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Locust Dynamics: Behavioral Phase Change and Swarming

Chad M. Topaz, Maria R. D'Orsogna, Leah Edelstein-Keshet, Andrew J. Bernoff

arXiv:1207.4968v1q-bio.QMnlin.PS

TL;DR

The paper asks how individual-level crowding and social interactions produce population-level hopper-band formation. It constructs a continuum model coupling nonlocal movement with density-dependent phase conversion, then analyzes and simulates its dynamics. The model predicts a critical density beyond which dense, traveling gregarious clusters emerge and mass gregarization occurs, with implications for outbreak control.

  • Problem

    The paper addresses how crowding-driven conversion between solitarious and gregarious phases produces marching hopper bands at the population level.

  • Method

    The authors construct a continuum partial integrodifferential model coupling nonlocal social movement with density-dependent interconversion between the two locust phases.

  • Results

    Beyond a critical population density, the model predicts dense, traveling gregarious patches, rapid phase transitions, and formation of hopper-band-like clusters.

  • Takeaways & Limitations

    The calculated instability threshold links individual interaction parameters to outbreak risk and identifies a population level below which gregarious outbreaks may be avoided.

  • Takeaways & Limitations

    The model omits vegetation, resource gradients, and other environmental cues, limiting direct biological relevance and preventing representation of long-range band motion.

Abstract

from arXiv · show

Locusts exhibit two interconvertible behavioral phases, solitarious and gregarious. While solitarious individuals are repelled from other locusts, gregarious insects are attracted to conspecifics and can form large aggregations such as marching hopper bands. Numerous biological experiments at the individual level have shown how crowding biases conversion towards the gregarious form. To understand the formation of marching locust hopper bands, we study phase change at the collective level, and in a quantitative framework. Specifically, we construct a partial integrodifferential equation model incorporating the interplay between phase change and spatial movement at the individual level in order to predict the dynamics of hopper band formation at the population level. Stability analysis of our model reveals conditions for an outbreak, characterized by a large scale transition to the gregarious phase. A model reduction enables quantification of the temporal dynamics of each phase, of the proportion of the population that will eventually gregarize, and of the time scale for this to occur. Numerical simulations provide descriptions of the aggregation's structure and reveal transiently traveling clumps of gregarious insects. Our predictions of aggregation and mass gregarization suggest several possible future biological experiments.

Model equations

The model couples spatial movement with density-dependent conversion between solitarious and gregarious locust phases. Solitarious locusts repel others, whereas gregarious locusts aggregate through short-range repulsion and long-range attraction.

  • Model equations: The continuum model tracks total density ρ(x,t)=s(x,t)+g(x,t) through separate solitary and gregarious density fields.The phase densities move with phase-specific velocities and interconvert through density-dependent rates.
  • Model equations: Solitarious movement is generated by the negative gradient of a nonlocal social-interaction potential.The convolution formulation represents interactions distributed over finite spatial ranges rather than only local derivatives.
  • Model equations: The two-phase equations generalize classic swarming models from one population density to interacting phase-specific populations.Classic models can produce steady swarms, spreading populations, or contracting groups depending on the social-interaction potential.
  • Model equations: Gregarious interactions combine short-range repulsion with long-range attraction, producing the clumping regime used to model aggregation.The potential uses interaction magnitudes Rg and Ag and length scales rg and ag, with parameter inequalities enforcing this structure.
  • Model equations: The phase-conversion rates depend on total density, with δ1,2 setting maximal rates and k1,2 setting half-maximal transition densities.Crowding favors conversion toward the gregarious phase in the model.

Parameter selection and estimation

The simulations use biologically motivated social-interaction parameters estimated from locust speeds and then adapt the remaining parameters for a one-dimensional periodic domain.

  • Parameter selection and estimation: Locust speed ranges provide the basis for estimating social-interaction strengths for solitary and gregarious individuals.The solitary speed when alone is 72–216 m/hr, while group speed is 144–216 m/hr.
  • Parameter selection and estimation: A critical group density of 65 locusts/m^2 is used to estimate the solitary interaction magnitude.The estimate considers a solitary locust near a semi-infinite group-density field.
  • Parameter selection and estimation: For one-dimensional simulations, the authors choose k1,2 = k = 8 locusts/m and dimension-adjusted social-interaction strengths.The one-dimensional parameter choices include Rs = 6.83, Rg = 6.04, and Ag = 12.9 m^2/(hr · locust).

Homogeneous steady states

Homogeneous steady states are parameterized by total density, and their phase composition shifts toward gregarious locusts as density increases. Under the default parameters, low-density states are mostly solitarious and high-density states mostly gregarious.

  • Homogeneous steady states: For a known mean density ρ0, the homogeneous steady state is obtained by setting time and space derivatives to zero in the phase equations.This produces solitary and gregarious steady-state components s0 and g0.
  • Homogeneous steady states: The phase fractions satisfy φs + φg = 1 and are computed from the homogeneous steady-state components.These fractions simplify later stability analysis.
  • Homogeneous steady states: The gregarious fraction φg increases monotonically with total density ρ0.The scaled variables are γ = δ1/δ2, K = k1/k2, and ψ = ρ0/k2.

