Three-dimensional random walk models of individual animal movement and their application to trap counts modelling

Background Random walks (RWs) have proved to be a powerful modelling tool in ecology, particularly in the study of animal movement. An application of RW concerns trapping which is the predominant sampling method to date in insect ecology, invasive species, and agricultural pest management. A lot of research effort has been directed towards modelling ground-dwelling insects by simulating their movement in 2D, and computing pitfall trap counts, but comparatively very little for flying insects with 3D elevated traps. Methods We introduce the mathematics behind 3D RWs and present key metrics such as the mean squared displacement (MSD) and path sinuosity, which are already well known in 2D. We develop the mathematical theory behind the 3D correlated random walk (CRW) which involves short-term directional persistence and the 3D Biased random walk (BRW) which introduces a long-term directional bias in the movement so that there is an overall preferred movement direction. In this study, we consider three types of shape of 3D traps, which are commonly used in ecological field studies; a spheroidal trap, a cylindrical trap and a rectangular cuboidal trap. By simulating movement in 3D space, we investigated the effect of 3D trap shapes and sizes and of movement diffusion on trapping efficiency. Results We found that there is a non-linear dependence of trap counts on the trap surface area or volume, but the effect of volume appeared to be a simple consequence of changes in area. Nevertheless, there is a slight but clear hierarchy of trap shapes in terms of capture efficiency, with the spheroidal trap retaining more counts than a cylinder, followed by the cuboidal type for a given area. We also showed that there is no effect of short-term persistence when diffusion is kept constant, but trap counts significantly decrease with increasing diffusion. Conclusion Our results provide a better understanding of the interplay between the movement pattern, trap geometry and impacts on trapping efficiency, which leads to improved trap count interpretations, and more broadly, has implications for spatial ecology and population dynamics.


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Modelling individual animal movement and navigation strategies using random walks has long been a 29 successful tradition in movement ecology (Nathan et al., 2008). The earliest models considered animal 30 paths as uncorrelated and unbiased, e.g. Simple Random Walks (SRW) (Lin and Segel, 1974; Okubo, 31 1980). A natural extension known as the Correlated Random Walk (CRW), firstly conceived by Patlak 32 (θ i+1 , φ i+1 ), between locations x i and x i+1 , can be modelled as an orthodromic (or great circle) arc, char-108 acterized by two angles: the initial arc orientation β i , measured between −π and π in the frontal plane 109 with respect to the horizontal level, and the arc size ω i , measured between 0 and π in the plane defined by 110 the two headings: (2.1.2) For a balanced CRW (including SRW as a special case) or BRW, the random variable β is independent 112 of ω, and its distribution must also be centrally symmetric so that its mean sine and cosine are both null. 113 Whether short or long term directional persistence is incorporated into the RW can be realised through the 114 mean cosine of ω, c ω : one gets c ω > 0 for a balanced CRW and BRW and c ω = 0 for a SRW. CRW and 115 BRW can be further distinguished based on how the heading at any step is determined. For both types of 116 walks it is drawn at random around a predefined 3D direction µ. For a CRW, µ corresponds to the heading 117 at the previous step, whereas for a BRW, µ corresponds to the target direction. In this case, the arc size 118 corresponding to the angular discrepancy between a given heading and the target direction will be referred 119 to as ν, which is statistically related to the arc size between successive headings ω through the relationship c ω = c 2 ν as occurs with 2D BRW (Marsh and Jones, 1988;Benhamou, 2006;Codling et al., 2008). 121 The Mean Squared Displacement (MSD), E R 2 n , which is defined as the expected value of the squared 122 beeline distance between an animals' initial and final locations after n steps, serves as a useful metric to 123 analyse movement patterns. The general MSD formulation for 2D CRW (Kareiva and Shigesada, 1983;124 Benhamou, 2006), in which left and right turns are not necessarily balanced, is extremely complex. We 125 will consider here its extension in 3D space only for balanced CRW, developed by Benhamou (2018), and 126 which reads: For a large step number n, the MSD approaches: For a 3D SRW, with c ω = 0, the MSD reduces to E R 2 n = nE l 2 , whatever the step number. It is 130 readily seen from equation (2.1.4) that the MSD is asymptotically proportional to n, and therefore the 131 walk becomes isotropically diffusive in the long term. The subscript 'a' is included here to represent the 132 asymptotic value to which the MSD tends when n increases indefinitely. For an isotropically diffusive 133 RW, the MSD is related to the diffusion coefficient D as follows: E R 2 n = 2qDT n where T n is the duration 134 of the n step RW and q = 1, 2, 3 corresponds to the number of dimensions (Crank, 1975 where s is the mean speed, with q = 3 for a random walk in 3D space (Benhamou, 2006(Benhamou, , 2018.

