Modelling the movement and survival of the root-feeding clover weevil, Sitona lepidus, in the root-zone of white clover

Ecological Modelling 190 (2006) 133–146

Modelling the movement and survival of the root-feeding cloverweevil,Sitona lepidus, in the root-zone of white clover

Xiaoxian Zhanga,∗, Scott N. Johnsonb, Peter J. Gregoryb, John W. Crawforda,Iain M. Younga, Philip J. Murrayc, Steve C. Jarvisc

a SIMBIOS Centre, University of Abertay Dundee, Bell Street, Dundee DD1 1HG, UKb School of Human and Environmental Sciences, Department of Soil Science, University of Reading,

Whiteknights, P.O. Box 233, Reading RG6 6DW, UKc Institute of Grassland and Environmental Research, North Wyke Research Station, Okehampton EX20 2SB, Devon, UK

Received 18 May 2004; received in revised form 20 January 2005; accepted 31 January 2005Available online 15 June 2005

Abstract

White clover (Trifolium repens) is an important pasture legume but is often difficult to sustain in a mixed sward because,among other things, of the damage to roots caused by the soil-dwelling larval stages ofS. lepidus. Locating the root nodules onthe white clover roots is crucial for the survival of the newly hatched larvae. This paper presents a numerical model to simulatethe movement of newly hatchedS. lepidus larvae towards the root nodules, guided by a chemical signal released by the nodules.T peed of thel nd only tot s used tos urement ofl the effectso ging of thel rs for larvals n noduled©

K

f

ntalmentms,

0

he model is based on the diffusion–chemotaxis equation. Experimental observations showed that the average sarvae remained approximately constant, so the diffusion–chemotaxis model was modified so that the larvae respohe gradient direction of the chemical signal but not its magnitude. An individual-based lattice Boltzmann method waimulate the movement of individual larvae, and the parameters required for the model were estimated from the meas

arval movement towards nodules in soil scanned using X-ray microtomography. The model was used to investigatef nodule density, the rate of release of chemical signal, the sensitivity of the larvae to the signal, and the random fora

arvae on the movement and subsequent survival of the larvae. The simulations showed that the most significant factourvival were nodule density and the sensitivity of the larvae to the signal. The dependence of larval survival rate oensity was well fitted by the Michealis–Menten kinetics.2005 Elsevier B.V. All rights reserved.

eywords: White clover;Sitona lepidus; Modelling; Chemotaxis; Lattice Boltzmann

∗ Corresponding author. Tel.: +44 1382 308 611;ax: +44 1382 308 117.

E-mail address: [email protected] (X. Zhang).

1. Introduction

There is mounting concern about the environmeand economic consequences of current managepractices in many agricultural production syste

304-3800/$ – see front matter © 2005 Elsevier B.V. All rights reserved.doi:10.1016/j.ecolmodel.2005.01.063

134 X. Zhang et al. / Ecological Modelling 190 (2006) 133–146

leading to increasing interest in developing more sus-tainable management strategies (Thrupp, 1996). In pas-ture systems, for example, it is widely recognised thatwhite clover (Trifolium repens) in mixed swards ishighly desirable because of its high nutritional quality(Leach et al., 2000) and the nitrogen enrichment it pro-vides to the system via nitrogen fixation by symbioticRhizobium spp. bacteria (Newbould, 1982; Ledgardand Steele, 1992). Whilst a high clover content is desir-able in mixed swards to reduce dependency on nitrogenfertilisation, it is often difficult to either establish orsustain white clover. Many reasons are put forwardto explain why white clover fails to thrive, includingcompetition with grasses, either directly for light andnutrients, or because grasses have a competitive advan-tage when soil is rich in mineral nitrogen (Schwinningand Parsons, 1996; Schulte et al., 2003). Another fac-tor is the damage caused to clover by invertebrate pests(Murray, 1991; Clements, 1995), amongst which theclover root weevil,S. lepidus, is considered to be oneof the more destructive.S. lepidus is widely distributedthroughout the northern hemisphere. It was acciden-tally introduced to New Zealand recently, where it hasflourished in the absence of effective natural enemies,and continues to threaten white clover which underpinsmuch of New Zealand’s pastoral industries (Barratt etal., 1996; Phillips et al., 2000).

