François G Schmitt
email: francois.schmitt@univ-lille1.fr

Laurent Seuront

Intermittent turbulence and copepod dynamics: increase in encounter rates through preferential concentration

Keywords: Turbulence, copepods, preferential concentration, scaling, contact rate

Turbulence has a strong influence on plankton contact rate, which is a crucial parameter for plankton ecology. In the field of particle-turbulence interactions, it is now well known that fully developed turbulence does not always homogenise particle distributions, but instead creates, in some well-defined conditions, preferential concentrations of heavy particles. Many studies have considered the influence of this type of preferential concentration on particle contact rate. We consider here the possible application of these results to copepods, assuming that some results obtained for heavy particles are still valid for light particles. Using parameter values representative of copepod species in coastal waters and open ocean, we numerically estimate the possible enhancement of copepod contact rates due to the preferential concentration by turbulence. The assessment is done by using data from a laboratory experiment: we find from the trajectory analysis of small neutrally buoyant particles, that the preferential concentration effect increases the contact rate up to 140%. We argue that this effect may be more pronounced for higher Reynolds numbers, and may have important ecological applications.

Introduction

Turbulence is now widely recognized to influence dispersal, feeding and reproduction of copepods. Two of the major aspects of copepod dynamics involve contact rates between particles in turbulent flows: predator-prey encounter rates, and mating encounter rates. In copepod ecology, a large body of literature has considered encounter rate enhancement through remote detection ability and behavioural activity. This includes chemoreception and mechanoreception [START_REF] Buskey | Swimming pattern as an indicator of the roles of copepod sensory systems in the recognition of food[END_REF][START_REF] Weissburg | Following the invisible trail : kinematic analysis of mate-tracking in the copepod Temora longicornis[END_REF], prey movement detection [START_REF] Visser | Hydromechanical signals in the plankton[END_REF][START_REF] Jiang | Hydrodynamic interaction between two copepods: a numerical study[END_REF], feeding currents (Marrassé et al., 1990) and swimming behaviour (Gerritsen and Strickler, 1977;[START_REF] Rothschild | Small-scale turbulence and plankton contact rates[END_REF][START_REF] Evans | The encounter speed of moving predator and prey[END_REF][START_REF] Hill | Encounter rate by turbulent shear of particles similar in diameter to the Kolmogorov scale[END_REF][START_REF] Visser | Turbulence-induced contact rates of plankton: the question of scale[END_REF][START_REF] Titelman | Swimming and escape behavior of copepod nauplii : implications for predator-prey interactions among copepods[END_REF]. The effect of the behaviour is, however, only effective within a limited distance between organisms. The encounter rate can thus be modelled as a combination of two independent, though complementary, physically-and biologically-driven processes. First, turbulence brings two particles (either a predator and its prey, or two mates) below a threshold detection distance. Then, when the detection distance is reached, behavioural processes are triggered, leading to attempts to capture the encountered particles (i.e. a prey or a mate). While we only consider the effect of turbulence on copepod encounter rates, without considering the subsequent behavioural processes, the behavioural contribution would further enhance the encounter rates (see e.g. [START_REF] Kiørboe | Planktivorous feeding in calm, and turbulent environments with emphasis on copepods[END_REF]; [START_REF] Seuront | Turbulence intermittency, small-scale phytoplankton patchiness and encounter rates in plankton : where do we go from here?[END_REF]). In the plankton ecology literature, turbulence is implicitly assumed to homogenise particle distributions at small scales. In contrast, in the community of turbulence physics, turbulence is widely acknowledged to increase inhomogeneity at small scales, which is referred to as the intermittency effect [START_REF] Frisch | Turbulence : the legacy of A. N. Kolmogorov[END_REF].

Concerning particles, three types of effect are expected, depending on the inertia of these particles (their Stokes number, considered in the next section). Zero inertia particles behave as passive tracers, purely passively advected by the flow, whereas heavy particles (infinite inertia) have their own trajectories and are not influenced by turbulence. In contrast, intermediate inertia particles are only influenced by some portions of turbulent motions as they cannot be influenced by the largest vortical motions due to their finite inertia [START_REF] Wang | Statistical mechanical description and modelling of turbulenct collision of inertial particles[END_REF]. As a consequence, particles denser than then fluid cannot stay in high vorticity regions, and have larger concentrations in low vorticity and high strain rate regions. The result is known as the phenomenon of preferential concentration of heavy particles due to turbulence [START_REF] Maxey | The gravitational settling of aerosol particles in homogeneous turbulence and random flow fields[END_REF][START_REF] Squires | Preferential concentration of particles by turbulence[END_REF][START_REF] Wang | Settling velocity and concentration distribution of heavy particles in homogeneous isotropic turbulence[END_REF][START_REF] Fessler | Preferential concentration of heavy particles in a turbulent channel flow[END_REF]. This "inertia difference" effect can also be interpreted as being due to the difference in inertia of a particle and the fluid it displaces, for particles having a density larger than the density of the fluid (density ratio > 1). However, particles may tend to migrate towards the center of vortices because a small difference in acceleration can be amplified when the particle approach a hyperbolic point in the fluid flow [START_REF] Babiano | Dynamics of a small neutrally buoyant sphere in a fluid and targeting Hamiltonian systems[END_REF]. This occurs at all density ratios, but is most evident when the density ratio is close to 1. This has only been shown in models, and in the following we will not consider this case and focus on density ratio larger than 1, although in copepod applications this ratio is close to 1.

The effects of preferential concentration have mainly been studied on the coagulation of colloidal particles, with aerosols, sprays and cloud physics applications in mind (Brunk et al., 1988;[START_REF] Vaillancourt | Review of particle-turbulence interactions and consequences for cloud physics[END_REF][START_REF] Wang | Statistical mechanical description and modelling of turbulenct collision of inertial particles[END_REF][START_REF] Zhou | Modelling turbulent collision of bidisperse inertial particles[END_REF][START_REF] Falkovich | Acceleration of rain initiation by cloud turbulence[END_REF]. Its application to copepod ecology has been, to our knowledge, only considered in one paper [START_REF] Squires | Preferential concentration of marine particles in isotropic turbulence[END_REF], using Direct Numerical Simulation (DNS) of flow equations (see also a discussion by Jimenez [START_REF] Jimenez | Oceanic turbulence at millimeter scales[END_REF]). This paper showed that preferential concentration of particles presenting material characteristics close to those encountered in marine environments can occur, with peak particle densities ranging from 10 to 60 times the mean value. This early work was published before the precise effect of preferential concentration on particle contact rates was extensively studied in a series of papers by several teams of researchers [START_REF] Sundaram | Collision statistics in an isotropic particle-laden turbulent suspension. Part 1. Direct numerical simulations[END_REF][START_REF] Zhou | On the collision rate of small particles in isotropic turbulence. II. Finite inertia case[END_REF][START_REF] Wang | Statistical mechanical descriptions of turbulent coagulation[END_REF][START_REF] Wang | Statistical mechanical description and modelling of turbulenct collision of inertial particles[END_REF][START_REF] Reade | Effect of preferential concentration on turbulent collision rates[END_REF][START_REF] Collins | Reynolds number scaling of particle clustering in turbulent aerosols[END_REF]. We discuss here the application of these results to copepods, and show its potential importance for the estimation of copepod encounter rates. In section 2, we review the most salient results of the particle turbulence literature concerning preferential concentration and encounter rates. In section 3 these results are considered in the framework of copepod ecology. In particular, Stokes numbers, characterizing the inertia of particles relative to local turbulence properties, are estimated for characteristic flow and copepod properties, and the relevance of the previous inertial particle framework to copepods is discussed. We also investigate the possible effects on radial distribution function and enhancement of copepod encounter rates. An experimental database of polystyrene particle dispersion by turbulence is analyzed as an example of preferential concentration of buoyancy free particles. Finally, we discuss the potential importance of this phenomenon for the estimation of encounter rates in aquatic ecology.

Preferential concentration in turbulence and contact rate

Contact rate in homogeneous prey fields

The encounter rate and its relevant estimation in various experimental conditions is a critical issue in plankton ecology. Models of encounter rates have been originally proposed for motile predators and preys in non-turbulent conditions. For a predator moving at a constant velocity V in an homogeneous field of non-motile prey (concentration n) without turbulence, the encounter rate is written as:

Z = nπR 2 V (1)
where Z is the number of encounters per unit volume per unit time, R is the encounter, or perceptive, radius of the predator. This expression becomes more complex when the prey have a non-zero velocity. Considering a homogeneous field of prey moving in random directions at a velocity U , and a predator moving at a constant velocity V (we consider here the case V ≥ U ) Eq. ( 1) becomes (Gerritsen and Strickler, 1977):

Z = n πR 2 3 U 2 + 3V 2 V (2)
To take turbulenceinto account, Eq. ( 2) has been modified by [START_REF] Rothschild | Small-scale turbulence and plankton contact rates[END_REF], with further contributions from [START_REF] Evans | The encounter speed of moving predator and prey[END_REF] and [START_REF] Visser | Turbulence-induced contact rates of plankton: the question of scale[END_REF] (see a review in [START_REF] Lewis | Planktonic contact rates in homogeneous isotropic turbulence: theoretical predictions and kinematic simulations[END_REF]). For zero velocity predator and prey (U = V = 0) in a homogeneous turbulence medium, the formulae given by [START_REF] Rothschild | Small-scale turbulence and plankton contact rates[END_REF] involves only turbulent velocity and recovers the standard expression of particle encounter rates in turbulence [START_REF] Saffman | On the collision of drops in turbulent clouds[END_REF][START_REF] Delichatsios | Coagulation in turbulent flow: theory and experiment[END_REF][START_REF] Hill | Encounter rate by turbulent shear of particles similar in diameter to the Kolmogorov scale[END_REF][START_REF] Wang | Statistical mechanical description and modelling of turbulenct collision of inertial particles[END_REF]:

Z = nπR 2 < w r (R) > (3)
where < . > means ensemble average, w r (R) = |W (x + R) -W (x)| is the radial turbulent velocity between two particles separated by a distance R, and < w r (R) > is the mean amplitude of the relative radial velocity (the subscript "r" is for radial). This expression is analogous to Equation (1), with the predator constant velocity replaced by the turbulent term < w r (R) >.

