Swarming Insects May Have Finely Tuned Characteristic Reynolds Numbers

Abstract

Over the last few years, there has been much effort put into the development and validation of stochastic models of the trajectories of swarming insects. These models typically assume that the positions and velocities of swarming insects can be represented by continuous jointly Markovian processes. These models are first-order autoregressive processes. In more sophisticated models, second-order autoregressive processes, the positions, velocities, and accelerations of swarming insects are collectively Markovian. Although it is mathematically conceivable that this hierarchy of stochastic models could be extended to higher orders, here I show that such a procedure would not be well-based biologically because some terms in these models represent processes that have the potential to destabilize insect flight dynamics. This prediction is supported by an analysis of pre-existing data for laboratory swarms of the non-biting midge Chironomus riparius. I suggest that the Reynolds number is a finely tuned property of swarming, as swarms may disintegrate at both sufficiently low and sufficiently high Reynolds numbers.

1. Introduction

Mating swarms of flying insects are a form of collective animal behaviour that typically displays a high degree of spatial cohesion but lacks the ordered collective movements that are a defining feature of herds, flocks, and schools [1,2,3,4,5]. As the insects circulate within the swarm, they execute erratic flight patterns [1,2,6,7]. Over the last few years, there has been much effort put into the development and validation of stochastic models of the trajectories of swarming insects [8,9,10,11,12]. These models account for numerous observations, including the emergence of dynamical scaling and correlations in perturbed swarms, the emergence of macroscopic mechanical properties like tensile strength, and the ability of swarms to be driven through ‘thermodynamic cycles’ by the application of external perturbations [13,14,15,16,17,18]. These models typically assume that the positions and velocities of swarming insects can be represented by continuous jointly Markovian processes, or more rarely, that the positions, velocities, and accelerations of swarming insects are collectively Markovian. Mathematically, these models can be seen to be the lowest levels in a hierarchy that could be extended to higher orders. Physically, the hierarchy corresponds to the inclusion of a timescale representative of the largest scales of motion at first order, and to the addition of a second timescale representative of the smallest scales of motion at second order. This is directly analogous to stochastic models of the trajectories of tracer particles in high Reynolds number turbulence, wherein the Reynolds number, $R={\left({\displaystyle \frac{T}{t_2}}\right)}^2$, which is determined by the ratio of a timescale representative of the energy-containing scales, T, and the Kolmogorov time scale, $t_2$, representative of the dissipative scales of motions appears as a parameter at second order [19]. This Lagrangian Reynolds number is proportional to the better-known Eulerian Reynolds number [19]. Although it is mathematically conceivable that this hierarchy of stochastic models could be extended to higher orders [20,21,22], in the case of high Reynolds number turbulence is it not apparent that such a procedure would be well-based physically since there is no obvious relevant timescale smaller than the Kolmogorov timescale [19]. Here in the case of swarming insects, I show that the procedure is not well-based biologically.

2. Materials and Methods

Third-order one-dimensional models for the positions, $x$, velocities, $u$, accelerations, $A$, and jerks, $J$, of swarming insects are given by

$$dJ=a\left(J,A,u,x\right)dt+bdW\left(t\right)$$

(1)

$$dA=Jdt$$

$$du=Adt$$

$$dx=udt$$

where $a\left(J,A,u,x\right)$ is featured in the Fokker–Planck equation:

$${\displaystyle \frac{\partial P_3}{\partial t}}+u{\displaystyle \frac{\partial P_3}{\partial x}}+A{\displaystyle \frac{\partial P_3}{\partial u}}+J{\displaystyle \frac{\partial P_3}{\partial A}}=-{\displaystyle \frac{\partial }{\partial J}}\left(a{P}_3\right)+{\displaystyle \frac{b^2}{2}}{\displaystyle \frac{{\partial }^2{P}_3}{\partial J^2}},$$

(2)

