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A primer of swarm equilibria
Andrew J. Bernoff, Chad M. Topaz
TL;DR
The paper investigates equilibrium configurations of swarming organisms and the conditions under which they minimize an energy functional. It derives continuum and analytical solutions across force laws and domains, reproducing diverse compactly supported equilibria and a locust-like ground–air configuration in a quasi-two-dimensional model.
Problem
The paper investigates how swarm models produce equilibrium configurations, including the unresolved conditions for bubble formation that separates concentrated ground and airborne locust groups.
Method
The authors derive a continuum variational formulation from a discrete model, characterize equilibria through an energy functional and Fredholm equation, and use differential-operator methods to analyze minimizers.
Results
Exact one-dimensional equilibria include compactly supported densities with δ-concentrations or jump discontinuities, while a quasi-two-dimensional model produces minimizers with disconnected ground and airborne components.
Takeaways & Limitations
The framework captures a wide range of equilibrium morphologies, including locust-like separated ground and airborne populations that one-dimensional solutions do not reproduce.
Takeaways & Limitations
A global minimizer need not be a global attractor, because other multi-component local minimizers may coexist; the model also assumes position-dependent velocities and neglects inertia.
Abstract
from arXiv · showhide
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 $δ$-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.
1. Introduction.
The paper develops a general framework for swarm equilibria shaped by endogenous interactions, exogenous forces, boundaries, and dimensionality, linking discrete models to continuum densities. It applies this framework to exact one-dimensional examples and locust swarms, showing that quasi-two-dimensionality supports separated ground and airborne groups.
- 1. Introduction.: The locust model produces a bubble-like swarm with dense ground and airborne groups separated by an unoccupied gap, while wind induces rolling migration.The model uses a two-dimensional position aligned with wind and vertical height, and includes endogenous forces, gravity, and wind.
- 1. Introduction.: The authors formulate a framework connecting discrete interacting swarm members with continuum density models to study equilibrium distributions and their stability.The framework addresses endogenous interactions, exogenous forces, boundaries, and the correspondence between discrete and continuum systems.
- 1. Introduction.: Exact equilibria for several one-dimensional settings include compact support, boundary δ-concentrations, and discontinuous density edges.Examples use repulsive endogenous forces with bounded, half-line, and unbounded domains, plus exogenous forces such as gravity and quadratic potentials.
- 1. Introduction.: The continuum analytical solutions agree closely with numerical equilibria of the underlying discrete system, even when the discrete swarm contains few members.This comparison is reported across the example equilibria developed in Sections 3 and 4.
- 1. Introduction.: A one-dimensional vertical-slice model cannot separate ground and airborne groups, whereas a quasi-two-dimensional model has minimizers with the observed disconnected morphology.The quasi-two-dimensional model accounts for the swarm’s horizontal extent and yields a ground concentration plus a disconnected classical airborne component.
2. Mathematical formulation.
The paper develops discrete and continuum swarm models with pairwise endogenous and exogenous forces, then uses an energy functional and variational conditions to characterize equilibria and their stability.
- 2.1. Discrete model.: The discrete model assumes overdamped particle motion with symmetric, additive pairwise endogenous forces and exogenous forces.The endogenous force is scaled by social mass m as N grows, while M denotes total social mass.
- 2.1. Discrete model.: Integrable forces are written as gradients of even mutual and exogenous potentials, producing a gradient flow that evolves toward energy minimizers.The interaction potential is Q and the exogenous potential is F.
- 2.2. Continuum model.: The continuum formulation represents particles by a density, defines continuum velocity and mass conservation, and introduces an energy functional analogous to the discrete potential.The no-flux boundary condition permits mass concentration at domain boundaries, and energy dissipation links stable equilibria to minimizers.
- 2.2. Continuum model.: The continuum–discrete correspondence matches cumulative densities at particle positions, interpolates the discrete cumulative density, and differentiates it to approximate continuum density.This correspondence is used to compare analytical continuum equilibria with numerical discrete simulations.
- 2.3. Local and global minimizers.: For perturbations supported within the swarm, positivity of the second variation W2 guarantees both local and global minimality.For broader zero-mass perturbations, positivity of the first variation condition is necessary for local minimality, while positive W2 additionally gives global minimality.
- 2.3. Local and global minimizers.: Equilibria satisfy the Fredholm integral equation and mass constraint, with candidate solutions tested through Λ(x) and positivity of W2.For compactly supported perturbations, positivity of the Fourier-transformed interaction potential is sufficient for W2 positivity; operator eigenvalue positivity provides another criterion.
- 2.4. Swarm minimizers.: A swarm minimizer is an equilibrium satisfying Λ(x) ≥ λ_i near each swarm component, corresponding to infinitesimal Lagrangian mass redistributions.This criterion evaluates energy minimality locally around each component rather than only under perturbations confined to the support.
- 2.4. Swarm minimizers.: Multiple local multi-component swarm minimizers may coexist with a global minimizer, so the global minimizer need not be a global attractor.The framework therefore distinguishes energetic global minimality from dynamical attraction.
3. Examples with a repulsive social force.
For repulsive Laplace interactions, variational analysis yields compactly supported one-dimensional equilibria whose structure depends on domain boundaries and exogenous potentials. Exact solutions include boundary δ-concentrations and agree closely with discrete numerical equilibria.
- Examples with a repulsive social force.: Exact equilibria are obtained for bounded, gravitational half-line, and quadratic-well domains, with bounded swarms observed in each case.The solutions can be classical in the interior while containing boundary concentrations or discontinuities.