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In the case of a BRW, headings are drawn independently of each other in the target direction. This 139 leads to the following expression for the MSD: where ν is the arc size between an heading and the target direction, which is statistically related to the arc 141 size between successive headings ω through the relationship c ω = c 2 ν . This expression highlights that a 142 BRW is essentially a combination of the diffusive random walk and a drift, and its MSD is dominated in 143 the long-term by the contribution of the drift. It is worth noting that the MSD expressions for balanced 144 CRW and BRW in 3D space are similar to those obtained in 2D space (Hall, 1977; Marsh and Jones, 145 1988). The only difference is that the mean cosine of turning angles that is used in 2D space is replaced by 146 the mean cosine of orthodromic arcs corresponding to the reorientations between successive 3D headings.

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We can derive the conditions under which two 3D balanced CRWs are 'equivalent', in the sense that 148 they have the same MSD after n steps, given that n is sufficiently large. In particular, if we consider a SRW 149 with step length l * and mean cosine c * ω = 0, assuming the same coefficient of variation of step length and 150 the same mean path length L, we obtain the following 'condition of equivalence': Now consider a SRW and a BRW with step lengths l * and l and mean cosines c * ω = 0 and c ω , respectively.

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The condition of equivalence between these RWs in terms of diffusion is obtained with: (2.1.8) 154 We relied on a distribution of step length so that the distributions of increments ∆x, ∆y, and ∆z, when 155 reorientations are purely random (SRW), are zero-centred Gaussian distributions with the same standard-156 deviation σ , which represents the mobility of the animal. For a SRW with such increments, the probability

Mathematical bases for simulations of 3D RW
(2.2.1) As l, θ and φ are mutually independent random variables, one gets the following probability distribution 160 functions for these variables: The mean step length is E [l] = 4σ √ 2π and mean squared step length E l 2 = 3σ 2 . The coefficient of 162 variation is therefore γ = 3π 8 − 1. Note that, as expected, λ (l) can be considered a transformation of the 163 Chi distribution with 3 degrees of freedom, for re-scaled step lengthsl = l σ (Walck, 2007).

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To specify the distributions of initial arc orientation β and arc size ω in our simulations, we used the S q−1 in R q of the unit random vector z = (z 1 , z 2 , ..., z q ) is given by: where µ is the mean direction with norm ||µ|| = 1 and κ > 0 is a measure of the concentration about Given that µ and z are two unit vectors which deviate by ζ from each other, one gets µ · z = cos(ζ ), with 174 ζ = ω for a balanced 3D CRW, where µ corresponds to the previous heading, or ζ = ν for a 3D BRW, 175 where µ corresponds to the target direction. Furthermore, by setting the pole of the sphere at the endpoint 176 of µ, the infinitesimal surface element ds can be rewritten without loss of generality as sin(ζ )dβ dζ as it 177 appears that ζ then behaves as a co-latitude and β as a longitude. With β uniformly distributed between 178 -π and π, one gets: where ψ and η correspond to the probability distribution functions of the initial arc orientation and arc In the limit κ → 0, the distribution of the arc size simplifies to η(ζ ) = sin (ζ ) 2 with mean cosine c ζ = 0, as 182 expected for a SRW.  To tune the scale parameters of various CRW so that they are equivalent in terms of diffusion, we can 189 express equation (2.1.7) as: In the long term, a CRW with scale parameter σ behaves as a SRW with scale parameter σ * . The sinuosity Similarly, for a BRW with step length distribution parameter σ and long term persistence parameter κ 193 one gets:

Modelling trapping 196
In 3D trapping scenarios, consider a population of N individuals moving independently of each other. The

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path of each individual is modelled as a 3D RW in unbounded space, with initial location in proximity of a 3D trap. Each subsequent step is determined by the recurrence relation 199 resulting in a RW which is governed by the type of probability distribution for the step vector (∆x), and its In this study, we consider three shapes of 3D traps, namely the spheroid, cylindrical and rectangular 209 cuboid types with trap geometry D defined by the following: with the specific case r s = h s reduces to a spherical shaped trap.
with the specific case e b = h b reduces to a cube shaped trap.