Modification of the contact rate in the case of preferential concentration of particles

In fully developed turbulence, it has been recognized since [START_REF] Maxey | The gravitational settling of aerosol particles in homogeneous turbulence and random flow fields[END_REF] that inertial particles are inhomogeneously distributed; see also [START_REF] Squires | Preferential concentration of particles by turbulence[END_REF], Wang and[START_REF] Wang | Settling velocity and concentration distribution of heavy particles in homogeneous isotropic turbulence[END_REF][START_REF] Fessler | Preferential concentration of heavy particles in a turbulent channel flow[END_REF]. Due to their inertia, spherical particles of diameter σ are ejected from highly vortical regions, and the resulting local particle concentration is higher than the average, leading to a correction term for the encounter rate expression. This leads to the following expression [START_REF] Sundaram | Collision statistics in an isotropic particle-laden turbulent suspension. Part 1. Direct numerical simulations[END_REF][START_REF] Wang | Statistical mechanical descriptions of turbulent coagulation[END_REF][START_REF] Reade | Effect of preferential concentration on turbulent collision rates[END_REF][START_REF] Collins | Reynolds number scaling of particle clustering in turbulent aerosols[END_REF]:

Z = nπσ 2 g(σ) < w r (σ) > (4)
where the encounter (i.e. contact) between particles occurs at a distance r = σ. The term < w r (σ) > expresses the radial relative velocity between particles, estimated at contact and is the dynamical aspect of the encounter rate. In Equation ( 4), the important new term is the correction factor g(σ) related to preferential concentration. The function g(r) is the radial distribution function, also called the pair correlation function. It measures the level of particle pair accumulation with interparticle distance r, and thus directly represents the accumulation effect. This is a standard function in statistical mechanics [START_REF] Pathria | Statistical Mechanics[END_REF], widely used in various fields [START_REF] Bernal | Radial distribution of the random close packing of equal spheres[END_REF][START_REF] Bishop | The pair correlation function: a probe of molecular order[END_REF][START_REF] Lock | Correlated light scattering by a dense distribution of condensation droplets on a window pane[END_REF] to characterize the distribution of particles. An introduction to the concept and its implementation can be found in [START_REF] Younge | A model for examining the radial distribution function[END_REF]. It may be written as the probability of finding a particle in a given volume dV at distance r, normalized by the probability of finding a particle in dV if the particles were uniformly distributed. Hence it is also a local measure of how uniform the particle distribution is, it is close to 1 for homogeneous distributions and is larger for inhomogeneous distributions.

This function depends on the Reynolds number and on the Stokes number St.

The Stokes number is the non-dimensional ratio of an inertial characteristic time scale τ p (also called the aerodynamic response time, see [START_REF] Fessler | Preferential concentration of heavy particles in a turbulent channel flow[END_REF]), to a turbulent characteristic time (the Kolmogorov time scale) τ η . This gives [START_REF] Reade | Effect of preferential concentration on turbulent collision rates[END_REF]:

St = τ p τ η = C p σ η 2 (5)
with the constant:

C p = β 18 (6)
where β = ρ p /ρ is the ratio of the particle density to the fluid density, and

η = ν 3 ǫ 1/4 (7)
is the Kolmogorov length scale. The latter can also be written using the Reynolds number and the integral length scale L as η = LRe -3/4 . The Stokes number characterizes the inertia of particles and is one of the fundamental parameters characterizing particle-turbulence interactions: for St ≪ 1, particles follow passively the flow, whereas for St ≫ 1, large inertia particles are not influenced by turbulence and follow their own trajectories.

Let us consider the St dependence, rewriting the function g as g(r, St). The function G(St) = g(σ, St) has been studied numerically through DNS; r = σ corresponds to a contact at particle diameter distance [START_REF] Sundaram | Collision statistics in an isotropic particle-laden turbulent suspension. Part 1. Direct numerical simulations[END_REF][START_REF] Wang | Statistical mechanical descriptions of turbulent coagulation[END_REF][START_REF] Wang | Statistical mechanical description and modelling of turbulenct collision of inertial particles[END_REF][START_REF] Reade | Effect of preferential concentration on turbulent collision rates[END_REF]. We reproduce in Figure 1 [START_REF] Sundaram | Collision statistics in an isotropic particle-laden turbulent suspension. Part 1. Direct numerical simulations[END_REF] or 0.8 [START_REF] Reade | Effect of preferential concentration on turbulent collision rates[END_REF], it is justified to consider here that G(St) is maximal for values ranging from 0.4 to 1. Furthermore, the maximal value obtained for G(St) can be as high as 12 to 200, depending on the simulation considered using various microscale Reynolds numbers and particle diameters. The encounter rate enhancement factor due to preferential concentration thus ranges between 1 to 2 orders of magnitude.

The function g(r, St) has also been studied numerically for fixed St values [START_REF] Reade | Effect of preferential concentration on turbulent collision rates[END_REF][START_REF] Collins | Reynolds number scaling of particle clustering in turbulent aerosols[END_REF], in this case we keep below the notation g(r). It has been shown that the residual radial distribution function h(r) = g(r) -1 has a power-law r dependence, of the form [START_REF] Reade | Effect of preferential concentration on turbulent collision rates[END_REF][START_REF] Collins | Reynolds number scaling of particle clustering in turbulent aerosols[END_REF]:

h(r) = c 0 r η -c 1 (8)
with c 0 = 1.75 ± 0.1 and c 1 = 0.65 ± 0.1 for values of St between 0.7 and 1 [START_REF] Collins | Reynolds number scaling of particle clustering in turbulent aerosols[END_REF]. This power-law fit has been shown to be very close to data for r/η values between 5 × 10 -3 and 1 [START_REF] Collins | Reynolds number scaling of particle clustering in turbulent aerosols[END_REF]), but it is still approximately valid for r/η = 2 (see [START_REF] Reade | Effect of preferential concentration on turbulent collision rates[END_REF]). Rewriting Equation ( 8) for a distance r > σ and for r = σ leads to:

h(r) = h(σ) r σ -c 1 (9)
This provides a correction factor for the radial distribution function when the encounter distance is larger than the particle diameter. This will be particularly useful for the copepod encounter rate, as their perceptive distance increases their effective size.

3 Application to copepods

Copepod Stokes number and applicability of DNS results

For the previous framework to be applicable to copepods, we consider a hypothetical non-motile copepod population uniform in size. Let us consider the Stokes number for copepods. First, the constant C p (Equation ( 5)) is estimated as: C p = 5.7 ± 0.1 × 10 -2 (10) considering a copepod density ρ p bounded between 1.02 to 1.08 × 10 3 kg/m 3 (McCave, 1984), the sea water density of ρ = 1.027 × 10 3 kg/m 3 , and the subsequent ratio β = ρ p /ρ close to 1. Considering Eq.( 5) and writing the optimal Stokes number as St = 0.7 ± 0.3, the optimal copepod diameter, relative to the Kolmogorov turbulent scale η is:

σ η = 3.5 ± 1 (11)
As a consequence, to have a Stokes number bounded between 0.4 and 1, as a copepod needs to be larger than the Kolmogorov scale. The studies discussed in the previous section referred to inertial particles having a much larger density ratio than copepods (i.e. β ≈ 800 for rain droplets, and β ≈ 800 to 1500 for aerosol particles), so that the particle diameter could be much smaller than the Kolmogorov scale.

Since we consider here particles having a size larger than the Kolmogorov scale, there is another alternative for the choice of a relevant Stokes number associated to such particles. We can consider that larger particles distort the turbulence at scales below the particles' diameter: they are not sensitive to velocity fluctuations at scales below σ, and hence a new Stokes number shall be considered. The previous Stokes number may be called "Kolmogorov Stokes number" [START_REF] Loth | Numerical approaches for motion of dispersed particles, droplets and bubbles[END_REF], being a ratio of particle relaxation time to the Kolmogorov time scale. A new Stokes number may be considered as relevant if one considers the time scale associated to eddies of spatial scale σ. It may be written:

St σ = τ p τ σ ( 12
)
where τ σ = τ η (σ/η) 2/3 is due to dimensional arguments in the inertial range, for scales σ > η. This gives (using Eq. ( 5)):

St σ = St η σ η -2/3 = C p σ η 4/3 (13)
In this case the previous expression for the optimal copepod diameter associated to the optimal Stokes number of St = 0.7 ± 0.3 is modified, and becomes:

σ η = 6.4 ± 2 (14)
which is still a distance larger than the Kolmogorov scale, and is about twice the previous optimal diameter.