$P_3\left(J,A,u,x,t\right)$ is the joint distribution of $J, A, u, x$ and time, $t$, $b$ is the magnitude of the driving noise, and $d{W}_i\left(t\right)$ is an incremental Wiener process with correlation property $\overline{dW\left(t\right)dW\left(t+\tau \right)}=\delta \left(\tau \right)dt$ [20]. The prescription of $a\left(J,A,u,x\right)$ guarantees that the statistical properties of the simulated trajectories are distributed according to $P_3\left(J,A,u,x,t\right),$ which a model input. The deterministic term $a\left(J,A,u,x\right)$ takes the form:

$$a={\displaystyle \frac{b^2}{2}}{\displaystyle \frac{\partial }{\partial J}}lnJ+{\displaystyle \frac{\varphi }{P_3}}$$

(3)

where

$${\displaystyle \frac{\partial \varphi }{\partial J}}={\displaystyle \frac{\partial P_3}{\partial t}}+u{\displaystyle \frac{\partial P_3}{\partial x}}+A{\displaystyle \frac{\partial P_3}{\partial u}}+J{\displaystyle \frac{\partial P_3}{\partial A}}$$

(4)

Integrating Equation (2) over all $J$ gives an equation for the average jerk strength:

$$0={\displaystyle \frac{\partial P_2}{\partial t}}+u{\displaystyle \frac{\partial P_2}{\partial x}}+A{\displaystyle \frac{\partial P_2}{\partial u}}+\langle J\rangle {\displaystyle \frac{\partial P_2}{\partial A}}$$

(5)

where $P_2\left(A,u,x,t\right)$ is the joint distribution of $A, u, x$ and time, $t$.

Integrating Equation (6) over all $J$ gives an equation for the average acceleration:

$$0={\displaystyle \frac{\partial P_1}{\partial t}}+u{\displaystyle \frac{\partial P_1}{\partial x}}+\langle A\rangle {\displaystyle \frac{\partial P_1}{\partial u}}$$

(6)

where $P_1\left(u,x,t\right)$ is the joint distribution of $u, x$ and time, $t$.

The least biased choice for $P_3\left(J,A,u,x,t\right)$ and the one adopted here is a multivariant Gaussian. The resulting stochastic models for the trajectories of swarming insects are minimally structured (maximum entropy) models. It follows from Equations (3)–(6) that for stationary swarms with Gaussian statistics:

$$\langle A\rangle =-{\displaystyle \frac{{\sigma }_u^2}{{\sigma }_x^2}}x$$

(7)

$$\langle J\rangle =-\left({\displaystyle \frac{{\sigma }_A^2}{{\sigma }_u^2}}+{\displaystyle \frac{{\sigma }_u^2}{{\sigma }_x^2}}\right)u$$

(8)

$$\langle S\rangle \equiv {\displaystyle \frac{\varphi }{P_3}}=-\left({\displaystyle \frac{{\sigma }_J^2}{{\sigma }_A^2}}+{\displaystyle \frac{{\sigma }_A^2}{{\sigma }_u^2}}+{\displaystyle \frac{{\sigma }_u^2}{{\sigma }_x^2}}\right)A-{\displaystyle \frac{{\sigma }_J^2}{{\sigma }_A^2}}{\displaystyle \frac{{\sigma }_u^2}{{\sigma }_x^2}}x$$

(9)

Insect trajectories were simulated by numerically integrating the stochastic models. Other predicted quantities, such as velocity spectra, were obtained analytically, as detailed in the Appendix A, Appendix B and Appendix C. A full description of the experimental setup that provided data for the model validation can be found in Sinhuber et al. [23].

3. Results

The predicted average acceleration towards the centre is a defining feature of insect swarms that keeps the swarms intact [2]. As observed, this effective force increases linearly as the distance from the swarm centre increases. Individuals in real and simulated swarms therefore behave on average as if they are trapped in elastic potential wells. Model predictions, Equation (8), for the average strength of the jerks, are in quantitative agreement with observations of asymptotically large swarms [5] containing on average 15 to 94 individuals [24]. This correspondence indicates that swarming insects are described by second or higher-order models.

