Hilliard model as a testbed, we investigate the degree to which an external potential modeling food sources can be used to suppress peak population density, which is essential for controlling locust outbreaks. A random distribution of food sources tends to increase peak density above its intrinsic value, while a periodic pattern of food sources can decrease it

Mathematics in the Mountains: The Park City Mathematics Institute

It\u27s noon. A Fields medalist, master high school teachers from the US and abroad, aspiring undergraduate and graduate students, gifted expositors of mathematics, and mathematical artists gather at tables under a tent. Lunch and so much more is served at these meetings of the minds

A model for rolling swarms of locusts

We construct an individual-based kinematic model of rolling migratory locust swarms. The model incorporates social interactions, gravity, wind, and the effect of the impenetrable boundary formed by the ground. We study the model using numerical simulations and tools from statistical mechanics, namely the notion of H-stability. For a free-space swarm (no wind and gravity), as the number of locusts increases, it approaches a crystalline lattice of fixed density if it is H-stable, and in contrast becomes ever more dense if it is catastrophic. Numerical simulations suggest that whether or not a swarm rolls depends on the statistical mechanical properties of the corresponding free-space swarm. For a swarm that is H-stable in free space, gravity causes the group to land and form a crystalline lattice. Wind, in turn, smears the swarm out along the ground until all individuals are stationary. In contrast, for a swarm that is catastrophic in free space, gravity causes the group to land and form a bubble-like shape. In the presence of wind, the swarm migrates with a rolling motion similar to natural locust swarms. The rolling structure is similar to that observed by biologists, and includes a takeoff zone, a landing zone, and a stationary zone where grounded locusts can rest and feed.Comment: 18 pages, 11 figure

Continuum Model of Thin-Film Deposition and Growth

A continuum theory for the deposition and growth of solid films is presented. The theory is developed in a coordinate-independent manner and so incorporates the fully nonlinear physics. The evolution of the film is modeled in three steps. First, the adsorption of atoms in the incident beam is modeled as a ballistic process. Second, the random motion of the adatoms is treated as a diffusive process. Finally, sticking of adatoms to the film occurs as a Poisson process. The resulting system of differential equations is examined in several parameter limits. The diffusively dominated limit appears similar to zone 1 of the structure-zone model. Generically the surface slope develops discontinuities; these ‘‘kinks’’ play the role of grain boundaries. In the ballistically dominated case these kinks may be advected along the surface giving rise to columnarlike microstructures, as is observed in zone 2

Groundswell: Winning in a World Transformed by Social Technologies

The revised and updated groundswell is a must-read for anyone who wants to take advantage of the opportunities of a newly emerging environment in which customers have a say in your business. Expanded with uo to the minute statistic and new chapters on Twitter and the organizational challenges as they emege, understand how customer participation can strengthen your business and create a process for building this vital marketing channel into your future stratefy

Scroll Waves in the Presence of Slowly Varying Anisotropy with Application to the Heart

We consider the dynamics of scroll waves in the presence of rotating anisotropy, a model of the left ventricle of the heart in which the orientation of fibers in successive layers of tissue rotates. By choosing a coordinate system aligned with the fiber rotation and studying the phase dynamics of a straight but twisted scroll wave, we derive a Burgers’ equation with forcing associated with the fiber rotation rate. We present asymptotic solutions for scroll twist, verified by numerics, using a realistic fiber distribution profile. We make connection with earlier numerical and analytical work on scroll dynamics