S. lepidus feed adults above-ground on white cloverleaves, where it lays eggs that fall to the soil surface,and hatch into soil-dwelling larvae that then attackt ef feedib elyl tiono ur-v ghtt inn

age,s edd ainsl reat-m ht,1 oil-d ntt und.F

that 30% of root materials were detached by larvae.Comparatively little is known about the ecology ofthe larval stages compared with the adult weevil, butnew techniques for studying soil-dwelling insects havehelped increase understanding of this part of the life-cycle. It was recently shown byJohnson et al. (2004a)that neonatalS. lepidus larvae could detect the pres-ence of white clover nodules and roots from a distanceof 60 mm and distinguish them from grass roots andother species of clover under laboratory conditions. Afurther non-invasive study using X-ray microtomogra-phy showed that neonatal larvae appeared to directlytarget the root nodules (Johnson et al., 2004b).

While there has been some progress in the studyof root-feeding insects likeS. lepidus, these studieshave been entirely empirical, with no attempt to math-ematically model this part of the life-cycle. Theoreticalmodelling could inform future empirical research andprovide predictive simulations that may ultimately leadto better management practices to control this pest, andtherefore has received increasing interest in many fields(Murry, 2002). In the community of ecological mod-elling, several models have been proposed to modelorganism movement ranging from microbial transportin porous media (Li et al., 1996) to cattle grazing ina pasture (Shiyomi and Tsuiki, 1999). The mathemat-ical approaches developed for modelling such move-ment include discrete random walk (Blackwell, 1997;Wu et al., 2000; Yamamura et al., 2003) and diffu-sion equation (Shiyomi and Tsuiki, 1999). AlthoughY ntd r-r onei ss ist theirs cs ont

perb rtedb itha f lar-v te am sur-v er.A el-o e ins lt ofa The

he root system (Bigger, 1930). Larvae that emergrom eggs (neonates) are ca. 1 mm in length andnitially on root nodules that contain N2-fixing Rhizo-ium spp. bacteria before moving onto progressivarger roots during larval development. Consumpf root nodules is a crucial determinant of larval sival (Gerard, 2001) because the nodules are thouo provide protection from predators and are richitrogen compounds.

Insect pests that have a root-feeding larval stuch asS. lepidus, often cause the most sustainamage to plants because their attrition rem

argely unseen, preventing early diagnosis and tent (Brown and Gange, 1990; Villani and Wrig990; Hunter, 2001). The damage caused by the swellingS. lepidus larvae is frequently more significa

han that caused by the adults feeding above-groor example,Murray and Clements (1998)reported

amamura et al. (2003)considered biased movemeue to wind andWu et al. (2000)considered the coelation betweens organisms’ consecutive jumps,mportant mechanism these models did not addrehe biased movement of the organisms towardsubstrate and the effects of the substrate dynamiheir foraging behaviour and consequent survival.

The mathematical modelling presented in this pauilds on the experimental findings recently repoy Johnson et al. (2004b). These results, together wrena experiments using time-lapse observations oal movement reported here, are used to formulaathematical model to describe the movement and

ival of neonatal larvae in the root-zone of white clovn individual-based lattice Boltzmann model is devped to simulate the movement of neonatal larvaoil, assuming that larval movement is the resurandom foraging and a chemotactic movement.

X. Zhang et al. / Ecological Modelling 190 (2006) 133–146 135

model utilizes the experimental observation that thesignal released by the nodules affects only the direc-tion and not the speed of the responsive movement ofneonatal larvae. The two parameters, random forag-ing coefficient and chemotactic coefficient, required inthe model are estimated from experimental data. Themodel is then used to investigate the movement andsubsequent survival of neonatal larvae under differentconditions in the root-zone of white clover.

