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5.3.3.1 Intra- & inter- individual variation

To determine if there were significant intra-individual differences in the basic summary path statistics, a repeated measures ANOVA was used to compare observed results across the three trials. Inter-individual variation across the study population was tested by a one-way ANOVA (or Kruskal-Wallis test for those summary statistics which violated the homogeneity of variance assumption) on the mean of each summary statistic across the three trials. These analyses were carried out using JASP v.0.8.0.0 (JASP Team, 2018) and R (R Core Team, 2018).

To measure the consistency of behaviour among individuals, the repeatability, r, (also known as the intra class coefficient, ICC (Lessels & Boag, 1987)) was calculated, where

ICC = Vind Vind+ V

(5.1)

with Vindthe variance among individuals and V the residual (error) variance (Nakagawa

& Schielzeth, 2010; Dingemanse & Dochtermann 2013; Houslay & Wilson 2017). There- fore, ICC tells us the relative strength of the variance between individuals compared to the total variance, with the total variance considered as the sum of the variance among individuals, Vind, and the total variance within individuals V – whilst this variance is

technically a measure of the residuals, it is commonly referred to as within-individual variation (Brommer 2013; Dingemanse & Dochtermann 2013; Brommer et al 2014; Dos- mann et al, 2015). These variances were found using Linear Mixed Effect Models using Restricted Maximum-Likelihood parameter estimation following the method described in Nakagawa & Schielzeth (2010) by use of the rptR package (Stoffel et al, 2017) in R (R Development Core Team, 2019).

5.3.3.2 Correlation in movement parameters Between individuals

To see if there were a correlation between any of the parameters at the between-individual level, (e.g. does a beetle which has higher displacement on average also spend more time moving than the average?) a bivariate (two-trait) mixed model was used (Houslay & Wilson, 2017), with the individual beetle as the random intercept, the experiment number (centred) as the repeat number, and the parameters (centred and scaled) as the random effects, as per Houslay & Wilson (2017).

The model was then implemented by the MCMCglmm package (Hadfield 2010) in R (R Development Core Team, 2019). In order to ensure auto-correlation was not an effect, a large number of iterations were run 500,000 with a ‘burn-in’ period of 15,000 and a thinning of 100. Results were deemed to be significant if the confidence intervals (95%) did not span 0, as is standard with Bayesian CI’s (Houslay & Wilson 2017).

The correlation between two parameters, rindα, rindβ, is then found by calculating their

between individual covariance, COVindα,indβ, and dividing by the square root of the prod-

uct of the between individual variances of the two parameters, Vindα, Vindβ (Dingemanse

& Dochterman 2013; Dosmann & Mateo, 2014; Dosmann et al, 2015).

rindα,indβ =

COVindα,indβ

pVindαVindβ

(5.2)

Within individuals

For any parameter whose residual variance V (equivalent to the variability within each

individual across the three trial runs) was seen to be high, we might wish to answer the question whether a correlation exists within individual’s trial runs for each pair of parameters e.g. did trials with a higher displacement also feature fewer bouts?

We use the same bivariate mixed effect model used to find the between individual correlation (described above) to find any such correlation between two parameters, rα,β,

using a similar calculation as in Eq. 5.2 except we now consider the covariance of the parameters at the within-individual level, COVα,β, divided by the square root of

the product of the variances of each parameter at the within-individual level Vα, Vβ

(Dingemanse & Dochterman 2013; Dosmann et al, 2014).

rα,β =

COVα,β

pVαVβ

(5.3)

One could ask why a simple correlation test (Spearman’s, Pearson’s, etc.) for each individual trial or for the mean of each statistic across the three trials would not suffice as a test of the relationship between parameters. However, such a method has shown to be over generous with the significance level of any resulting correlations (Hadfield et al, 2010; Dingemanse et al 2012; Houslay & Wilson, 2017).