Another important difference with the framework presented in section 2 is the existence of an encounter radius, R, for zooplankters (R > σ); they can grab an incoming mate or particle before the latter is in contact which corresponds to an increase of their effective size. This property has several implications on the framework presented above. First, Eq. ( 4) has to be modified, replacing the particle diameter σ by the encounter radius R, except for the Stokes number which is purely driven by the absolute size of the copepod. The resulting copepod contact rate with preferential concentration is then the following, using Eq. ( 4):

Z = nπR 2 g(R) < w r (R) > (15
) where g(R) is given by Eq. ( 9):

g(R) = 1 + (g(σ) -1) R σ -c 1 (16)
Considering a typical value for copepods of R/σ = 2 [START_REF] Lenz | Distal setal mechanoreceptors of the first antennae of marine copepods[END_REF] find R/σ ≈ 1, but other studies experimentally find values close to 3 [START_REF] Bagoien | Blind dating-mate finding in planktonic copepods. I. Tracking the pheromone trail of Centropages typicus[END_REF][START_REF] Doall | Locating a mate in 3D: the case of Temora longicornis[END_REF]) and c 1 = 0.65, the radial distribution function at contact g(σ, St) = G(St) is multiplied by a factor of about 0.6, so that g(R) still reaches high values. This shows that the effect of preferential concentration may be important for copepods: copepods with the adequate size will have a much higher contact rate than others with a different size.

The optimal size is given by Eq.( 11) and depends on the Kolmogorov scale.

The contact rate is given by Eq.( 15), with a correction factor relative to the classical expression (Eq.( 3)), provided in Eq.( 16).

We note here that the numerical studies discussed above are based on direct numerical simulations of the Navier Stokes equations, together with the use of Maxey-Riley-Gatignol equation for particle dynamics [START_REF] Maxey | Equation of motion of a small rigid sphere in a non-uniform flow[END_REF][START_REF] Gatignol | The Faxen formulas for a rigid particle in an unsteady non-uniform Stokes flow[END_REF], and are only valid for particles diameters much smaller than the Kolmmogorov scale (see [START_REF] Michaelides | Hydrodynamic force and heat/mass transfer from particles, bubbles, and drops-The Freeman scholar lecture[END_REF] for a review). This means that these numerical results cannot be applied to simulate the dynamics of copepods larger than the Kolmogorov scale. This is also an implicit weakness in the approach of [START_REF] Squires | Preferential concentration of marine particles in isotropic turbulence[END_REF]. The equation for such large particles in turbulent flow are still unknown, even for spherical particles. For numerical studies of the dynamics of large particles (typically of the size of the smallest eddies) one needs to explicitly take into account the shape of the particles [START_REF] Pan | Numerical investigation of the effects of large particles on wall-turbulence[END_REF]. Nevertheless, this does not invalidate the equations for contact rates, which are still valid (Eqs.( 4) and ( 15)) since they have been obtained analytically using very general assumptions involving an appropriate statistical average of the collision operator over all possible particle locations and velocities [START_REF] Sundaram | Collision statistics in an isotropic particle-laden turbulent suspension. Part 1. Direct numerical simulations[END_REF]. The numerical values of G(St) and g(r) have been obtained numerically, and need to be confirmed by experiments with particles having a density and size close to the ones of copepods. An experimental study conducted in this specific framework is presented in the next section.

Application to experimental dispersion data

We consider here the experimental data presented in [START_REF] Mann | Experimental study of relative, turbulent diffusion[END_REF] and [START_REF] Ott | An experimental investigation of the relative diffusion of particle pairs in three-dimensional turbulent flow[END_REF]. A water tank of inner dimension 320 × 320 × 450 mm 3 , with two oscillating grids at the top and the bottom, was used to generate a flow with no mean velocity and approximately homogeneous and isotropic turbulence at the central region. The flow was seeded with polystyrene particles with diameters in the range 500 -600 µm. A heat treatment was applied to the particles in order to adjust their density to the water density. The resulting density was also very close to that of the copepods. Video images were taken by four synchronized CCD cameras. A light source was provided through a Xenon discharge tube synchronised with the cameras, with flashes at a rate of 25 flashes per second. Raw images were treated using an image analysis software, in order to provide particle positions in time; recorded positions were restricted to a spatial region of 140 × 140 × 120 mm 3 . About 700 to 1000 particles were used for each experiment. Among the available databases, we have considered here the one corresponding to the largest Reynolds number. The Taylor microscale Reynolds number is R λ = 104, the dissipation ǫ = 2.79 10 -4 m 2 s -3 , the kinematic viscosity of the water is ν = 0.89 10 -6 m 2 /s. Using Eq. ( 7) giving the Kolmogorov length scale η and the analogous expression for the Kolmogorov time scale τ η = (ν/ǫ) 1/2 , one obtains η = 0.22 mm and τ η = 0.056 s. Using Eq.( 5) and C p given by Eq.( 10), one obtains a Stokes number for these particles in this flow of St = 0.4. On the other hand, if one considers the Stokes number at scale σ (Eq. ( 13)), one obtains St = 0.2, which is slightly outside the possible range of the preferential concentration effect.

While [START_REF] Ott | An experimental investigation of the relative diffusion of particle pairs in three-dimensional turbulent flow[END_REF] and [START_REF] Mann | Turbulent particle flux to a perfectly absorbing surface[END_REF] have considered dispersion properties using these particles, we focus here on the preferential concentration effect. Since the particle and flow characteristics are close to those corresponding to the copepod framework: the value of β is close to 1, and the Stokes number is close to the range (0.4, 1) corresponding to the preferential concentration effect, according to the numerical results reported above. The Stokes number values St = 0.2 -0.4 are at the lower end of the range of values corresponding to preferential concentrations. Additional experiments are needed to reach a Stokes number close to 1. This database nevertheless shows a slight preferential concentration effect. A 2D projection of a 3D snapshot (Fig. 2) shows a slight accumulation of particles. This visual impression is quantitatively confirmed by the radial distribution function g(r), estimated from 3D positions of particles, and as an ensemble average over all the particles and for all the successive discrete times available. There are 1438 times, corresponding to a record duration of about 57 s, and there is an average number of 774 particles for each time. The number of particles taken into account in the estimation of g(r) is hence 1438 × 774. There is a mean particle density over the whole data set of 0.13 particles per cm 3 . The radial distribution function is estimated as described in [START_REF] Younge | A model for examining the radial distribution function[END_REF]: bins were chosen going from the minimal interparticle distance (d min /η = 2.7) to a maximum distance of d max /η = 347 with a size exponentially increasing. 37 bins have been taken, with a width going from 1.14η to 17η. The number of particle couples in each bin is variable, with a minimum value of 8 10 4 . The result is shown in Figure 3. The function peaks to a value of about 2.4 for small distance while the optimal value corresponds to a distance of twice the diameter of particles, taken here as σ = 0.55 µm. This maximum value of g(r) is compatible with the numerical results, taking into account the fact that the Stokes number here is not the optimal one and that the distances considered are larger than the ones considered for numerical simulations.

This shows directly that preferential concentration is possible for light and large particles, having properties (density, size) compatible with copepods.

We may mention here a discussion by [START_REF] Jimenez | Oceanic turbulence at millimeter scales[END_REF] where he showed, using a simplified equation of motion for a particle, that when particle density and fluid density are close to each other, preferential concentration cannot occur. However, this result may not be valid when the full particle equation is taken into account, as indicated by the experimental results presented here. Since the radial distribution function is a multiplicative factor in the encounter rate, the value close to 2.4 that was found here corresponds to an increase of 140% of the encounter rate, relative to the standard expression. This corrective effect is thus already very important even if the Stokes number is not optimal. It can then realistically be expected that the corrective factor reaches much higher values for Stokes numbers closer to 1, and for larger Reynolds numbers.

The optimal Stokes number for the preferential concentration effect, for particles slightly larger than the Kolmogorov scale, and of density close to the density of the fluid, is not known experimentally nor numerically, since most numerical and experimental papers on this topic have focused on heavy and smaller particles. We have seen here that the preferential concentration effect exists for larger particles but more thorough analysis is needed to determine the optimal Stokes number. In future work, we plan to estimate the effect of the Reynolds number on g(r) for the same type of flow where increasing the Reynolds number decreases the Kolmogorov scale, and hence increases the Stokes number, for the same particles.

We also mention here the possible influence of sedimentation on particle concentration. Indeed, using DNS data Wang and Maxey [START_REF] Wang | Settling velocity and concentration distribution of heavy particles in homogeneous isotropic turbulence[END_REF] have found an increase of the mean settling velocity of particles when their density is larger than the density of the fluid, due to a preferential sweeping effect. One may also find more recent studies on turbulence increasing the average settling velocity using numerical [START_REF] Bosse | Small particles in homogeneous turbulence: settling velocity enhancement by two-way coupling[END_REF] as well as experimental data [START_REF] Aliseda | Effect of preferential concentration on the settling velocity of heavy particles in homogeneous isotropic turbulence[END_REF][START_REF] Ruiz | Turbulence increases the average settling velocity of phytoplankton cells[END_REF]. In their experimental study of the average settling velocity of phytoplankton cells, [START_REF] Ruiz | Turbulence increases the average settling velocity of phytoplankton cells[END_REF] have obtained a very small average settling velocity. Thus, this phenomenon is not expected to have an influence over small time scales, as considered in the experiment outlined in the present paper. Furthermore, [START_REF] Mann | Experimental study of relative, turbulent diffusion[END_REF] have estimated the mean centre volume velocity for their data base and found a good isotropy with no significant vertical mean velocity.

Discussion

In their reviews, [START_REF] Granata | The fluid mechanics of copepod feeding in a turbulent flow: a theoretical approach[END_REF] and [START_REF] Hill | Encounter rate by turbulent shear of particles similar in diameter to the Kolmogorov scale[END_REF] already noticed that adult copepods may possess inertia, and hence may not be true passive tracers. [START_REF] Squires | Preferential concentration of marine particles in isotropic turbulence[END_REF] have also considered the question of preferential concentration in the framework of copepod studies. However, these papers have been published before several DNS papers could study this issue more thoroughly using for example precise results of contact rates associated with preferential concentration and numerical estimation of the function G(St). The quite high correction factor given by Eq.( 16) can have ecological applications that we discuss here, using available published results on the complex couplings observed between turbulence and plankton abundance.

