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Wildebeest herds on rolling hills: Flocking on arbitrary curved surfaces

Christina L. Hueschen, Alexander R. Dunn, and Rob Phillips
Phys. Rev. E 108, 024610 – Published 24 August 2023
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  • INTRODUCTION
  • FLOCKING THEORY FOR ARBITRARY CURVED…
  • FORMULATING THE SURFACE TONER-TU…
  • PARAMETERS AND DIMENSIONLESS RATIOS FOR…
  • SOLVING TONER-TU ON HIGHLY SYMMETRIC…
  • WILDEBEEST ON SPHERES AND HILLS: COMPLEX…
  • CONCLUSION
  • ACKNOWLEDGMENTS
  • APPENDICES
    • Supplemental Material
    References

    Abstract

    The collective behavior of active agents, whether herds of wildebeest or microscopic actin filaments propelled by molecular motors, is an exciting frontier in biological and soft matter physics. Almost three decades ago, Toner and Tu developed a continuum theory of the collective action of flocks, or herds, that helped launch the modern field of active matter. One challenge faced when applying continuum active matter theories to living phenomena is the complex geometric structure of biological environments. Both macroscopic and microscopic herds move on asymmetric curved surfaces, like undulating grass plains or the surface layers of cells or embryos, which can render problems analytically intractable. In this paper, we present a formulation of the Toner-Tu flocking theory that uses the finite element method to solve the governing equations on arbitrary curved surfaces. First, we test the developed formalism and its numerical implementation in channel flow with scattering obstacles and on cylindrical and spherical surfaces, comparing our results to analytical solutions. We then progress to surfaces with arbitrary curvature, moving beyond previously accessible problems to explore herding behavior on a variety of landscapes. This approach allows the investigation of transients and dynamic solutions not revealed by analytic methods. It also enables versatile incorporation of new geometries and boundary conditions and efficient sweeps of parameter space. Looking forward, the paper presented here lays the groundwork for a dialogue between Toner-Tu theory and data on collective motion in biologically relevant geometries, from drone footage of migrating animal herds to movies of microscopic cytoskeletal flows within cells.

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    • Received 11 June 2022
    • Accepted 10 July 2023

    DOI:https://doi.org/10.1103/PhysRevE.108.024610

    Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

    Published by the American Physical Society

    Physics Subject Headings (PhySH)

    1. Research Areas
    Biological self-organizationFlockingMulti-organism systems
    1. Physical Systems
    Dry active matter
    1. Techniques
    Finite-element method
    Polymers & Soft MatterPhysics of Living Systems

    Authors & Affiliations

    Christina L. Hueschen1,*, Alexander R. Dunn1, and Rob Phillips2,3,†

    • 1Department of Chemical Engineering, Stanford University, Palo Alto, California 94305, USA
    • 2Department of Physics, California Institute of Technology, Pasadena, California 91125, USA
    • 3Division of Biology and Biological Engineering, California Institute of Technology, Pasadena, California 91125, USA

    • *Corresponding author: chueschen@gmail.com
    • †Corresponding author: phillips@pboc.caltech.edu

    Article Text

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    Vol. 108, Iss. 2 — August 2023

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    • Figure 1
      Figure 1

      Building intuition for the physics of the velocity field in the Toner-Tu theory. The different terms in the Toner-Tu analysis are represented graphically for a one-dimensional herd. We can think of each term as providing an update to the current velocity. In each case, an example of a simple velocity profile (blue) or density profile [graphic in (b)] is shown alongside the corresponding velocity update (green). (b) The preferred speed term increases the velocity of wildebeests that are moving too slow and decreases the velocity of wildebeests that are moving too fast. (b) The pressure term punishes gradients in density, adjusting velocity to flatten these gradients. Given mass conservation in a finite system, the pressure term alone would lead to a steady state of uniform density ρ=ρ0. (c) The neighbor coupling term captures the velocity adjustment made by a wildebeest to better match its neighbors, smoothing out differences in velocity. A given wildebeest (middle black dot) adjusts its velocity by an amount represented by the green arrow: an average of the difference between its velocity and its two neighbors' velocities. This averaging of two differences is mathematically analogous to taking a local second derivative or local Laplacian, ∇2v. (d) Finally, the advection term ensures that the filament velocity field is swept along according to its own velocities, just as fluid velocity is self-advected in the Navier-Stokes equations. Mathematically, the velocity adjustment needed to ensure velocity advection is a function of the spatial gradient of velocity (−∇v, how mismatched in velocity nearby wildebeests are) and the velocity itself (v, how fast that mismatch is carried along by the herd).