2. Mathematical models

2.1. Model for larval movement

The model for simulating larval movement is basedon the diffusion–chemotaxis equation (DCE) with amodification (see Section4). The continuum form DCEhas been widely used to model such asE. coli move-ment towards bacteria, tiger hunting (Murry, 2002),transport of soil-borne bacteria in aquifers (Ginn etal., 2002) and movement of nematodes in soils (Huntet al., 2001). An individual-based model derived fromthe DCE was given inAnderson et al. (1997)to modelnematode movement, which was also used bySchofieldet al. (2002)to model parasitoids movement. The modelassumes that the larval movement is the result of a ran-dom forage and a chemotactic responsive movement tothe signal released by the nodules. The random forageis a diffusive movement and the chemotactic movementi g thed sedb n is(

w rof ento ib-i icals ft Mort

ctsa dt ging

coefficient in the absence of the signal. The chemotac-tic coefficientχ, on the other hand, is more complexand less understood. A discussion on the form ofχ

for cell migration was given inPainter et al. (2000).Two commonly used chemotactic coefficients are:χ isa constant, andχ is a function of the concentration of thechemical signal asχ =χ0/(1 +αc), whereχ0 andα areconstants so that the chemotactic coefficient decreasesas the concentration increases.

A constant chemotactic coefficient, or a chemotac-tic coefficient that is a function of the chemical signal,in Eq.(1) results in the speed of the organism increas-ing with the concentration gradient, as in Eq.(1) thedistance an organism travels during time periodt isgiven by (Dgan, 1989):

l = χ

√(∂c

∂x

)2

+(

∂c

∂y

)2

+(

∂c

∂z

)2

t

+ N(0, 2DL t), (2)

where N(0, 2DL t) is a zero-mean normal distri-bution with variance 2DL t. Experimental observa-tions of how root-feeding insects move chemotacticallytowards host plants and/or attractant(s) are scarce, soa dedicated experiment (seeJohnson et al., 2004bandbelow) was designed to investigate the random forageand chemotactic movement of neonatalS. lepidus lar-vae towards white clover roots and root nodules.

2n

a)s l lar-v tityo otsr c-o ughs ownt emi-ca s ag tancer -n to5 -i d to

s a directed movement towards the nodules alonirection of the gradient of the chemical signal releay the nodules. The diffusion–chemotaxis equatioMurry, 2002)

∂n

∂t= ∇(DL∇n) − ∇(χn∇c) + Mor, (1)

here is the gradient operator,t time,n the numbef neonatal larvae per unit volume of soil,DL a random

oraging coefficient describing the random movemf neonatal larvae,χ a chemotactic coefficient descr

ng the sensitivity of neonatal larvae to the chemignal released by the nodules,c the concentration ohe chemical signal released by the nodules, andhe mortality/hatch rate of neonatal larvae.

In most existing models for the movement of insend organisms, the coefficientDL is usually assume

o be a constant and its value is the random fora

.2. Model for the chemical signal released by theodules

The experiments described inJohnson et al. (2004howed that signalling exists between the neonataae and white clover nodules. However, the idenf the signal is unclear at present. Most plant roelease substances such as CO2, amino acids and sendary plant volatiles from behind the root tips throloughing off cells and wounding. Studies have shhat soil-inhabiting insects are attracted to such chals released by the roots, most notably CO2 (Bernklaund Bjostad, 1998a,b). We assume that the signal iaseous chemical that is a compound of the subseleased by the nodules becauseS. lepidus can recogise white clover roots in soil from a distance up0 mm (Johnson et al., 2004a). Movement of the chem

cal signal emanating from the nodules is assume


be a diffusive process and is modelled by the followingdiffusion equation:

∂c

∂t= ∇[Dc∇(c)] + sa − wa, (3)

whereDc is the effective diffusion coefficient of thechemical signal in soil,sa the rate that the root nodulesrelease the chemical signal, andwaa term to account forthe loss of chemical due to physical/chemical and bio-logical reactions. The ratesa might vary with time andfrom nodule to nodule to reflect the impact of geneticdifference, nodule age and nodule damage.

Soil is a tri-phasic matrix comprising water, air andsolid, and the gaseous chemical can only move in theair-filled space. As a result, the effective diffusion coef-ficient Dc is a function of soil moisture content. Inthis paper, the dependence ofDc on soil moisture isdescribed by the following equation (Van Genuchten,1980):

Dc(s) = D√

s[1 − (1 − s1/m)m

]2, s = 1 − θ − θr

θs − θr

(4)

whereθ is volumetric water content,θs andθr the sat-urated and residual volumetric water contents, respec-tively, m a parameter andD the diffusion coefficientat θr, i.e. when all the connected pores are effectivelyfilled by air.