Consider the optimal turbulent dissipation rate ǫ opt corresponding to a particle Fig. 2. A 2D projection of particle positions at a given time, as obtained from the experimental study of [START_REF] Ott | An experimental investigation of the relative diffusion of particle pairs in three-dimensional turbulent flow[END_REF].

of diameter σ. The related Stokes number writes as St opt = 0.7 ± 0.3. Using Eqs.( 5), ( 7) and ( 13), we obtain:

ǫ opt = A σ 4 (17)
with A = ν 3 (St opt /C p ) 2 for the choice of the Kolmogorov Stokes number, Eq.

(5), and A = ν 3 (St opt /C p ) 3 for the choice of the σ-Stokes number, Eq. ( 13).

In both cases, A is a constant depending on the temperature through the viscosity. Equation ( 17) provides the optimal turbulent dissipation rate for the accumulation of a particle of diameter σ. Since G(St) peaks when St is close to its optimal value, Eq. ( 17) also provides the turbulence intensity acting as a "resonance" for a given particle size. This is represented in Fig. 4, with σ in mm, ǫ opt in cm 2 /s 3 and the constant A estimated for two extreme situations: a lower estimate associated to St = 0.4 and high temperature waters It is computed using available 3D positions of particles An effect of preferential concentration is visible at small distances; the larger value corresponds to twice the diameter of particles.

respectively for the Kolmogorov Stokes number and the σ-scale Stokes number. There is one order of magnitude difference in the constants A, and hence one order of magnitude difference for the optimal dissipation rate associated with the two different choices of Stokes numbers. Figure 4 also shows typical marine environments with their related ranges of dissipation rates taken from [START_REF] Granata | The fluid mechanics of copepod feeding in a turbulent flow: a theoretical approach[END_REF][START_REF] Kiørboe | Planktivorous feeding in calm, and turbulent environments with emphasis on copepods[END_REF] and [START_REF] Druet | The fine structure of marine hydrophysical fields and its influence on the behaviour of plankton: an overview of some experimental and theoretical investigations[END_REF]: coastal zone, corresponding to ǫ ranging from 10 -3 to 10 0 cm 2 /s 3 , oceanic shelf with ǫ ranging from 10 -3 to 10 -2 cm 2 /s 3 , and the open ocean with ǫ being as small as 10 -6 cm 2 /s 3 . It is interesting to notice from this graph that there is a hierarchy of particle sizes for the different seas. In the coastal zone, only small particles may display preferential concentration, with sizes ranging from 0.8 to 4 mm: on the oceanic shelf particles range from 2 to 10 mm, and in the open ocean smaller dissipation rates correspond to particle diameters ranging from 1 to 4 cm.

These theoretical results may be interpreted in terms of their consequences for plankton ecology. We first notice that buoyancy free particles with diameters smaller than 0.5 mm do not seem to be concerned with the preferential concentration effect, since the required turbulence dissipation rate would be too high to be realistic of any marine environment. Only locally, and in very spe-cific situations (e.g. breaking waves) the turbulence intensity may reach values high enough to create a preferential concentration effect on such small particles. The same applies to phytoplankton cells. Cells of size such as 50 µm may be accumulated only for extreme levels of turbulence such as those induced by strong winds. In addition, this effect might be even more enhanced when plankton blooms are triggered by elevated turbulence intensities [START_REF] Reigada | Plankton blooms induced by turbulent flows[END_REF]. However, a local enhancement of copepod concentration may cause a local enhancement of prey capture rate, and in turn increase the small-scale patchiness of the prey. In future work, one may thus also consider the possible influence of copepod behaviour, and the interplay between behavior and turbulence, as initiated in previous works [START_REF] Rothschild | Small-scale turbulence and plankton contact rates[END_REF]; [START_REF] Lewis | Planktonic contact rates in homogeneous isotropic turbulence: theoretical predictions and kinematic simulations[END_REF].

The preferential concentration effect may also affect planktonic copepods and larval fishes. Indeed, this accumulation effect may be ecologically extremely relevant for copepods as a way to enhance their contact rates with potential sexual partners. However, copepods may have developed behavioral traits to avoid such turbulence intensities since it increases their predation risk, while predators like fish larvae may, in turn, be attracted by the same turbulence intensity as it strongly increases the concentration of their prey. This hypothesis is consistent with the "optimal environmental window" that results in a dome-shaped relation of larval fish ingestion with turbulence intensity [START_REF] Cury | Optimal environmental window and pelagic fish recruitment success in upwelling areas[END_REF][START_REF] Mackenzie | Turbulence, larval fish ecology and fisheries recruitment: a review of field studies[END_REF]. This also provides an alternative explanation to the turbulence avoidance strategy proposed by [START_REF] Franks | Turbulence avoidance: an alternate explanation of turbulence-enhanced ingestion rates in the field[END_REF] to explain the enhances in prey ingestion by larval fish within wind-driven mixed-layers.

Conclusion

We have reviewed here the main results obtained in the field of particle turbulence studies, and their influence on contact rates. These studies have been conducted for inertial particles, mainly in aerosol physics, and correspond to a high density and small particle size relative to the Kolmogorov scale. These studies have shown that the phenomenon of preferential concentration is the largest for Stokes numbers close to 1. They have also provided an important multiplicative correction factor to the encounter rate kernel, which is the radial distribution function at contact g(σ). We have considered here the possible application of this effect to copepods. The main difference with aerosols comes from the fact that copepods have a density ratio close to 1, whereas for aerosols it is of the order of 10 3 . In order to obtain a similar Stokes number, the particle size must be much larger for copepods than for aerosols. We then assumed here that the numerical results obtained for a density ratio β ≫ 1 are still valid for lighter particles with a density ratio β ≈ 1. This assumption needs experimental validation. Using an experimental data base on particle dispersion with buoyancy free particles of size similar to copepods, we have subsequently shown that copepods may also display the effect of preferential concentration when they have a size which is slightly larger than the Kolmogorov scale. Considering that the optimal accumulation effect is obtained for a narrow range of values of the Stokes number (St = 0.7±0.3), a one-to-one correspondence between optimal particle size and turbulent dissipation rate can be drawn. This shows that for a given flow configuration, associated to a mean turbulent dissipation rate ǫ, there is an optimal plankton size for which there will be a noticeable preferential concentration effect. The shape of the function G(St) shows that this optimal size may be selected as a resonance effect: it is much larger than 1 only for a short window of St values. For other St values, it is close to 1 and there is no accumulation and no increase of the contact rate. Using typical ranges of values of the mean dissipation rate in different marine environments, we have shown that this relation is effective to select large particles (1 to 4 cm) in the deep seas, and smaller particles (0.8 -4 mm) in surface and coastal waters. This might provide an alternative explanation to earlier works suggesting that copepod optimal size was driven by the balance between inertial and viscous forces [START_REF] Yen | Life in transition: balancing inertial and viscous forces by planktonic copepods[END_REF]. We have also used this relation to interpret some published results on the different behavior of larval fishes in variable turbulent environments. Finally since the mean abundance of prey may be too low to maintain a population of predators, our results potentially have some implications on zooplankton tophodynamics as predators would critically benefit from dense aggregations of prey. While some behavioral mechanisms have been suggested to explain the formation of patches [START_REF] Genin | Swimming against the flow: a mechanism of zooplankton aggregation[END_REF], our results show that high intensity turbulence can also create accumulation patches that will be very useful for predators.

  one of the typical curves (adapted from Wang et al. (2000)), showing G(St) as a function of the Stokes number St for three different values of the microscale Reynolds number R λ . In this example, G(St) is maximal for St values close to 1. As previous studies found this function maximal for St close to 0.4

Fig. 1 .

 1 Fig. 1. Representation of the radial distribution function at contact G(St) versus the Stokes numbers, and for various microscale Reynolds numbers R λ . Data from DNS simulation kindly provided by L. P. Wang[START_REF] Wang | Statistical mechanical description and modelling of turbulenct collision of inertial particles[END_REF].

  (ν = 1.05 10 -6 m 2 /s at 20 deg C), and a larger estimate associated to St = 1 and low temperature waters (ν = 1.8 10 -6 m 2 /s at 0 deg C). The corresponding range of values for the constant A are: A η ∈ [0.6, 18] and A σ ∈[4, 300]

Fig. 3 .

 3 Fig.3. The function g(r) estimated for the experimental data of[START_REF] Mann | Experimental study of relative, turbulent diffusion[END_REF]. It is computed using available 3D positions of particles An effect of preferential concentration is visible at small distances; the larger value corresponds to twice the diameter of particles.

Fig. 4 .

 4 Fig. 4. Estimation of the optimal turbulent dissipation rate corresponding to different values of the particle diameter σ: the dotted lines represent the high and low hypothesis for the constant A in Eq.(17), associated to different Stokes number definitions. Typical values of ǫ in the coastal zone, shelf and open ocean are displayed.

Acknowledgements

We thank professor L.P. Wang for providing the data for Figure 1, and professor J. Mann for sending the dispersion data analysed in section 3, and for interesting email exchanges. We also thank the participants of the WTD 2005 conference for interesting comments and discussions, and especially H. Yamazaki, A. Visser, A. Dominguez, and the organizers of the conference for this very nice event. Patricia "Trish" Lavery kindly improve the language. We acknowledge substantial improvements of the manuscript following the advice of the referees.