5.3.3.3 Analysis of population level movement dynamics

Global orientation of movement directions (corresponding to bouts of movement only) were considered at both the population and individual level, to ascertain whether there was a global or an individual preference in direction. As global orientations form circu- lar data, the Watson Test was used to test if the distribution of orientations could be described by the circular uniform distribution (see section 3.6). The Rayleigh test then determined whether the distribution corresponded to a unimodal wrapped distribution with specific resultant vector, where a resultant vector close to 1 would indicate a strong preference in movement direction, while a resultant vector close to 0 would indicate no preference in movement direction (Mardia & Jupp, 2000; Jammalamadaka & SenGupta, 2011).

The observed turning angles (corresponding to bouts of movement only) were fitted to two standard circular probability distributions: the von Mises (which is a close ap- proximation to the normal distribution on a circle) and the wrapped Cauchy (which is a heavy-tailed circular distribution). These distributions were fitted using the CircStats package in R. The Kuiper and the Watson-U2 tests were used to check the validity of both models, with the Akaike Information Criterion (AIC) used to indicate the closer fitting distribution. Evidence of unimodal turning angle distributions centred around 0 would indicate persistence in the beetles’ movements.

Four distributions were considered for fitting the observed distribution of step lengths (instantaneous speeds; corresponding to bouts of movement only), with the same dis- tributions also considered for the movement and non-movement bout durations: power- law, exponential, Weibull and log-normal. Distributions were fitted using the fitdistrplus package in R, except for the power-law that was fitted using the power.law.fit function in the iGraph package in R. We fit the power-law in two ways. Firstly, in order to ensure the power-law fitted all the data, we used a restricted power-law where the xminvalue was

set at the smallest non-zero value of the data rather than the value for xmincalculated by

power.law.fit function (Virkar & Clauset, 2014). Secondly, we also consider a power-law fit to only the tail of the data, as this is one of the features described in the literature that is indicative of L´evy walk behaviour (Sims et al, 2007; Edwards et al, 2007; Reynolds et al, 2013; Ahmed et al, 2018). The tail of the data was calculated by using the best fit xmin value calculated by the power.law.fit function, and then the potential distributions

were fitted only to the data points which were greater than this minimum value. As the fitting algorithm for the power-law utilised a maximum likelihood estimation (MLE) method to maximise the p-value for the Kolmogorov–Smirnov (K-S) test, a G-test was also used to consider the fit of the distributions (Edwards et al, 2007).

Data for turning angles and step lengths (speeds) were fitted at the population level (10045 data points from 66 movement paths) and at the individual path level (between 37 and 298 data points for each movement path).

5.3.3.4 Comparison of paths as a CRW or BRW

To further investigate whether the characteristics of the beetle movement paths could be best classified as either a correlated random walk (CRW; i.e. movement is persistent but not globally directed) or a biased random walk (BRW; i.e. movement is globally directed), we measured the ∆ statistic from (Marsh & Jones, 1988):

∆ = 1 n2  X cos φi 2 +Xsin φi 2 − 1 (n − 1)2  X cos θi 2 +Xsin θi 2 (5.4)

where, φi is the global orientation and θi is the turning angle, at time i. The ∆ statistic

gives a relative measure of how well the observed data fits each of the two types of random walk movement model by returning a positive value for a BRW and a negative value for a CRW (see details in Appendix B5).

The ∆ statistic was calculated for each individual movement path separately and also for all turning and global orientation angles aggregated at the population level. Data for bout durations were fitted only at the population level due to the limited number of data points from each individual path (326 data points from 66 movement paths).

An additional method was also used to determine evidence of either CRW or BRW behaviour. This involved sub-sampling the observed movement data across a range of steps and determining how the estimated mean cosine of turning angles changes with the sub-sampling step used (Bovet & Benhamou, 1988; Codling & Hill, 2005; Benhamou, 2006). For a CRW the observed mean cosine of turning angles is expected to decrease to 0 as correlation between increasingly distant steps diminishes. However, for a BRW, the observed mean cosine is expected to increase towards 1 as the movement path appears increasingly more linear due to the global directional bias present.

5.4

Results