3

3m

bedi reb

)a la n too mitst sw on-i

lt lonc ted

Fig. 1. A two-dimensional snapshot of the X-ray tomographic imageof larval movement towards a white clover nodule.

towards the centre. A neonatal larva was placed cen-trally at the soil surface (approximately 5 mm fromthe plant) initially, and each column was then scannedsequentially after 3, 6 and 9 h. For illustration,Fig. 1shows a singleS. lepidus larva in close proximity to rootnodules on a white clover root. The average burrowingspeed of the larvae was calculated from their initialpositions and locations scanned at the three times. Theresult is shown inFig. 2. Clearly, the speed was approx-imately constant at 1.8 mm h−1.

3.2. Video-camera measurement of larvalmovement in open arenas

Because of the time required to scan a soil column,the X-ray technique is insensitive to subtle larval move-ments. To complement this study, we therefore exam-ined the responsive movement of larvae in open arenasto freshly macerated white clover roots with and with-

F soilm

. Experiments

.1. X-ray microtomography scanning of larvalovement in a soil matrix

The X-ray microtomography procedure is descrin Johnson et al. (2004b), but is summarised heecause the results underpin the present model.

This system, described inGregory et al. (2003nd Johnson et al. (2004b), relies on the differentiattenuation of X-rays passing through a soil columbserve roots and insects within the soil. This per

he sequential movement ofS. lepidus larvae towardhite clover roots and root nodules to be studied n

nvasively.Six Perspex columns (∅: 20 mm, H: 30 mm, wal

hickness: 1 mm) were set-up with a single stoutting of three stemmed white clover plants plan

ig. 2. The average speed vs. time for the larval movement inatrix.


out root nodules. The system comprised an open arena(∅: 90 mm Petri dish lined with dampened glass fibrefilter paper) suspended beneath a high-resolution dig-ital camera (AxioCam MRc, Zeiss Vision, Germany)

Fwn

connected to a personal computer. A single larva wasintroduced to the centre of the arena under three dif-ferent conditions: (a) an empty arena, (b) an arenacontaining 10 mg of nodulated white clover roots atone end, and (c) an arena containing 10 mg of non-nodulated white clover roots at one end. Each test wasrepeated for 30 different larvae, using fresh filter paperand root mashes on each occasion. Images of the arenawere captured at 10 s interval for 15 min (90 imagesin total) using image capture software (Axio Vision,v 3.1, Imaging Associates Ltd., Oxfordshire, UK) totrack larval movement. Images were then processedusing Tracker Vision software (developed by SIMBIOScentre and Imaging Associates Ltd. to operate withinKSRUN v 3.0, Carl Zeiss Vision, Germany) to deter-mine the locations of the larvae, from which the speedand turning angle were calculated.Fig. 3 shows theaverage speed with time under the three conditions.Although the average speed showed fluctuation underall the three conditions, it did not obviously decreaseor increase with time. This indicates that the signalreleased by the nodules affects only the direction andnot the speed of neonatal larvae when they forage.

The assumption that the random foraging of neona-tal larvae in the absence of roots and root nodules can bemodelled as a diffusive process requires that the larvalturning angle should be uniformly distributed over allangles. The turning angles of the larvae in the emptyarenas were calculated with different sampling inter-vals from 20 to 150 s.Fig. 4shows the results, where then d thep rn-i ging

ig. 3. Average speed vs. time for the larval movement in open arenasith: (A) nodulated white clover roots, (B) no roots and (C) non-odulated white clover roots.

f erow uni-f 0 s.T e ofr dif-f al isl

4

inga speedo tant.T nod-

egative values represent the larvae turning left anositive values turning right. The distribution of tu

ng angles was sampling-interval dependent, chanrom a symmetrical distribution with a peak at zhen sampling interval was 20 s to approximately

orm when the sampling interval increased to 15he random foraging of the larvae in the absencoots and nodules can therefore be modelled as ausion process, provided that the sampling intervong enough.