Aude Verwilghen
email: aude.verwilghen@cirad.fr

Marie-Agnès Rabillard

Yannick Chaval

Dominique Rieffel

Martua Hasiholan Sinaga

Mohd Naim

Jean-Pierre Caliman

Patrick Giraudoux

Francis Raoul

Relative age determination of Rattus tiomanicus using allometric measurements

Keywords: allometry, barn owl pellet, eye lens weight, oil palm plantation, rodent

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Oil palm is one of the world's most rapidly expanding crops; the main producer countries are Indonesia and Malaysia (Oil World 2012). Expanding oil palm plantations (OPPs) over large areas support rodent population outbreaks, and certain rodents are considered invasive pests that cause significant damage to production (Liau 1990, Turner and[START_REF] Turner | Oil palm cultivation and management[END_REF]. Pest control typically includes treating a field using anticoagulant rodenticides and/or reinforcing barn owl predation (Tyto alba, Scopoli) in the plantation by providing nest boxes [START_REF] Wood | A critical review of the development of rat control in Malaysian agriculture since the 1960s[END_REF]. For more sustainable oil palm production, natural predation by barn owls should be enhanced, which requires that we better understand its impact on rodent population dynamics [START_REF] Hafidzi | Natural predation: a viable option for controlling vertebrate pest in Malaysia[END_REF], notably for selective predation based on rodent age or size [START_REF] Saint Girons | L'âge des micromammifères dans le régime de deux rapaces nocturnes, Tyto alba et Asio otus[END_REF][START_REF] Kittlein | Assessing the impact of owl predation on the growth rate of a rodent prey population[END_REF].

In Southeast Asian OPPs, barn owls primarily feed on rats [START_REF] Lenton | The feeding and breeding ecology of barn owls Tyto alba in Peninsular Malaysia[END_REF][START_REF] Small | Research on the use of barn owls Tyto alba for biological control of rats in oil palm plantations[END_REF][START_REF] Puan | Absence of differential predation on rats by Malaysian barn owls in oil palm plantations[END_REF]; their prey spectrum is almost entirely confined to the three species, which are the major pests for oil palm crop: Rattus tiomanicus Miller, the Malaysian field rat; Rattus argentiventer Robinson and Kloss, the ricefield rat; and Rattus tanezumi Temminck (synonym: Rattus rattus diardii Jentink), the oriental house rat [START_REF] Liau | Rat species composition and distribution in West Malaysian plantations (with particular reference to oil palm plantations)[END_REF][START_REF] Corley | The oil palm[END_REF][START_REF] Wood | A critical review of the development of rat control in Malaysian agriculture since the 1960s[END_REF]. In most mature estates, R. tiomanicus predominates [START_REF] Hafidzi | Status of rat infestation and recent control strategies in oil palm plantations in Peninsular Malaysia[END_REF][START_REF] Wood | A critical review of the development of rat control in Malaysian agriculture since the 1960s[END_REF][START_REF] Chia | Rat control in oil palm plantations: a review[END_REF], wherein it composes most of the barn owl prey [START_REF] Lenton | The feeding and breeding ecology of barn owls Tyto alba in Peninsular Malaysia[END_REF]. Although barn owl predation on rats on oil palm estates is widely used and has been primarily studied in Malaysia for a long time [START_REF] Duckett | Owls as major predators of rats in oil palm estates with particular reference to the barn owl (Tyto alba)[END_REF], 2008[START_REF] Lenton | Biological control of rats in oil palm by owls[END_REF][START_REF] Small | Research on the use of barn owls Tyto alba for biological control of rats in oil palm plantations[END_REF]), the real effect of barn owl predation on the rat population is inconclusive (Chia et al. 1995, Wood and[START_REF] Wood | A critical review of the development of rat control in Malaysian agriculture since the 1960s[END_REF]. In addition, there is no clear pattern for selective predation based on size, weight, or age [START_REF] Small | Research on the use of barn owls Tyto alba for biological control of rats in oil palm plantations[END_REF][START_REF] Lim | Prey selection by Barn Owls (Tyto alba) in an oil palm plantation in Malaysia[END_REF][START_REF] Hafidzi | Prey selection by barn owl in rice fields in Malaysia[END_REF][START_REF] Puan | Absence of differential predation on rats by Malaysian barn owls in oil palm plantations[END_REF], and certain authors suggest that differential barn owl predation may be site specific [START_REF] Trejo | Owl selection on size and sex classes of rodents: activity and microhabitat use of prey[END_REF]Guthmann 2003, Leveau et al. 2006). Therefore, predation must be locally investigated to adapt pest control strategies to each plantation/site. Thus, as suggested by [START_REF] Sherfy | A reexamination of age-related variation in body weight and morphometry of Maryland Nutria[END_REF], prediction models based on morphometrics are valuable for reconstructing the prey population age structure, which is a useful tool for monitoring the demographic consequences of population control. Eye lens weight (ELW) is considered to be the most accurate age indicator for mammals because it varies much less with environmental conditions than other body measurements [START_REF] Lord | The lens as an indicator of age in cotton-tail rabbits[END_REF][START_REF] Friend | A review of research concerning eye lens weight as a criteria of age in animals[END_REF][START_REF] Morris | A review of mammalian age determination methods[END_REF][START_REF] Pucek | Age criteria in small mammals[END_REF][START_REF] Leirs | Population ecology of Mastomys natalensis (Smith, 1984). Implications for rodents control in Africa[END_REF][START_REF] Gliwicz | Aging and cohort dynamics in Sorex shrews[END_REF][START_REF] Lalis | Polymorphism of the age population structure of two wild Mastomys natalensis (Rodentia: Muridae) Tanzanian habitat samples: a multicriteria comparison[END_REF][START_REF] Augusteyn | Growth of the lens: in vitro observations[END_REF]. ELW is widely used to estimate rodents' exact or relative age (Le [START_REF] Louarn | Détermination de l'âge par la pesée du cristallin chez quelques espèces de rongeurs[END_REF][START_REF] Poulet | Détermination de l'âge par la pesée des cristallins chez cinq espèces de rongeurs Muridés et Gerbillidés de l'ouest de l'Afrique[END_REF][START_REF] Barnett | Expedition field techniques: small mammals (excluding bats)[END_REF][START_REF] Burlet | Efficient age determination: how freezing affects eye lens weight of the small rodent species Arvicola terrestris[END_REF]. Eye lens growth patterns from known-age individuals bred in a laboratory have primarily been investigated using vole and mice (Rowe et al. 1985, Yabe and[START_REF] Yabe | Eye-lens weight curve for estimation of age in the Japanese grass vole, Microtus montebelli Milne-Edwards (Rodentia: Muridae)[END_REF]. A few Rattus species also exhibit a quantitative relationship between age and ELW [START_REF] Williams | Determination of age of Polynesian rats[END_REF][START_REF] Hardy | Estimation of age in the Norway rat (Rattus norvegicus Beerkenhout) from the weight of the eye lens[END_REF][START_REF] Tanikawa | An eye-lens weight curve for determining age in black rats, Rattus rattus[END_REF][START_REF] Shrestha | Eye-lens weight curve for estimation of age in rats, Rattus rattus brunneus and Bandicota bengalensis in Nepal[END_REF]; however, Rattus tiomanicus has not been studied. The eye lens cannot be retrieved from raptor pellets; however, other macroremains can be collected, such as skulls and bones. Therefore, models have been developed to predict the ELW from osteometric measurements. Variables related to skull and jaw dimensions are typically considered most appropriate for small-mammal age estimations using macroremains (Quéré et al. 1994, Granjon and[START_REF] Granjon | Prey selection by barn owls in relation to small-mammal community and population structure in a Sahelian agro-ecosystem[END_REF]. However, because of prey decapitation, the absence of skull was frequently reported for barn owl pellets from Southeast Asian OPPs [START_REF] Medway | Barn owl pellets from Kulai, Johore[END_REF][START_REF] Lenton | The feeding and breeding ecology of barn owls Tyto alba in Peninsular Malaysia[END_REF][START_REF] Lim | Prey selection by Barn Owls (Tyto alba) in an oil palm plantation in Malaysia[END_REF][START_REF] Hafidzi | Prey selection by barn owl in rice fields in Malaysia[END_REF][START_REF] Puan | Absence of differential predation on rats by Malaysian barn owls in oil palm plantations[END_REF]); thus, it was necessary to use variables other than cranial and dental variables. Very few authors have investigated the allometric relationships between osteometric measurements and age or size/body weight for R. tiomanicus [START_REF] Small | Research on the use of barn owls Tyto alba for biological control of rats in oil palm plantations[END_REF][START_REF] Lim | Prey selection by barn owls (Tyto alba) and its impact on rat control in oil palm plantations[END_REF][START_REF] Tanikawa | An eye-lens weight curve for determining age in black rats, Rattus rattus[END_REF], although this species is the most studied rodent pest in OPPs [START_REF] Wood | A long-term study of Rattus tiomanicus populations in an oil palm plantation in Johore, Malaysia: I. Study methods and population size without control[END_REF], Wood and Liau 1984a,b, Buckle et al. 1997[START_REF] Chia | Rat control in oil palm plantations: a review[END_REF]. Moreover, only two authors, [START_REF] Lim | Prey selection by Barn Owls (Tyto alba) in an oil palm plantation in Malaysia[END_REF] and [START_REF] Puan | Absence of differential predation on rats by Malaysian barn owls in oil palm plantations[END_REF], have explored the relationship between femur length and relative age, respectively, for R. tiomanicus and Rattus tanezumi, to overcome the absence of skull elements in pellets.

In this study, we investigated the correlations between ELW and a range of osteometric measurements for Rattus tiomanicus trapped in a Sumatran (Indonesia) OPP. Our objectives were to assess the following: (i) the frequency of different types of bones and their physical integrity in barn owl pellets and (ii) the best combination of osteometric variables to predict R. tiomanicus lens weights theoretically (i.e., considering all variables) and practically (i.e., considering the bones most often observed in owl pellets).

