Bat Motion can be Described by Leap Frogging

Abstract

We present models of bat motion derived from radio-tracking data collected over 14 nights. The data presents an initial dispersal period and a return to roost period. Although a simple diffusion model fits the initial dispersal motion we show that simple convection cannot provide a description of the bats returning to their roost. By extending our model to include non-autonomous parameters, or a leap frogging form of motion, where bats on the exterior move back first, we find we are able to accurately capture the bat's motion. We discuss ways of distinguishing between the two movement descriptions and, finally, consider how the different motion descriptions would impact a bat's hunting strategy.

Keywords: Bat motion; Partial differential equations.

Fig. 1

Greater horseshoe bats with radio transmitters glued to their backs. The radio transmitters have very thin antennae, and are highlighted in white. Photographs taken by Professor Fiona Mathews

Fig. 2

The locations of the same bat over two nights during the survey. The roost has been normalised to be at the origin in each case. The circles represent detections and the numbers next to them represent the time in hours after sunset that the bat was detected at the given location

Fig. 3

A histogram of the time intervals between consecutive recordings

Fig. 4

a The mean-squared distance (MSD) for all radio tracked bats interpolated at $\mathrm{\Delta }t=200$s. b The MSD for all radio tracked bats interpolated at $\mathrm{\Delta }t=1000$, 2000 and 3000 s, from left to right, respectively. The red line is the MSD trajectory and the ribbon represents the mean±standard error of the squared displacement trajectory data

Fig. 5

a Simulation of Eq. (2), the diffusion in polar coordinates, and b derived MSD from Eqs. (6) and (8) compared with approximations from Eqs. (10) and (12). Parameter values are $D=100$ m²/s and $R=2000$ m (Color Figure Online)

Fig. 6

Population density simulations (left) and MSD plots (right) from Eq. (13) for different convection functions, $\mathit{v}$. All simulations occur on a circle of radius $R=2000$m with $D=100$m²/s. In a and b $\mathit{v}=\chi \widehat{\mathit{r}}$, c and d $\mathit{v}=\chi r^2\widehat{\mathit{r}}$ and e and f $\mathit{v}=\chi /(1+r^2)\widehat{\mathit{r}}$. In the left-hand figures $\chi =1$, whilst $\chi$ is specified in the legend of the right-hand figures (Color Figure Online)

Fig. 7

a Population density simulations and b MSD from Eq. (13) using temporally evolving diffusion and convection terms from Eq. (17) fitted to the data in Fig. 4a. The parameters and 95% confidence intervals are $D_c=100$m²/s, [94.4 106]m²/s; $t_c=8.23$h, [7.44, 9.00]h; and ${\chi }_c=0.83\times {10}^{-5}$m/s², [0.75, 0.91]$\times {10}^{-5}$m/s². In b The red line is the MSD trajectory and the ribbon represents the mean±standard error of the squared displacement trajectory data. The black solid line represents the simulated data with best-fit parameters. The dashed lines represent simulations using parameters from the upper and lower limits of the confidence intervals (Color figure online)

Fig. 8

Schematic diagram illustrating the leap frogging strategy of movement in one-dimension. The black bats are all moving randomly. The furthest out, highlighted in blue convects until it is no longer the furthest out. Its motion then becomes diffusive. The bat that is now furthest out starts to convect towards the roost and the process repeats (Color figure online)

Fig. 9

a Results of a diffusion simulation on a shrinking domain at different times given in the legend. The dashed line presents the boundary location at the given times. b The extracted values of MSD, with $R{(t)}^2/2$ for comparison. The initial condition is a delta function at $r=0$ and the time-dependent radius of the domain is $R(t)=R_0$ for $t<t_s$ and $R(t)=R_0\sqrt{1-{\left((t-t_s),/,(8-t_s)\right)}^2}$, with $R_0=2000$m, $t_s=1.5$ hours and $D=100$m²/s (Color Figure Online)

Fig. 10

Fitting the two phases of bat MSD data. a Phase 1 presents the data for $t<1.5$hrs after sunset. The data is fitted with a straight line. b Phase 2 presents the data for $t>1.5$hrs after sunset. The data is fitted with a quadratic polynomial. The ribbons represent the standard error

Fig. 11

MSD of diffusing agents on a shrinking domain. The red line is the MSD trajectory and the ribbon represents the mean±standard error of the sqaured displacement trajectory data. The black lines represent simulated data. The diffusion cofficient is $D=a_{11}/4$ and the radius of the domain, R(t), is given by Eq. (21). The solid line uses the best-fit parameter values, whilst the upper and lower dashed lines use the upper and lower values from the 95% parameter confidence intervals. The parameters are specified in Table 1. a The parameters are fitted with $t_s=1.5$ hours. b The parameters are refitted with $t_s$ included in the fitting (Color figure online)

Fig. 12

Comparing the density of returning bats within 100 m of the roost across the two simulated mechanisms of non-autonomous convection-diffusion and leap frogging with the trajectory data (Color Figure Online)