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For any trap type, we can specify its shape by introducing dimensionless elongation parameters. For the 217 spheroid, we considered the ratio of polar to equatorial radii ε s = h s r s , where ε s < 1 corresponds to an oblate 218 spheroid and ε s > 1 to a prolate spheroid. For the cuboid we considered the ratio of height to base side where ε b = 1 corresponds to a cube, and for the cylinder we considered the ratio of height 220 to base diameter ε c = h c 2r c .

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We can then write expressions for the total surface area as: and for volume: (2.3.6) We can also express volume as a function of area as: (2.3.7) The initial distribution of individual location is considered to be uniform over a vicinity, which is 226 defined as the space between the trap and some fixed outer distance R, measured from the centre of the where U is a random variable drawn from the uniform distribution between 0 and 1.

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In the case of other trap shape, the vicinity no longer has an infinite number of symmetry axes and 234 therefore, to simulate a homogeneous population is not as straightforward. In these cases, we drawn the 235 initial locations at random in the whole sphere of radius R, and removed those occurring within the trap.  (2.3.7). The values noted alongside each curve are the squared correlation coefficients. The range of volumes/area considered are found from the upper bounds in (3.1.2). Simulation details: the movement type used is a SRW with σ * = 1 (S = 1.79). Trap counts are recorded after a maximum path length of L = 500 has been reached.
We considered a cube trap h b = e b (ε b = 1), and a 'normalized' cylinder where the height is equal to the 244 base diameter h c = 2r c (ε c = 1). The normalized cylinder and the cube lie within a sphere of radius R 245 provided that the following inequalities apply: which we use to determine the range of trap dimensions, areas and volumes.

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The simulated trap counts are shown in Fig. 3.1.2. It is readily seen that the cumulative trap count 249 is a monotonously increasing, nonlinear function of trap surface area and volume. Note that the order of 250 trap shapes, in terms of capture efficiency, is reversed depending on whether we consider the traps to have 251 equal total area or volume. Trap counts for a given volume and a given trap shape (Fig. 3.2.1a) varies a lot, but the variation as a 256 function of the elongation parameter is mainly due to a variation of area. Indeed, the sharp increase in the 257 trap count seen in Fig, 3.2.1a for small ε is an immediate consequence of the fact that the decrease in ε to 258 values ε 1 makes the shape almost flat. In order to preserve the volume, the area then becomes large.

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On the contrary, when the area is kept constant for all trap shape types and elongation parameter, we found 260 that the number of captures does not vary much (Fig. 3.2.1b). In this context, spheroidal traps slightly 261 outperform cylindrical and cuboid traps in terms of capture efficiency. As elongation has no noticeable 262 effect (for each type of trap) whereas this factor changes the volume for a given area, it makes sense to 263 consider traps with the same area for subsequent analyses of the possible effects of short-term persistence, 264 long-term directional bias and diffusion of the walk.   Contrary to what occurs in Fig. 3.3.1, the scaling parameter was the same for all walks (σ * = σ = 1) so that the diffusion increases with c ω .    of an elevated trap is essentially performed in 3D space, and hence it should be modelled as such.

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Understanding the efficiency of trapping resulting from the interplay between the movement pattern 310 (as described by the SRW, CRW and BRW) and the shape of the trap was the focus of this study. We  Fig. 3.1.2). On considering trap elongation, we found that trap counts do not vary 318 much given that the surface area is fixed, and that there is a clear hierarchy in terms of which traps are more 319 efficient, with the spheroidal trap outperforming the cylindrical trap, followed by the cuboidal trap (see 320 Fig. 3.2.1). Also, rather counter-intuitively, we showed that the short-term persistence of the individual In conclusion, these issues notwithstanding, we have shown how different trap geometries and the 3D 378 movement of individuals can bias trapping efficiency. Understanding how diffusion, directed movement 379 and trap shape can affect counts, estimates and observations has critical implications for spatial ecology 380 and for understanding the distribution and abundance of species. These individual based, geometric ap-381 proaches warrant further investigation and application in problems in contemporary spatial ecology. The 382 next natural step that we hope to see in the near future, is analyses of real flying animal movements using 383 3D RW models.