Materials and methods

Study site and sampling design

The research was conducted at the Libo Estate, which is a large-scale OPP (approximately 4370 ha) in the Riau Province, Sumatra. The climate is typically humid equatorial, with an average temperature of 28°C, a mean annual rainfall of 2500 mm, and two dry seasons in February and June through August. The palm plantation is old (it was planted between 1986 and 1990) and has not been treated using rodenticide for > 10 years after the introduction of barn owls for rodent pest control in the 1990s. The rodents were trapped in May 2011. Nine trap lines with 25 cage traps were set up approximately every 9 m at the base of palm trees. The traps were set for two or three consecutive nights (seven trap lines were only set for two consecutive nights) and controlled each morning. We used locally constructed cage traps (32 × 12 × 15 cm) that were baited with freshly cut half of a palm fruit. The traps were located in the plantation core at a minimum distance of 600 m from human settlements to minimise trapping species other than Rattus tiomanicus.

Species identification and osteometric measurements

The rats were euthanised using chloroform (outdoors with safety precautions for the operator's health) and then sexed, and classic external body measurements were collected (weight, head/body, and tail length; see [START_REF] Herbreteau | Protocols for field and laboratory rodent studies[END_REF]. The head and left leg were severed for further dissection, boiled in an autoclave at 121°C for 30 min, and then submerged in water for 1 week to ease bone and flesh separation. For eye lens processing, various similar techniques are described in the literature [START_REF] Poulet | Détermination de l'âge par la pesée des cristallins chez cinq espèces de rongeurs Muridés et Gerbillidés de l'ouest de l'Afrique[END_REF][START_REF] Rowe | Relationship between eye lens weight and age in the wild house mouse (Mus muculus)[END_REF][START_REF] Shrestha | Eye-lens weight curve for estimation of age in rats, Rattus rattus brunneus and Bandicota bengalensis in Nepal[END_REF][START_REF] Jánová | Is the eye lens method of age estimation reliable in voles?[END_REF], and as noted by [START_REF] Lalis | Polymorphism of the age population structure of two wild Mastomys natalensis (Rodentia: Muridae) Tanzanian habitat samples: a multicriteria comparison[END_REF], the accuracy is rarely assessed. We adapted the protocol to tropical conditions. The eyeballs were removed and fixed in 4% formalin for at least 2 weeks in a refrigerator. The lenses were then excised, washed with distilled water, and oven dried at 40°C in two batches for 9 and 7 days (given the humid conditions, a preliminary trial was used for the first batch to estimate the drying duration necessary for weight stabilisation). Once out of the oven, the eyeballs were immediately weighed in pairs to the nearest 0.1 mg. The mean weight of each pair was used for analysis.

One hundred and sixty-one presumably Rattus tiomanicus individuals (identified according to external morphology; [START_REF] Corbett | The mammals of the Indomalay region: a systematic review[END_REF]Hill 1992, Aplin et al. 2003) were used as the reference collection for our measurements. A subsample composed of 120 individuals was selected using the following process. A principal component analysis with log-transformed osteometric measurements (see below) was used to select three variables that were least correlated between each other and least correlated for animal size (BP, CML1-3, and JawL1; see Figure 1). The Adapted from [START_REF] Quéré | Détermination de la structure en âge relatif d'une population de Microtus arvalis (Rongeurs, Arvicolidés) par mesures ostéométriques crâniennes ou mandibulaires[END_REF]. Skull: GSL, greater skull length; IB, interorbital breadth; LR, length of rostrum; CLM1-3, molar row length; LD, length of diastema; BP, breadth of palate; and upper incisors. Jaw: L1, mandible length; L2, mandible height; L3, diastema height; L4, jaw height; L5, molar row length; and lower incisor. Femur, short femur and long femur.

animals were then selected at the periphery and centre of the data cloud from a triangular graph, wherein the axes were the three variables. The animals were then identified using a molecular barcoding method, with species assignation using the RodentSEA reference database available at http://www.ceropath.org/barcoding_tool/rodentsea [START_REF] Clairon | Rodent SEA: a web service for rodent species identification in South-East Asia[END_REF].

The following 15 osteometric variables (Figure 1) were measured to the nearest 0.1 mm for each specimen by the same person using digital calipers: (i) in accordance with [START_REF] Dickman | Differential predation of size and sex classes of mice by the barn owl, Tyto alba[END_REF] and [START_REF] Quéré | Détermination de la structure en âge relatif d'une population de Microtus arvalis (Rongeurs, Arvicolidés) par mesures ostéométriques crâniennes ou mandibulaires[END_REF], 11 variables for the skull and jaw that relate well to age and are typically collected intact from bones and easily found in pellets; (ii) two femur variables, one composing the entire length of the bone, and the other excluding the distal epiphysis cartilage that is often absent (hereafter, the "short length"); the femur was used because certain authors note that heads are frequently absent from the pellets in OPPs, likely because barn owls decapitate their prey before carrying them to the nest [START_REF] Lenton | The feeding and breeding ecology of barn owls Tyto alba in Peninsular Malaysia[END_REF][START_REF] Lim | Prey selection by barn owls (Tyto alba) and its impact on rat control in oil palm plantations[END_REF]; and (iii) two incisor variables were measured to use small-carnivore scat in the model because incisors are typically the only intact rat remnant in a scat [START_REF] Bonnaud | Measuring rodent incisors from scats can increase accuracy of predator diet studies: an illustration based on island cats and rats[END_REF].

Material used for measurement (skull, femur, and eye lenses of each of the 161 specimens) is available at the Centre de coopération internationale en recherche agronomique pour le développement (CIRAD), Montpellier, France. Specimens are labelled LI1105XXxx (xx being the number of the individual trapped on the line XX; traplines are labelled AA, AB, AC, AD, AE, AR, AS, AT, AU).

The final aim of this study was to propose a model with a combination of variables (osteometric measurements) that best predicts the ELW and thus the relative rat age based on pellet remains. Therefore, concurrent with trapping, we collected the intact barn owl pellets inside nest boxes in our study area and the vicinity. One hundred pellets were collected in 57 nest boxes, covering an area of approximately 1600 ha (typically, one nest box in each block of 30 ha). It has been suggested that the presence or absence of skulls in pellets is associated with an owl's breeding cycle (Lenton 1984, Hafidzi and[START_REF] Hafidzi | Prey selection by barn owl in rice fields in Malaysia[END_REF]. Subsequently, 20 additional pellets were retrieved in August 2010, during the barn owl breeding peak, to encompass a potential behavioural change in prey consumption. We examined the occurrence and integrity of the rodent skull, jaw, and femur in pellets to assess which of the 15 variables were most likely to be collected from the pellets. We also estimated the imprecision for measurements from the same observer over two measurement sessions; the 15 variables from a sample composed of 30 rats were measured twice by the same person.

Statistical analysis

The percentage of measurement error (%ME; i.e., the percentage of sample variation due to measurer imprecision) can be defined as "the ratio of the within-measurement component of variance to the sum of the within-and among-measurement component" [START_REF] Bailey | A new, old method for assessing measurement error in both univariate and multivariate morphometric studies[END_REF]: where m is the number of repeated measurements [START_REF] Yezeriniac | Measurements errors and morphometric studies: statistical power and observer experience[END_REF][START_REF] Claude | Morphometrics with R[END_REF].

%ME = [
Two sets of models were computed based on two sets of variables: (i) all osteometric variables without considering whether they can be frequently collected from the pellets ("theoretical" model) and (ii) variables most commonly found in pellets due to high bone integrity (i.e., likely to be most useful for researchers and practitioners working with pellets; "practical" models). The practical models were constructed using the osteometric variables selected on the basis of three criteria: (i) the frequency in pellets is > 10%; (ii) the Pearson correlation coefficient for the given variable with an ELW is ≥ 0.8; and (iii) the ME percentage is < 10%. Because we did not have a biological a priori for the order of variables in the model, a stepwise procedure (backward and forward) was used to select the best set of predictors based on the Akaike information criterion (AIC) [START_REF] Burnham | Model selection and multi-model inference: a practical information -theoretic approach[END_REF]. Visual analyses of bivariate graphs among the ELW and osteometric measurements suggested that linear modelling was an acceptable option. The normality of model residuals was determined using the Kolmogorov-Smirnov goodness of fit test. We used a 10-fold cross-validation procedure to estimate the coefficient of determination (R 2 ) and the root mean squared error (Rmse), which is a good measure of model fit and accuracy.

The prey's sex cannot be defined in pellets using our measurements. However, we investigated a potential difference in ELW between sexes from our trapped sample using a linear model. In addition, we also investigated the performance of models, including the variable sex as a covariate, to assess whether a bias was induced by using only one equation for both sexes.

The data were statistically analysed using the R software (version 2.15.0; R Foundation for Statistical Computing, Vienna, Austria) and the packages ade4, bootstrap, DAAG, epicalc, lattice, pgirmess, plotrix, and ipred.

Results

Rattus tiomanicus characteristics

Among the 120 individuals that were subsampled for molecular identification, 110 were successfully sequenced, 100% of which were confirmed as Rattus tiomanicus. Summary statistics on the body and bone measurements (Table 1) as well as ELW distribution (Figure 2) suggest that both juvenile and adult specimens were sampled with minimum and maximum body weights at 17 and 139 g, respectively. The sex ratio of our sample was 1.36 males per female. We did not detect a significant difference in ELW between males and females (F-value = 0.0723, p = 0.7883; Kolmogorov-Smirnov test of

Bone frequency in pellets and measurement errors

As reported in Table 2, full osteometric measurements were impossible in at least 70% of pellets because of the low integrity of the skull and jaw bones in barn owl pellets. However, we measured the short femur in 80.8% of pellets. Between the two different sessions of measurements (by the same person), the percentage of measurement errors was low for the short femur (%ME = 0.23). The L1 and L2 values were also among the lowest at 1.8% and 1.4%, respectively. Seven variables, L3, L4, L5, Lower incisor, IB, CLM1-3, and Upper incisor, were > 10% ME.