. The modified model for larval movement

The experimental results from both X-ray scannnd the arenas showed that the average foragingf neonatal larvae remained approximately conshis suggests that the signal released by the root


Fig. 4. The distribution of turning angles calculated with a samplinginterval: (A) 20 s, (B) 100 s and (C) 150 s, for the larval movementin empty dishes; the negative values represent the larvae turning leftand the positive values turning right.

ules affected the direction but not the speed of larvalmovement. In essence, the signal is only a guide and nota “driving force” implicit in Eq.(1). To appropriatelymodel this, the chemotactic term in Eq.(1) was modi-fied as follows so that the larvae responded only to thedirection and not the magnitude of the concentrationgradient of the chemical signal:

∂n

∂t= ∇[DL∇n] − ∇

(χ

|∇c|n∇c

)+ Mor. (5)

The unit of the chemotactic coefficientχ in Eq. (5) isthe same as the speed and can be interpreted as theaverage speed of neonatal larvae moving towards thenodules guided by the chemical signal.

5. Three-dimensional lattice Boltzmannsimulation

The movement of the chemical signal and neona-tal larvae were simulated with the lattice Boltzmannequation (LBE) model. The LBE model is a numericalmethod developed over the past two decades based onkinetic theory to simulate fluid dynamics (Chen andDoolen, 1998). The idea of the LBE is to simulatecomplex phenomena by tracking individuals. The LBEmodel developed in this paper for larval movement isindividual-based, which in the limit, i.e. as the numberof larvae increases, converges to the continuum formD

5

byt hed ingB

w epm -r ll asw ist in

CE given by Eq.(5).

.1. LBE model for the chemical signal

The LBE model simulates the chemical signalracking the movement of chemical particles. Tynamics of each particle is described by the followoltzmann equation:

∂f

∂t+ ξ∇f = Ω, (6)

here f, called particle distribution function, is throbability of finding a particle at positionx and timetoving with velocityξ, andΩ is a collision term rep

esenting the collision between the particles as weith soil. A simple approach to the collision term

he so-called single-relaxation-time approximation


which the collision is approximated by

Ω = f eq − f

µ, (7)

wherefeq is the value off in equilibrium, that is, whenthe particle density reaches the equilibrium state;µ arelaxation time measuring how farf is from feq, or inother words, how fastf approachesfeq. If the mass of thechemical particle at positionx and timet is c, to ensurethat the collision does not result in mass change, therelationship ofc with f, as well as withfeq, must be

c =∫∫∫

f dξ =∫∫∫

f eqdξ. (8)

The LBE model is to discretize the velocity spaceξ

appropriately so that the particle moves only with a fewvelocities while at macroscopic scale it gives a resultthat is same as what is described by a given equation.The velocity-discretized form of Eq.(6) is

∂fi

∂t+ ξi∇fi = f

eqi − fi

µi = 0, 1, 2, . . . , N, (9)

wherefi is the probability of finding a particle at posi-tion x and timet moving with velocityξi, andN is thenumber of velocities with which the particles move.The consequent discrete form of Eq.(8) is

c =N∑

fi =N∑

feqi . (10)

E thisp if-f

f

w sr

eenu dingc porti ls ist ee ings citys we

assume that its movement is diffusive and the chem-ical particles move with seven velocities in a cubiclattice:ξ0 = (0, 0, 0),ξ1 = (x/t, 0, 0),ξ2 = (0,x/t, 0),ξ3 = (0, 0,x/t), ξ4 = (−x/t, 0, 0),ξ5 = (0,−x/t, 0),ξ6 = (0, 0,−x/t), wherex is the length of the cubes.The equilibrium function under this descretization isf

eqi = c/7 (i = 0, 1, 2, 3, 4, 5, 6).

The above LBE model assumes that initial distribu-tion of the chemical signal is known. In implementa-tion, two steps, a collision step and a streaming step,are needed to advance one time step. The collisionstep is to calculate the term on the right-hand side ofEq.(11), f ∗

i (x, t) = fi(x, t) + [f eqi (x, t) − fi(x, t)]/λ,

using the data available at timet; and the streamingstep is to movef ∗

i (x, t) from its position atx to a newposition atx + ξi δt to becomefi(x + ξi t, t + t) =f ∗

i (x, t). Once the streaming step is completed, themass of the chemical particle at each cube is updated

by c(x, t + t) =6∑

i=0fi(x, t + t).