Linear stability analysis

Linear stability analysis perturbs homogeneous steady states, decomposes the perturbations into Fourier modes, and identifies when a positive eigenvalue causes instability. Under the stated interaction assumptions, instability first appears in the long-wavelength mode q = 0 and occurs above a threshold gregarious fraction or critical density.

  • Linear stability analysis: The analysis writes s=s0+s1 and g=g0+g1, then linearizes the phase equations around the homogeneous steady state.The resulting perturbation equations describe growth or decay of small deviations.
  • Linear stability analysis: Fourier expansion converts the perturbation dynamics into independent ordinary differential equations for each wavenumber q.The mode amplitudes are represented in matrix form using Fourier-transformed interaction potentials.
  • Linear stability analysis: Instability requires the leading eigenvalue λ1 to be positive because λ2 is negative.The gregarious fraction enters through a decreasing factor (1−φg)/φg, so sufficiently large φg favors instability.
  • Linear stability analysis: Under ag ≥ rs and ag > rg, instability first occurs at q = 0, corresponding to long-wavelength perturbations.The relevant positive terms increase with q, so their minimum occurs at q = 0.
  • Linear stability analysis: Instability occurs when the gregarious fraction exceeds a threshold, which can be converted into a critical scaled density.The positive quadratic-formula branch yields the biologically meaningful density and is used to produce K–γ instability contours in Fig. 2.

Numerical simulation method

The simulations solve the phase-change model in one dimension on a periodic, finely resolved grid, using Fourier methods for social-interaction convolutions and finite differences for fluxes. Numerical diffusion and velocity thresholding stabilize computation while preserving the reported instability and late-stage dynamics.

  • Numerical simulation method: The model is simulated in one spatial dimension with periodic boundary conditions on a domain of length L and N = 1024 grid points.The fine grid resolves the steep edges of forming clusters.
  • Numerical simulation method: Social-interaction convolutions are evaluated as products in Fourier space, and velocities are computed pseudospectrally.The periodic formulation provides computational savings for convolution-based velocity calculations.
  • Numerical simulation method: The flux term is evaluated with a fourth-order accurate central finite difference.
  • Numerical simulation method: Small numerical diffusion reduces ringing caused by discontinuities in the density fields and also represents macroscopic random motion.
  • Numerical simulation method: Velocity thresholding limits speeds to vmax_g and affects transient clump speeds but not the initial instability or late-stage bulk dynamics.Without thresholding, individual locusts reach approximately 1.5 times vmax_g; the initial instability has small velocity and late dynamics are nearly stationary.

Locust Dynamics: Behavioral Phase Change and Swarming

The paper is titled “Locust Dynamics: Behavioral Phase Change and Swarming” and is authored by Chad M. Topaz, Maria R. D’Orsogna, Leah Edelstein-Keshet, and Andrew J. Bernoff.

  • Locust Dynamics: Behavioral Phase Change and Swarming: The listed authors are Chad M. Topaz, Maria R. D’Orsogna, Leah Edelstein-Keshet, and Andrew J. Bernoff.
  • Locust Dynamics: Behavioral Phase Change and Swarming: The document is identified as arXiv:1207.4968v1 in the q-bio.QM category, dated 20 Jul 2012.

Author Summary

The paper addresses how behavioral phase change and social interaction jointly drive the transition from dispersed solitarious locusts to aggregated gregarious hopper bands. It develops and analyzes a mathematical model, identifying a critical density threshold and quantifying collective phase change over time.

  • Author Summary: The study asks how phase change and social interaction contribute to the transition from dispersed solitarious locusts to destructive aggregated hopper bands.
  • Author Summary: The authors construct a mathematical model of the interplay between behavioral phase change and spatial dynamics.
  • Author Summary: Analysis and numerical simulations determine a critical density threshold for gregarious band formation.
  • Author Summary: The model quantifies collective phase change over time and discusses implications for preventative management and future biological experiments.

Introduction

Locust outbreaks cause extensive agricultural damage, while crowding can trigger a self-reinforcing shift from solitary to social behavior. The paper extends prior modeling by linking individual phase change and social interactions to population-scale hopper-band formation.

  • Introduction: Locust swarms can span up to a thousand square kilometers and travel a few hundred kilometers per day while consuming vegetation.
  • Introduction: A 2003–2005 West African plague destroyed $2.5 billion in crops, while control efforts totaling $400 million accompanied losses exceeding 50% in some regions.
  • Introduction: Overcrowding at resource-rich sites promotes transition to a social state through a self-reinforcing process, after which nymphs may exhibit mass migration.
  • Introduction: Locusts can reversibly switch between solitarious and gregarious states, with sparse surroundings favoring solitarization and crowded environments favoring gregarization.
  • Introduction: For Schistocerca gregaria, repetitive hind-leg stroking is a potent crowding stimulus that can induce gregarization after 5 seconds per minute for 4 hours.
  • Introduction: Unlike many individual-based models, this work uses a density-based Eulerian framework linking individual-level phase change and interactions to whole-band predictions.
  • Introduction: At low densities both phases remain uniformly spread with solitarious locusts dominant, whereas sufficiently large populations develop a dense traveling gregarious patch as solitarious locusts become scarce.