Predictive models for the relative age estimation

The Pearson correlation coefficients (r) between the ELW and the variables CLM1-3 and L5 were < 0.4 (Table 2). Other variables were strongly correlated with lens weight; r ranged from 0.79 to 0.92, and the most correlated variables were the short and long femur. Using criteria based on frequency in pellets ( > 10%), the measurement errors ( < 10%) and correlation with lens weight (r ≥ 0.8), only six variables, namely, L1, L2, LD, and BP, as well as the short and long femur, were selected for the first predictive model (see Table 2). The model that best predicted the ELW was the model that included L2, LD, BP, and the short femur with an AIC = 435.62 and R 2 = 0.847 (model 1, Table 3). Because most pellets are headless, we investigated a model based on the short femur measurement only; the AIC for this model (model 2, Table 3) was greater than model 1 (458.04 vs. 435.62). The accuracy evaluation for the two "practical" models through cross validation showed that model 1 was slightly more accurate at estimating ELW than the model that only included the femur (Rmse = 3.86 vs. Rmse = 4.14, only considering the femur). A "theoretical" model was constructed on the basis of all of the jaw, skull, and femur variables. The seven-variable combination that best predicted the ELW is shown in Table 3 (model 3). This model performed better than both of the best "practical" models (AIC = 422.05 and R 2 = 0.865); however, we only detected a slight improvement in the prediction precision (Rmse = 3.78). The standard errors for the parameter estimates in the three models are presented in Table 4. Including the variable sex as a covariate slightly improved the performance of the models (models 1sex, 2sex, and 3sex; Table 3). Our assumptions on the homogeneity of variance, independence, linearity, and normality of error were examined for each model. The normality of residuals was confirmed for each model (Kolmogorov-Smirnov test, p = 0.843 for model 3, p = 0.426 for model 1, and p = 0.331 for model 2).

Discussion

We presented a set of best equations for the prediction of Rattus tiomanicus' ELW, based on various skull, jaw, and femur measurements. We demonstrate here that, because rodent heads are frequently absent from barn owl pellets in Malaysian and Indonesian OPPs, the predictive model based only on the femur was the most useful, although slightly less precise than the models based on skull and jaw variables. We also argue that we can confidently attribute most, if not all, femurs in the pellets studied herein to R. tiomanicus.

Species identification

As has been reported since the 1970s, Rattus tiomanicus is still the dominant species in mature OPPs in Southeast Asia; however, it has progressively been replaced by Rattus tanezumi (previously Rattus rattus diardii) in several localities, notably the Malaysian peninsula [START_REF] Wood | A critical review of the development of rat control in Malaysian agriculture since the 1960s[END_REF]. The R. tiomanicus predominance in our study area was confirmed through our trapping experiments. The specimens successfully sequenced herein (n = 110) were confirmed as R. tiomanicus. Additional investigations at the same estate confirmed that R. tiomanicus was predominant; the 19 specimens identified using the same molecular barcoding method were R. tiomanicus [START_REF] Andru | Coumatetralyl resistance of Rattus tanezumi infesting oil palm plantations in Indonesia[END_REF]. Therefore, although we cannot completely exclude sibling species [START_REF] Bunn | The barn owl[END_REF]Reeder 2005, Pagès et al. 2010), we can assume that virtually all of the 161 specimens were R. tiomanicus.

Predictive modelling, a compromise between parsimony, accuracy, and practicability

Except for the upper and lower molar row length (CLM1-3 and L5), the monovariate correlations (r ≥ 0.80) between the jaw or skull variables and ELW were strong; length of rostrum (LR) and L2 yielded the best correlations (both r = 0.89). We could not find similar investigations on Rattus species in the international literature, which precludes a comparison. Our results differ from the Microtus results. [START_REF] Shi | Studies on age and reproduction of Tianshan vole (Microtus ilaeus)[END_REF] found that the length of the Microtus ilaeus Thomas upper molar row correlated best with age, and [START_REF] Quéré | Détermination de la structure en âge relatif d'une population de Microtus arvalis (Rongeurs, Arvicolidés) par mesures ostéométriques crâniennes ou mandibulaires[END_REF] found that the Microtus arvalis Pallas great skull length, LR, and molar row length (CLM1-3) correlated best with Moreover, the breadth of palate yielded one of the lowest correlations for M. arvalis, whereas in our results, it correlated well with ELW and was included in the final equation. We found a high measurement error for certain variables, such as CLM1-3 (26.9%), which may be due to the difficulty in finding precise landmarks on bones; therefore, these measurements were discarded from "practical" modelling. We did not detect a significant difference in ELW between males and females. These data are consistent with [START_REF] Williams | Determination of age of Polynesian rats[END_REF] and [START_REF] Shrestha | Eye-lens weight curve for estimation of age in rats, Rattus rattus brunneus and Bandicota bengalensis in Nepal[END_REF], who did not detect a difference in ELW based on sex for other Rattus species, such as Rattus exulans Peale and Rattus brunneus Hodgs, respectively. The latter species is currently regarded as a Rattus tanezumi synonym (Wilson and Reeder 2005). Certain authors have reported that males have slightly heavier lenses than females, such as for Rattus norvegicus Berkenhout [START_REF] Hardy | Estimation of age in the Norway rat (Rattus norvegicus Beerkenhout) from the weight of the eye lens[END_REF]) and other rodents (Rowe et al. 1985, Yabe and[START_REF] Yabe | Eye-lens weight curve for estimation of age in the Japanese grass vole, Microtus montebelli Milne-Edwards (Rodentia: Muridae)[END_REF]. Our results indicate that the models perform only slightly better when the variable sex is used as a covariate. Therefore, using models 1-3 to predict age may produce a minor sex-related bias. However, [START_REF] Hardy | Estimation of age in the Norway rat (Rattus norvegicus Beerkenhout) from the weight of the eye lens[END_REF] noted that considering the sex variable in modelling may not be justified because the improvement therefrom is slight.

Many studies aimed at estimating the age, size, or body mass of prey from pellets prefer only lower jaw variables because this organ is highly frequent and well preserved in macroremains [START_REF] Morris | Rats in the diet of the Barn owl (Tyto alba)[END_REF][START_REF] Trejo | Owl selection on size and sex classes of rodents: activity and microhabitat use of prey[END_REF][START_REF] Granjon | Prey selection by barn owls in relation to small-mammal community and population structure in a Sahelian agro-ecosystem[END_REF][START_REF] Bueno | Small mammal prey selection by two owl species in southeastern Brazil[END_REF]. However, we found that approximately 70% of pellets were headless. This result is consistent with other reports [START_REF] Medway | Barn owl pellets from Kulai, Johore[END_REF][START_REF] Lenton | The feeding and breeding ecology of barn owls Tyto alba in Peninsular Malaysia[END_REF][START_REF] Lim | Prey selection by Barn Owls (Tyto alba) in an oil palm plantation in Malaysia[END_REF][START_REF] Hafidzi | Prey selection by barn owl in rice fields in Malaysia[END_REF][START_REF] Puan | Absence of differential predation on rats by Malaysian barn owls in oil palm plantations[END_REF]. Considering the frequent absence of heads in barn owl pellets on OPPs, we focused on elements other than the skull, jaw, and teeth for modelling. In addition, the tendency of the barn owl to decapitate larger prey may yield a bias if the skull or jaw is used to estimate the age or size distribution and further assess the impact of barn owl predation [START_REF] Lim | Prey selection by Barn Owls (Tyto alba) in an oil palm plantation in Malaysia[END_REF]. The femur was a good candidate for ELW prediction because it was well represented in our pellet sample, and the variable short femur highly correlated with the ELW. Moreover, the femur measurement error was the lowest compared with the other variables used in the final equations. Thus, the final equation with only the short femur would be more accurate, even if the prediction precision was not greatly improved by considering all the variables.

Model limits and extrapolation

Age prediction is more accurate in younger animals [START_REF] Tanikawa | An eye-lens weight curve for determining age in black rats, Rattus rattus[END_REF][START_REF] Shrestha | Eye-lens weight curve for estimation of age in rats, Rattus rattus brunneus and Bandicota bengalensis in Nepal[END_REF]. William (1976) found that the ELW was only a useful indicator of Rattus exulans age for up to 3 or 4 months. [START_REF] Myers | Determination of age of indigenous rats in Australia[END_REF][START_REF] Hardy | Estimation of age in the Norway rat (Rattus norvegicus Beerkenhout) from the weight of the eye lens[END_REF]) studied indigenous rats in Australia and found a similar lack of precision at this threshold, which clearly limits age determination using the ELW.

Our measurements are consistent with the maximum weight of adult rats reported in the literature [START_REF] Corbett | The mammals of the Indomalay region: a systematic review[END_REF]Hill 1992, Aplin et al. 2003). However, according to [START_REF] Wood | A long-term study of Rattus tiomanicus populations in an oil palm plantation in Johore, Malaysia: I. Study methods and population size without control[END_REF], young rats up to approximately 10 g cannot be trapped by the cage traps used in our study. As the body weight of our specimens ranged from 14 to 139 g, we may not have included certain juvenile individuals in our models.