The average strength of the snaps, $\langle S\rangle$, is seen to enter the model formulation at third order. One contribution to this quantity is aligned with the acceleration vector, which itself tends to be aligned with the position vector (Equation (7)). The other contribution to $\langle S\rangle$ is manifestly aligned with the position vector. This contrasts with the average strength of the jerks, which enter the model at second order and are aligned with the velocity vector, i.e., aligned with the direction of travel and so aligned with the major axis of the insect. Such alignment minimizes the impact that jerks can have on flight dynamics. This is not the case with snaps, which can momentarily be aligned with the minor axis of the insect, thereby maximizing their disruptive impact on flight dynamics. This suggests that swarming insects are at most described by second-order models.

Experimental support for this prediction hinges on the fact that the velocity spectra for swarming insects are compatible with predictions from second-order models. Free-roaming trajectories are predicted by first- and second-order models to be characterised by velocity spectra that decrease respectively as ${\omega }^{-2}$ and as ${\omega }^{-4}$at high frequencies, whereas spectra decreasing faster than ${\omega }^{-4}$can only be captured by third- or higher-order models [19,20]. Confinement within a swarm does not change these scaling behaviours (Appendix A [25]). Instead, the quantity ${\displaystyle \frac{{\sigma }_u}{{\sigma }_x}}$ determines the position of the peak in the velocity spectra. The velocity spectra characterising the trajectories of swarming non-biting midge Chironomus riparius recorded in quiescent conditions in the laboratory decreases approximately as ${\omega }^{-3}$ at the highest frequencies accessible (Figure 1). This scaling cannot arise in first-order models but, as illustrated, does arise at low frequencies in second-order models when $R~O\left(100\right)$. This scaling is obtained for 19 different containing on average 15 to 94 individuals. Third-order processes are not evident.

4. Discussion

Herein it was argued that the trajectories of swarming insects, like the trajectories of tracer particles in turbulence, are at most described by second-order models in which the position, velocity, and acceleration of an insect are collectively Markovian, since higher-order processes, even if present, are not significant. This strengthens previously identified correspondences between swarming insects and the Lagrangian properties of high Reynolds number turbulence [24]. Their acceleration statistics have similar conditional dependencies on velocity. These conditional dependencies only become apparent for $\left|u\right|>2{\sigma }_u,$ and their occurrences may be attributed to occasional energetic rotations. The small size of this Reynolds number (as determined by matching predicted and observed velocity spectra), $R~O\left(100\right)$, may be a consequence of the fact that the average strength of the jerks, Equation (8), increases with increasing Reynolds number. Equation (8) can be rewritten as $\langle J\rangle =-\left({\displaystyle \frac{R^{1/2}}{T^2}}+{\displaystyle \frac{{\sigma }_u^2}{{\sigma }_x^2}}\right)u$. At sufficiently high Reynolds numbers, jerks, like snaps, may have the potential to destabilize flight dynamics, thereby causing the swarm to disintegrate. Indeed, the smallness of the estimate for the Reynolds number may be indicative of the susceptibility to swarming midges to the disruptive impact of jerks. The swarm may also disintegrate at sufficiently low Reynolds numbers following a disordered-order phase transition [12] if the confining potential, a collective emergent property of disordered swarms [1,2], cannot emerge in ordered swarms, or it may lose its collective properties if individuals remain in the vicinity of the swarm marker (a visually prominent marker over which swarm form). If this line of reasoning is correct, then the Reynolds number may be the result of fine-tuning, as are other emergent properties of swarming [12,26]. Jerks may also be particularly disruptive in swarms that are not asymptotically large (Appendix B [27,28]).

To summarise, even though the collective animal motions can exhibit fluidic behaviours, Reynolds numbers, which are perhaps the most ubiquitous quantity in the literature on fluid dynamics, have barely been featured in the literature on collective animal behaviours [29]. Herein, I addressed this shortcoming by showing that the trajectories of swarming insects are analogous to the trajectories of tracer particles in fluidic turbulence and that consequently swarming insects can be assigned Reynolds numbers. This new result strengthens ongoing attempts to describe collective animal behaviours as active matter [30] Moreover, the Reynolds numbers were shown to convey important new biological information, as all theoretical advances into collective animal behaviours should strive to do [31].