Model

The model couples spatial movement of solitarious and gregarious locust densities with density-dependent phase conversion. It uses phase-specific social interactions, periodic boundaries, and excludes reproduction and death because they operate on longer timescales.

  • Model equations: Nonlocal interaction forces are computed by convolving population density with the negative gradient of social potentials.This formulation represents spatially distributed attraction and repulsion rather than only local differential interactions.
  • Model equations: Separate density fields s(x, t) and g(x, t) track solitarious and gregarious locusts, with total density ρ = s + g.Their velocities arise from social interactions, while phase transitions connect the two populations.
  • Social interactions: Solitarious locusts are modeled as purely repulsive, whereas gregarious locusts experience short-range repulsion and longer-range attraction.The potentials use exponentially decaying interactions with amplitudes and length scales specifying force strengths and sensing distances.
  • Social interactions: The gregarious interaction parameters are restricted to a clumping regime so that attraction dominates at longer distances and macroscopic aggregation occurs.This restriction is needed because the balance distance between attraction and repulsion does not by itself determine population-level behavior.
  • Phase conversion: Density-dependent rational rates make gregarization faster and solitarization slower as local density increases.The rates have maximal values δ1,2 and characteristic half-maximal densities k1,2; f1 decreases with density while f2 increases and saturates.
  • Scope and implementation: The complete model uses periodic spatial boundaries and omits reproduction and death because those processes occur on much longer timescales than phase change.The framework is intended to be extendable to species differences, environmental heterogeneity, reproduction, and differing activity levels.

Results

The model predicts that sufficiently dense locust populations become unstable to clustering and mass gregarization, with critical thresholds shaped by phase-change and social-interaction parameters. Simulations and reduced theories describe transient gregarious clumps, eventual dominance of the gregarious phase, and hysteresis.

  • Homogeneous steady states: The gregarious fraction increases monotonically with total density, while low-density homogeneous states are mostly solitarious and high-density states mostly gregarious.Uniformly spread solitarious populations cannot be sustained when density becomes sufficiently high.
  • Linear stability analysis: For sufficiently large gregarious fractions, small perturbations grow and destabilize the homogeneous steady state.The instability condition is expressed through the critical gregarious fraction φ∗g.
  • Linear stability analysis: The critical scaled density ψ∗ increases with both γ and K, and the default parameters give instability above ρ∗0 = 62.3 locusts/m2.The unscaled threshold is obtained by multiplying ψ∗ by k2.
  • Linear stability analysis: Instability can begin even when solitarious locusts outnumber gregarious ones, and parameter variation produces substantially different critical densities.An alternative parameter set yields ρ∗0 = 15.9 locusts/m2, below both the individual gregarization density of 65 locusts/m2 and the solitarization density of 20 locusts/m2.
  • Linear stability analysis: Social-interaction parameters can permit instability and clumping with a critical gregarious fraction φ∗g much less than 1/2.Thus, only a few gregarious insects may be sufficient for instability in some parameter regimes.
  • Linear stability analysis: The most rapidly growing perturbations saturate at qmax ≈ 8.89 m−1, corresponding to a length scale of approximately 0.71 m.This identifies the characteristic scale of the fastest-growing spatial structures.
  • Numerical simulation: One-dimensional periodic simulations produce transient gregarious clumps that rapidly merge through long-range attraction, with the initial number of clumps depending on parameters.The default and alternative parameter sets produce two and three initial density peaks, respectively.
  • Spatially-homogeneous and spatially-segregated bulk theories: For large populations, the solitarious fraction decays exponentially at rate δ2 after segregation is nearly complete.The approximation is based on a segregated-state assumption and has leading-order corrections of O(1/M 2).

Discussion

The paper connects individual social interactions and density-dependent phase changes to population-level locust aggregation, identifying thresholds, hysteresis, and control implications while noting environmental limitations.

  • The model links intrinsic social interactions with density-dependent interconversion between solitarious and gregarious locusts.It complements models focused on environmental heterogeneity by representing both intrinsic and extrinsic influences on local density and gregarization.
  • Mathematical analysis connects individual sensing-range and interaction-strength parameters to a critical density for mass gregarization.This provides an explicit route from individual-level behavior to group-level transition thresholds.
  • Uniform distributions persist only up to a critical total population density, beyond which dense gregarious clusters form.Linear stability analysis relates the critical density to dimensionless biological-parameter ratios.
  • Population-level hysteresis makes the density required to form a gregarious aggregation significantly higher than the density required for it to break up.The same model parameters are used while average density is varied as the control parameter.
  • These results suggest preventing locust group formation may be easier than dispersing an already gregarized band.The control implication follows from the lower density needed for band annihilation than for initial aggregation formation.
  • The model excludes vegetation, resource gradients, and other environmental cues, limiting direct biological relevance and preventing reproduction of long-range band motion.Adding environmental factors and extending simulations to two spatial dimensions are identified as future directions.
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