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    • Figure 2
      Figure 2

      Illustration of the projection operator essential to the general surface formulation and finite element treatment of the Toner-Tu equations on curved surfaces. The surface of interest is represented by a collection of nodes (vertices of triangles) and a field of surface normal vectors, n (black arrows). An arbitrary vector v is projected onto the tangent plane (shown in gold), permitting a decomposition of the form v=v∥+v⊥.

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    • Figure 3
      Figure 3

      Estimates for key parameter values from tracking data and images of wildebeest herds. Mean density is estimated in the middle of a herd, while the critical density is estimated as in between the density of a region that does not show flocking structure (right white box) and the density of a region that does (left white box). For this paper, we use a wildebeest walking speed of 1 m/s, inferred from the plot of tracking data adapted from Ref. [49]. The scale of wildebeest acceleration is estimated from images of a turning herd, where a change in the vector v of roughly 1 m/s occurs over a timescale s/|v|, or roughly 10 s. Aerial photos from drones (mean density and acceleration image) and satellites (critical density image) are from Ref. [50].

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    • Figure 4
      Figure 4

      Toner-Tu herding in a 2D channel with periodic boundary conditions at its ends. The left panel shows the time evolution of the density field (colors) with the velocities superimposed as arrows. The right panel reports on the same simulation but uses a color map to show the speed. In the long-time limit, the density is uniform and the speed is constant throughout the channel except for a boundary layer near the walls. See also video 1 (SM). Here and in all subsequent figures, parameters used are those inferred in Sec. 4 unless explicitly stated otherwise.

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    • Figure 5
      Figure 5

      Steady-state velocity for active agents during channel flow as a function of position y, across the channel. Solutions were obtained both by the time-dependent finite element method (circles) and by numerically integrating the steady-state Toner-Tu equations (solid lines) for the highly symmetric channel geometry. Different choices of the neighbor coupling coefficient D result in boundary layers of different thicknesses. The width ℓFEM of the boundary layer is estimated by computing the initial slope of the velocity versus position curves and extrapolating out (dashed lines) to the preferred speed (vpref=1m/s), as shown in the top left of the figure. In Fig. 6, ℓFEM is compared to analytical calculations of the boundary layer thickness.

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    • Figure 6
      Figure 6

      Boundary layer thickness as a function of the neighbor coupling coefficient, D. A numerical result for the thickness of the boundary layer, ℓFEM, was estimated by computing dv/dy at the wall and using the condition (dv/dy)ℓFEM=vpref, where vpref=α(ρ0−ρc)/β. An analytical result for the thickness of the boundary layer, ℓanalytic, was calculated in Eqs. (74, 75, 76, 77).

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    • Figure 7
      Figure 7

      Solution of the Toner-Tu equations for a herd moving through a channel with a small circular obstacle (15 m obstacle radius; 100 m channel width). The left panel shows the time evolution of the density field (colors) with the velocities superimposed as arrows, from an initial condition of uniform density and a disordered velocity field. The right panel uses a color map to show the speed. For this choice of obstacle size and neighbor coupling coefficient D, a steady-state flow is achieved in which the active agents can reach their preferred speed even in the passage between the obstacle and the wall. See also video 1 (SM).

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    • Figure 8
      Figure 8

      Solution of the Toner-Tu equations for a herd moving through a channel with a large circular obstacle (30 m obstacle radius; 100 m channel width). The left panel shows the time evolution of the density field (colors) with the velocities superimposed as arrows. The right panel uses a color map to show the speed. For this choice of obstacle size and neighbor coupling coefficient D, no steady state motion is achieved. Most of the herd is reflected off the obstacle at each attempted passage, resulting in repeated reversals of direction. See also video 1 (SM).

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    • Figure 9
      Figure 9

      Phase diagram of herding behavior in a channel with an obstacle. Obstacle radius, R, and neighbor coupling coefficient, D, were systematically varied. Emergent herd dynamics were either unidirectional, reaching a steady-state as in Fig. 7 or displayed motion reversals at the obstacle indefinitely, as in Fig. 8. The transition between these two behaviors appears to occur when the width of the passage between obstacle and channel wall, L/2−R, is comparable to twice the boundary layer length scale ℓ obtained by dimensional analysis in Eq. (73).

