176 research outputs found
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
-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
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
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
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
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
Numerical Approximation of Diffusive Capture Rates by Planar and Spherical Surfaces with Absorbing Pores
In 1977 Berg and Purcell published a landmark paper entitled Physics of Chemore- ception, which examined how a bacterium can sense a chemical attractant in the fluid surrounding it [H. C. Berg and E. M. Purcell, Biophys J, 20 (1977), pp. 193–219]. At small scales the attrac- tant molecules move by Brownian motion and diffusive processes dominate. This example is the archetype of diffusive signaling problems where an agent moves via a random walk until it either strikes or eludes a target. Berg and Purcell modeled the target as a sphere with a set of small circular targets (pores) that can capture a diffusing agent. They argued that, in the limit of small radii and wide spacing, each pore could be modeled independently as a circular pore on an infinite plane. Using a known exact solution, they showed the capture rate to be proportional to the combined perimeter of the pores. In this paper we study how to improve this approximation by including interpore competition effects and verify this result numerically for a finite collection of pores on a plane or a sphere. Asymptotically we have found the corrections to the Berg–Purcell formula that account for the enhancement of capture due to the curvature of the spherical target and the inhibition of capture due to the spatial interaction of the pores. Numerically we develop a spectral boundary ele- ment method for the exterior mixed Neumann–Dirichlet boundary value problem. Our formulation reduces the problem to a linear integral equation, specifically a Neumann to Dirichlet map, which is supported only on the individual pores. The difficulty is that both the kernel and the flux are singular, a notorious obstacle in such problems. A judicious choice of singular boundary elements allows us to resolve the flux singularity at the edge of the pore. In biological systems there can be thousands of receptors whose radii are 0.1% the radius of the cell. Our numerics can now resolve this realistic limit with an accuracy of roughly one part in 108
Stability and Dynamics of Self-Similarity in Evolution Equations
A methodology for studying the linear stability of self-similar solutions is discussed. These fundamental ideas are illustrated on three prototype problems: a simple ODE with finite-time blow-up, a second-order semi-linear heat equation with infinite-time spreading solutions, and the fourth-order Sivashinsky equation with finite-time self-similar blow-up. These examples are used to show that self-similar dynamics can be studied using many of the ideas arising in the study of dynamical systems. In particular, the use of dimensional analysis to derive scaling invariant similarity variables is discussed, as well as the role of symmetries in the context of stability of self-similar dynamics. The spectrum of the linear stability problem determines the rate at which the solution will approach a self-similar profile. For blow-up solutions it is demonstrated that the symmetries give rise to positive eigenvalues associated with the symmetries, and it is shown how this stability analysis can identify a unique stable (and observable) attracting solution from a countable infinity of similarity solutions
- …