Environmental Fluid Mechanics

, Volume 8, Issue 2, pp 147–168 | Cite as

Modelling of concentrations along a moving observer in an inhomogeneous plume. Biological application: model of odour-mediated insect flights

Original Article
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Abstract

We develop a stochastic model for the time-evolution of scalar concentrations and temporal gradients in concentration experienced by observers moving within inhomogeneous plumes that are dispersing within turbulent flows. In this model, scalar concentrations and their gradients evolve jointly as a Markovian process. Underlying the model formulation is a natural generalisation of Thomson’s well mixed condition [Thomson DJ (1987) J Fluid Mech 180:529–556]. As a consequence model outputs are necessarily compatible with statistical properties of scalars observed in experiment that are used here as model input. We then use the model to examine how insects aloft within the atmospheric boundary-layer can locate odour sources by modulating their flight patterns in response to odour cues. Mechanisms underlying odour-mediated flights have been studied extensively at laboratory-scale but an understanding of these flights over landscape scales is still lacking. Insect flights are simulated by combining the stochastic model with a simple model of insect olfactory response. These simulations show the strong influence of wind speed on the distributions of the times taken by insects to locate the source. In accordance with experimental observations [Baker TC, Vickers NJ (1997) In: Insect pheromone research: new directions, pp 248–264; Mafra-Neto A, Cardé RT (1994) Nature 369:142–144], flight patterns are predicted to become straighter and shorter, and source location is predicted to become more likely as the mean wind speed increases. The most probable arrival time to the source decreases with the mean wind speed. It is shown that scale-free movement patterns arising from olfactory-driven foraging stem directly from the power-law distribution of concentration excursion times above/below a threshold level and are robust with respect to variations in Reynolds number. Flight lengths are well represented by a power law distribution in agreement with the observed patterns of foraging bumblebees [Heinrich B (1979) Oecologia 40(3):235–245].

Keywords

Odour-source location Inhomogeneous plume Intermittency Scale-free searching Lévy-walks 

List of Symbols

[[L]]

Symbol for the dimension of length

[[M]]

Symbol for the dimension of mass

[[T]]

Symbol for the dimension of time

[Sc]

Schmidt number [Dimensionless]

[Re]

Reynolds number [Dimensionless]

[P]

Eulerian probability density function for the pseudo concentration \({c^\prime}\) and the gradient of pseudo concentration \({{\dot{c}}}\) [L 6 M −2 T]

\({[P_{c^{\prime}}]}\)

Eulerian probability density function for the pseudo concentration \({c^\prime}\) [L 3 M −1]

\({{[P_{\dot{c}}]}}\)

Eulerian probability density function for the gradient of pseudo concentration \({c^\prime}\) [L 3 M −1 T]

\({[a(c^{\prime}, \dot{c}, {\bf r}, t)]}\)

Deterministic term in the stochastic differential equation, see Eq. 4 [ML −3 T −2]

\({[b(c^{\prime}, \dot c, {\bf r}, t)]}\)

Random term in the stochastic differential equation, see Eq. 4 [ML −3 T −3/2]

\({{[\dot{c}]}}\)

Concentration fluctuation gradient [ML −3 T −1]

[c]

Odour concentration [M L −3]

\({[c^\prime]}\)

Pseudo concentration [M L −3]

[cbase]

Concentration shift required to produce intermittency factor γ from the pseudo concentration \({c^\prime}\) in the simulated time series [M L −3]

[cfixed]

Concentration threshold [M L −3]

[cL]

Concentration threshold triggering cessation of movement [M L −3]

[cT]

Concentration threshold triggering upwind progression locating manoeuvres [M L −3]

[C]

Conditional mean concentration excluding zero concentration at the source height [M L −3]

[D]

Molecular diffusivity of the odour [L 2 T −1]

[h]

Height of the source above the ground [L]

[i]

Unconditional fluctuation intensity [Dimensionless]

[ip]

Conditional fluctuation intensity [Dimensionless]

[k]

Turbulent diffusivity [L 2 T −1]

[Q]

Mass flux of odour [M T −1]

[r]

Position of the insect relative to the source on a plane above the ground at the source height [L] [t] Time [T]

[Tarr]

Most probable arrival time to the source for the insect [T]

[Tc]

Integral timescale representative of the larger-scale eddies in the flow [T]

\({[T_c^{\prime}]}\)

Timescale representative of the larger-scale eddies in the flow for the second-order model. For the relation with the integral time scale T c , see Eq. 3 [T]

[Texp]

Exposure time of the insect to a concentration higher than \({c_{T}}\) [T]

[te]

Mean excursion time between and up-crossing and subsequent down-crossing [T]

[tL]

Time scale of fluid motions estimated as in Mylne and Mason [37] [T]

[tmax]

Maximum searching time before insect is assumed to have missed the source [T]

[Tmin]

Minimum time for an insect to reach the source in the limit of no cross-wind movement [T]

[U]

Mean speed of the carrier [L T −1]

[u*]

Friction velocity [L T −1]

[ux, uy]

Upwind and crosswind components of the velocity of the insects [L T −1]

[x, y]

Upwind and crosswind components of the position of the insects r [L]

[x0]

Spatial extent of plume meander [L]

[xs]

Location downwind the source where insects are released [L]

[xvs, yvs]

Visual range of the insect [L]

[z]

Height above the ground where the mean wind speed is measured [L]

[z0]

Typical roughness height of the ground [L]

[δ(t)]

Dirac delta function [Dimensionless]

\({[\bar {\epsilon}]}\)

Mean rate of dissipation of turbulence kinetic energy [L 2 T −3]

[γ]

Intermittency factor [Dimensionless]

[κ]

Von Karman constant [Dimensionless]

[ν]

Kinematic viscosity of the fluid [L 2 T −1]

[σ]

Standard deviation of the wind velocity fluctuations [L T −1]

\([\sigma_{c^\prime}^{2}]\)

Variance of the pseudo concentration \(c^{\prime}\) [M L −3]

\({[\sigma_{l}^{2}]}\)

Mean square plume width [L 2]

\({[\sigma_{s}^{2}]}\)

Mean square source size [L 2]

[σw]

Standard deviation of the vertical component of the wind velocity fluctuations [L T −1]

\({[\tau_{\eta}]}\)

Timescale representative of the dissipative eddies in the flow; Kolmogorov time scale [T]

[d c]

Infinitesimal increment in concentration [M L −3]

\({[{\rm d} c^{\prime}]}\)

Infinitesimal increment in pseudo concentration [M L −3]

\({{[{\rm d} {\dot c}]}}\)

Infinitesimal increment in concentration gradient [M L −3 T −1]

\({{[\sigma_{{\dot c} \mid c^{\prime}}^{2}]}}\)

Conditional variance of temporal gradients of pseudo concentrations [M 2 L −6 T −2]

[d t]

Infinitesimal time increment used in the computation of Eq. 4 [T]

[Δt]

Time interval used as bin width for histograms [T]

[d ξ]

Incremental Weiner process with mean zero and variance d t [T 1/2]

[d x, d y]

Upwind and crosswind increments of the position of the insect [L]

\({{[\langle c \rangle]}}\)

Ensemble average of the simulated concentration [M L −3]

\({{[\langle c^{2} \rangle]}}\)

Ensemble average of the square of the simulated concentration [M 2 L −6]

\({{[\langle {\dot c} {\mid} c^{\prime} \rangle]}}\)

Conditional mean temporal gradient of pseudo concentration [ML −3 T −1]

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Copyright information

© Springer Science+Business Media B.V. 2008

Authors and Affiliations

  1. 1.Rothamsted ResearchHarpendenUK

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