Because the rats' food supply does not vary greatly month to month owing to a relatively constant temperature and continuous oil palm fruiting throughout the year [START_REF] Lenton | The feeding and breeding ecology of barn owls Tyto alba in Peninsular Malaysia[END_REF], an effect from rat nutritional status or season on ELW and, thus, age estimations is unlikely.

The equations proposed here are theoretically only applicable for comparing the population structure in the same area during the same period [START_REF] Quéré | Détermination de la structure en âge relatif d'une population de Microtus arvalis (Rongeurs, Arvicolidés) par mesures ostéométriques crâniennes ou mandibulaires[END_REF]. However, even if the rats were trapped in May, extrapolating to another season or year should not be a problem in the tropics and specifically in OPPs without chemical control. Indeed, the rat population dynamics do not change markedly with the seasons (due to continuous oil palm tree fruiting; Lenton 1984), the population demography fluctuates slowly, and the weight spectrum is approximately constant [START_REF] Wood | A long-term study of Rattus tiomanicus populations in an oil palm plantation in Johore, Malaysia: I. Study methods and population size without control[END_REF]Liau 1984b). In contrast, the location is important because population-based genetic variations cannot be excluded.

Owing to the lack of skulls or presence of broken skulls, it is difficult to identify prey species in pellets, which may be an issue for attributing the femur to the correct species. However, the barn owl diet generally reflects the abundance of prey species in the hunting territories [START_REF] Bunn | The barn owl[END_REF][START_REF] Figueroa | Rodent prey of the barn owl Tyto alba and short-eared owl Asio flammeus during winter in agricultural lands in Southern Chile[END_REF]), as demonstrated for OPPs by [START_REF] Lenton | The feeding and breeding ecology of barn owls Tyto alba in Peninsular Malaysia[END_REF] and [START_REF] Puan | Absence of differential predation on rats by Malaysian barn owls in oil palm plantations[END_REF]. Therefore, because Rattus tiomanicus is the predominant species in our study area, we confidently assume that this species is the major, if not the only, prey in the barn owl pellets studied. In areas where several rat species are sympatric and abundant, new-generation molecular methods can be used to ensure that femurs from the pellets are from R. tiomanicus [START_REF] Galan | Next-generation sequencing for rodent barcoding: species identification from fresh, degraded and environmental samples[END_REF].

Conclusion

Our study demonstrates that the femur length is relevant to ELW predictions for prey from macroremains, specifically in OPPs, wherein most pellets are headless. Our models represent a preliminary step to establish a relative age structure for Rattus tiomanicus consumed by the barn owl and to compare it to the population structure in the field, with a view to further assessing the impact of barn owl predation and improving ecologically based pest control.

Figure 1

 1 Figure 1 Skull, jaw, and femur variables.Adapted from[START_REF] Quéré | Détermination de la structure en âge relatif d'une population de Microtus arvalis (Rongeurs, Arvicolidés) par mesures ostéométriques crâniennes ou mandibulaires[END_REF]. Skull: GSL, greater skull length; IB, interorbital breadth; LR, length of rostrum; CLM1-3, molar row length; LD, length of diastema; BP, breadth of palate; and upper incisors. Jaw: L1, mandible length; L2, mandible height; L3, diastema height; L4, jaw height; L5, molar row length; and lower incisor. Femur, short femur and long femur.

  of the one-way ANOVA were calculated to determine the components of variance: among -MSS within )/m,

Figure 2

 2 Figure 2 Population eye lens weight pyramid-stratified using sex for a Rattus tiomanicus sample (n = 161) trapped on an oil palm plantation in Riau (Indonesia). The numbers at the bar tops are percentages of the total.

Table 1

 1 Summary statistics for Rattus tiomanicus body and bone measurements (n = 161).

	Variable	Mean ± standard	Minimum	Maximum
		deviation
	Body weight (g)	72.73 ± 23.27	17.00	139.00
	Body length (mm)	139.9 ± 15.1	90.00	183.00
	Tail length (mm)	144.4 ± 14.6	85.00	189.00
	Eye lens weight	35.98 ± 7.02	13.20	64.00
	(mg)
	Jaw (mm)
	L1	16.82 ± 1.24	11.77	21.02
	L2	8.97 ± 0.97	4.77	12.18
	L3	5.27 ± 0.43	3.51	7.08
	L4	5.19 ± 0.55	3.21	7.52
	L5	6.16 ± 0.15	4.23	6.86
	Lower incisor	1.62 ± 0.20	0.87	2.35
	Skull (mm)
	GSL	36.27 ± 2.62	25.67	41.92
	IB	5.68 ± 0.45	4.30	7.10
	LR	12.74 ± 1.24	8.21	17.30
	CLM1-3	6.40 ± 0.19	5.21	7.12
	LD	9.87 ± 1.01	6.51	12.92
	BP	4.49 ± 0.40	2.60	6.49
	Upper incisor	1.97 ± 0.20	1.18	2.71
	Femur (mm)
	Long femur	23.57 ± 3.00	12.62	30.40
	Short femur	20.20 ± 2.63	10.97	26.45

L1, mandible length; L2, mandible height; L3, diastema length; L4, jaw height; L5, molar row length; GSL, greater skull length; IB, interorbital breadth; LR, length of rostrum; CLM1-3, molar row length; LD, length of diastema; and BP, breadth of palate.

Table 2

 2 Frequency in barn owl pellets (n = 120), measurement errors, and correlation with eye lens weight for each Rattus tiomanicus osteometric variable (n = 161).

	Osteometric	Frequency in	Measurement	Correlation with
	variable	pellets (%)	error (%)	eye lens weight
				Pearson	p-Value
				coef. r
	Jaw
	L1***	30.0*	1.8* 0.85* < 0.001
	L2***	30.8*	1.4* 0.89* < 0.001
	L3	32.5*	11 0.83* < 0.001
	L4	32.5*	10.8 0.87* < 0.001
	L5	32.5*	10.6	0.37 < 0.001
	Lower incisor	32.5*	14.5 0.86* < 0.001
	Skull
	GSL	2.5	0.2* 0.87* < 0.001
	IB	21.7*	26.4 0.79* < 0.001
	LR	8.3	4.9* 0.89* < 0.001
	CLM1-3	25.8*	26.8	0.27 < 0.001
	LD***	23.3*	1.9* 0.86* < 0.001
	BP***	23.3*	4* 0.83* < 0.001
	Upper incisor	22.5*	12.7 0.83* < 0.001
	Femur
	Long femur***	25.0*	1.2* 0.92* < 0.001
	Short femur***	80.8*	0.2* 0.92* < 0.001

L1, mandible length; L2, mandible height; L3, diastema length; L4, jaw height; L5, molar row length; GSL, greater skull length; IB, interorbital breadth; LR, length of rostrum; CLM1-3, molar row length; LD, length of diastema; and BP, breadth of palate. The values that fit with the pre-selection criteria (see Material and methods) are marked with an asterisk; the variables pre-selected for "practical" predictive modelling are indicated with three asterisks.

Table 3

 3 Comparison of predictive model fitness and accuracy for Rattus tiomanicus eye lens weight estimation using a stepwise procedure and cross validation.

	Pre-selection of variables	Model	Best combination of variables and associated	AIC	R 2	Rmse
			equation
	L1, L2, LD, BP, short and long femur	1	ELW = (1.5060 L2+0.8413 LD+2.7511 BP+1.4777	435.62	0.847	3.86
			Short femur)-28.0396
	L1, L2, LD, BP, short and long femur, sex	1sex	ELW = (-1.542 Sex+1.601 L2+0.909 LD+2.560	431	0.851	3.83
			BP+1.276 Long femur)-28.278
	Short femur	2	ELW = (2.806 Short femur)-20.705	458.04	0.859	4.14
	Short femur, sex	2sex	ELW = (-1.186 Sex+2.813 Short femur)-20.3405	456.72	0.851	4.16
	All variables	3	ELW = (0.8263 L2-2.4834 L5+0.7125 LD+2.6186	422.05	0.865	3.78
			BP+4.4933 Upper incisor+5.5058 Lower
			incisor+1.2114 Short femur)-17.1739
	All variables, including sex	3sex	ELW = (-1.260 Sex+1.101 L2-1.603 L3-2.425	419.1	0.851	3.71
			L5+1.030 LD+2.502 BP+4.146 Upper incisor+6.126
			Lower incisor+1.245 Short femur)-14.637
	Models 1 and 2 are "practical" models, and model 3 is the "theoretical" model.
	ELW, eye lens weight; L1, mandible length; L2, mandible height; L3, diastema length; L5, molar row length; LD, length of diastema; and
	BP, breadth of palate.

Table 4

 4 Parameter estimates and associated standard error (SE) for the Rattus tiomanicus eye lens weight prediction models.

	Variable	Estimate	SE
	Model 1
	Constant	-28.0396	2.4500
	L2	1.5060	0.5760
	LD	0.8413	0.5351
	BP	2.7511	0.9358
	Short femur	1.4777	0.2697
	Model 2
	Constant	-20.705	1.9790
	Short femur	2.806	0.0966
	Model 3
	Constant	-17.1739	6.6529
	L2	0.8263	0.5763
	L5	-2.4834	1.1539
	LD	0.7125	0.5094
	BP	2.6186	0.8939
	Upper incisor	4.4933	2.3100
	Lower incisor	5.5058	2.3749
	Short femur	1.2114	0.2710

L1, mandible length; L2, mandible height; L3, diastema length; L5, molar row length; LD, length of diastema; and BP, breadth of palate.

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Acknowledgments:

We thank the technicians from SMARTRI who helped with the fieldwork as well as PTSmart Tbk for their accommodation and for providing access to the facilities and plantation. We are grateful to the State Ministry of Research and Technology of Indonesia, which supported this research (0131/SIP:FRP/ SM/V/2011).