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    • Figure 10
      Figure 10

      Dynamics of Toner-Tu flow on a cylinder. (a) The initial condition is a uniform circumferential flow field with a very small initial speed. Over time, the speed relaxes to the steady-state value |vss|=α(ρ0−ρc)/β. The timescale for this relaxation is τ=1/(2α(ρ0−ρc)). (b) Comparison of analytic and numerical (COMSOL) results for dynamics of |v|2 for different choices of parameters. The asymptotic values of the curves correspond to the steady-state solution worked out earlier in this section. The ratio α/β is fixed such that the steady-state velocity of 1 m/s is the same for all four cases and |v|(0)=0.1m/s. (c) Data collapse in which |v|2 is plotted against dimensionless time, revealing that all curves are dictated by the same underlying timescale, set by α(ρ0−ρc).

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    • Figure 11
      Figure 11

      Comparison of analytic and numerical (COMSOL) solutions to the Toner-Tu equations on a sphere. (a) Example steady-state solution for Toner-Tu herds on the surface of a sphere, for η=1. (b) Steady-state herd density on the sphere. Different choices of the dimensionless parameter η elicit different density profiles. (c) Steady-state velocity v2 for Toner-Tu herds on the surface of a sphere. As in (b), different choices of the dimensionless parameter η result in different profiles. Parameter values used: α=2m2/s; σ=40m4/s2, 20 m4/s2, or 10m4/s2, respectively; others as in Sec. 4. In both (b) and (c), analytical results (curves) are in good accord with numerical results (open circles) obtained using the finite element method.

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    • Figure 12
      Figure 12

      Toner-Tu dynamics on the sphere. (a) Dynamics of herd density and velocity on the sphere, during the evolution from an initial condition of uniform density and circumferential velocity vϕ=vprefsinθ toward the steady-state band pattern shown at 2000 s and 10 000 s. In the long-time limit, density oscillations are damped out and a steady state emerges. (b) Density as a function of angular position, showing the damping of density oscillations over time. Parameters as defined in Sec. 4. See also video 2 (SM).

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    • Figure 13
      Figure 13

      Steady-state solutions on the sphere including the neighbor coupling term. (a) Steady-state density as a function of angular position θ for different choices of the neighbor coupling coefficient D. An analytic solution in the absence of the D term is plotted for comparison. (b) Steady-state velocity squared as a function of angular position θ for the same parameters values and analytic solution as shown in (a).

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    • Figure 14
      Figure 14

      Herds on the sphere can form rotating patch, oscillating band, and steady-state band patterns. In these examples, herd dynamics are seeded at t=0 s by confining wildebeest to an angular wedge between ϕ=0 and ϕ=π/2 and giving them a circumferential velocity vϕ=vprefsinθ. (a) Given a smaller pressure term (σ=5m4/s2), a rotating patch pattern of herd density emerges. (b) A larger pressure term (σ=15m4/s2) leads the density field to exhibit temporal oscillations about the equatorial plane (θ=π/2), while remaining axisymmetric in ϕ. (c) Given an ever larger pressure term (σ=20m4/s2), herd density is restricted to a narrow range around ρ0(0.0625m−2), and a steady-state banded pattern of density emerges. See also video 3 (SM).

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    • Figure 15
      Figure 15

      Increasing the Toner-Tu pressure term drives herd density from patch to band patterns. Plot shows density as a function of ϕ, angular position around the sphere circumference. The simulation was seeded with the initial condition depicted in Fig. 14 for varied σ that lead to rotating patch or steady-state band solutions in the long-time limit. Curves are aligned such that their maxima are at ϕ=π.

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    • Figure 16
      Figure 16

      Curvature induces local density and velocity changes. Steady-state herd density (colors) and velocity (black arrows) for wildebeest traversing a Gaussian hillside embedded in a racetrack straightaway. From an initial condition of random velocity orientation, a steady-state recirculation around the racetrack emerges in the long-time limit. In an effect reminiscent of lensing [54, 55], the Gaussian hill's curvature induces low and high density regions.

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    • Figure 17
      Figure 17

      Dynamics of a Toner-Tu herd on an undulating landscape, initialized with uniform density ρ0=0.25m−2 and randomly oriented velocities. Comparing the left column (ζ=0m/s2) and the right column (ζ=0.05m/s2) highlights the contribution of the gravitational force. In the presence of the gravitational force, the hills act as soft obstacles, favoring herd circulation around them. Other parameters used were presented in Sec. 4. See also video 4 (SM).

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