- δ-concentrations: The analysis excludes interior δ-concentrations under the assumed conditions, while boundary δ-concentrations remain possible because the replacement argument fails at domain boundaries.For short-range repulsion, replacing an interior δ-function by a narrow top-hat lowers the energy.
- Variational analysis: The minimization framework uses a differential-operator method to test candidate extrema and derive sufficient conditions for energy minimization.Global minimization follows from conditions involving Λ(x), W1, and W2.
- Gravitational potential on the half-line: On the gravitational half-line, weak gravity yields a ground δ-concentration plus a classical positive-height swarm, whereas M ≤ g produces only a δ-function at the origin.For M > g, the mixed solution is globally stable; for M ≤ g, the origin concentration is the global minimizer.
- Quadratic potential well: With a quadratic potential well on the infinite line, the minimizing solution has one connected component and is a global minimizer.The global-minimizer condition is verified using the upward concavity of sF(ln s).
4. Examples with a Morse-type social force.
The Morse potential yields exact continuum equilibrium solutions whose structure depends on whether interactions are catastrophic or H-stable and on the domain. These solutions are compared with discrete simulations and can include compact support, boundary δ-concentrations, and local or global minimizer properties.
- 4.1. Morse potential and stability regimes.: In the H-stable regime, constant-density states are stable to perturbations and density profiles spread, whereas catastrophic interactions favor finite-extent states after long-wave instability.The regime distinction follows from the sign of the Fourier transform at zero and its relation to linear stability.
- 4.1. Morse potential and stability regimes.: Equilibria are obtained by solving a Fredholm integral equation with a mass constraint, then checking nonnegativity and stability; boundary δ-amplitudes arise when support meets the domain boundary.The amplitudes are related to the classical density at the support boundaries through matching conditions.
- 4.2. Example: Catastrophic interactions in free space.: The catastrophic regime has mixed-sign Fourier transform, making global-minimizer status indeterminate even though the constructed equilibrium is a swarm minimizer and local minimizer.The paper leaves establishing global minimality through the combined first and second variations as an open problem.
- 4.3. Example: H-stable interactions on a bounded domain.: H-stable interactions on a bounded domain produce a global minimizer with boundary δ-concentrations, while catastrophic free-space interactions produce a compactly supported equilibrium.For the H-stable case, positivity of the Fourier transform ensures the second variation is positive; the displayed example includes boundary concentrations.
5. Modeling a locust swarm: Examples with a gravitational potential.
The locust model compares one-dimensional and quasi-two-dimensional gravitational swarms using repulsive interactions. Only the quasi-two-dimensional model supports minimizers with a separated grounded and airborne population, and these gap states become generic at sufficiently large mass.
- 5.2. Gravitational potential on the half-line with the quasi-2D-Laplace potential.: Unlike the one-dimensional model, the quasi-two-dimensional model can reproduce the observed locust morphology of a ground concentration, an unoccupied gap, and an airborne group.The quasi-two-dimensional construction assumes uniform horizontal structure while retaining vertical gravitational organization.
- 5.1. The quasi-two-dimensional Laplace potential.: The quasi-two-dimensional potential is horizontal at zero separation, decreases monotonically with |z|, and has a maximum force magnitude χmax that determines minimizer thresholds.Its positive Fourier transform implies that local minimizers are global minimizers.
- 5.2. Gravitational potential on the half-line with the quasi-2D-Laplace potential.: For M < M1 = g/χmax, the ground-concentrated solution is a global minimizer because Λ(x) increases strictly away from the ground.The authors believe this solution is the global attractor, but do not prove that claim.
- 5.2. Gravitational potential on the half-line with the quasi-2D-Laplace potential.: For M1 < M < M2, the ground-concentrated state remains a global minimizer but is not a global attractor, because a local minimum of Λ can support another dynamically stable minimizer.At M = M2 the local minimum matches the ground value; for M > M2, transferring mass toward that minimum lowers energy.
- 5.2. Gravitational potential on the half-line with the quasi-2D-Laplace potential.: For M > M1, quasi-two-dimensional minimizers can contain separated grounded and airborne components, and for M > M2 these gap states can have lower energy than ground-only states.The proportion of grounded mass varies across a continuum of minimizers, and simulations may select different members depending on initial conditions.
6. Conclusions.
The paper develops a continuum variational framework for swarm equilibria and identifies a broad range of compactly supported minimizers, including boundary concentrations and disconnected components. Its quasi-two-dimensional locust model reproduces a grounded concentration separated by an airborne swarm, a configuration not found in the one-dimensional examples.
- The framework connects discrete swarming models to continuum energy formulations and provides criteria for local, global, and swarm minimizers.Swarm minimizers are stable to infinitesimal Lagrangian deformations of mass.
- The analytical examples include compactly supported equilibria with boundary δ-concentrations or jump discontinuities, spanning bounded-domain, half-line, and attractive-repulsive settings.The conclusions describe constant profiles with endpoint δ-functions on bounded domains and additional Morse-potential cases.
- Numerical half-line simulations show a transition at M1 from all mass on the ground to an airborne swarm, while grounded mass decreases and airborne support expands as M increases.The airborne swarm exists only for M > M1, and its lower boundary remains approximately fixed.
- For the quasi-two-dimensional Laplace potential, simulations produce stable equilibria with grounded and airborne components separated by a gap, including a continuous family of such states.The examples use different grounded mass fractions, while both components remain dynamically stable.
- The disconnected grounded-and-airborne locust configuration suggests that quasi-two-dimensionality is essential because none of the discussed one-dimensional Laplace or Morse solutions contain a gap.The resulting equilibria resemble earlier discrete locust-swarm simulations.