It can be proven using the Chapman–Enskog mul-tiple expansion (Chen and Doolen, 1998; Zhang et al.,2002a) that the above LBE model converges to Eq.(3)in the limit with the diffusion coefficient given by

Ds = 2

7

x2

t

(λ − 1

2

). (12)

5.2. LBE model for movement of individual larvae

eda vaei E asg nti ento iceB

P

w atpi -s

sob icht

i=0 i=0

q. (9) is a convective equation and is solved inaper by the following first-order Lagrangian finite d

erence method:

i(x + ξi t, t + t) = fi(x, t) + feqi (x, t) − fi(x, t)

λ(11)

heret is time step andλ =µ/t is a dimensionleselaxation time.

The above LBE method is general and has bsed to simulate a wide range of phenomena incluomplex fluid dynamics and agrochemical transn soils. The key difference between these modehe equilibrium functionf eq

i . The exact form of thquilibrium function depends on the problem beimulated and discretization of the particle velopaceξ. For the chemical signal in this paper,

The LBM model for individual larvae is designiming at that in the limit, i.e. as the number of lar

ncreases, it converges to the continuum form DCiven in Eq.(5). The LBE model for larval moveme

s similar to that for the chemical signal. The movemf a single larva is described by the following lattoltzmann equation:

i(x + ζ t1, t+t1) = Pi(x, t) + Peqi (x, t) − Pi(x, t)

η,

(13)

herePi(x,t) is the probability of finding a larvaositionx and timet moving with velocityζi, P

eqi (x, t)

s the value ofPi(x, t) in equilibrium, andη is a dimenionless relaxation time.

The LBE model for individual larvae is alased on the 7-velocity cubic model in wh

he larvae move with seven velocities:ζ0 = (0, 0, 0),


ζ1 = (x/t1, 0, 0), ζ2 = (0,x/t1, 0), ζ3 = (0, 0,x/t1),ζ4 = (−x/t1, 0, 0), ζ5 = (0,−x/t1, 0), ζ6 = (0, 0,−x/t1). Notice that the time step for the larval move-ment ist1 instead of thet used for chemical particlemovement because the chemical signal moves muchfaster than the larvae (i.e.t1 > t).

The difference between the models for the larvae andfor the chemical signal is that the larvae move not onlyrandomly as for the chemical, but also directionallyalong the gradient of the chemical signal emanatingfrom the nodules. This is solved in the LBE modelusing the following equilibrium function:

Peqi = 1

7

(ε + 3.5χ t1

xζi

∇c

|∇c|)

i = 1, 2, 3, 4, 5, 6, Peq0 = 1 −

6∑i=1

Peqi , (14)

whereε is a free parameter introduced to adjust therandom foraging movement. The concentration gradi-ent of the chemical signal in Eq.(14) is estimated bythe following second-order finite difference approxi-mation:

∂c

∂x≈ cj+1,k,l − cj−1,k,l

2x,

∂c

∂y≈ cj,k+1,l − cj,k−1,l

2x,

∂c cj,k,l+1 − cj,k,l−1

w gti

| 1,l)2 +

I all ist-i nceo rob-ai stepi cal-c et l-

culated fromP∗i = Pi(x, t) + [Peq

i (x, t) − Pi(x, t)]/η.The seven calculated probabilities are then used togenerate seven probability ranges:R0 ∈ [0 ∼ P∗

0 ), and

RL ∈[

L−1∑n=0

P∗i ,

L∑n=0

P∗i

)(L = 1, 2, 3, 4, 5, 6). A ran-

dom number uniformly distributed between 0 and 1 isthen generated. Depending on the range in which therandom number falls, the larva staying at cube (j, k, l)at timet will continue to stay in cube (j, k, l) if the ran-dom number is in rangeR0, move to cube (j + 1,k, l)if in rangeR1, move to cube (j, k + 1,l) if in rangeR2,move to cube (j, k, l + 1) if in rangeR3, move to cube(j − 1,k, l) if in rangeR4, move to cube (j, k−l, l) if inrangeR5, and move to cube (j, k, l − 1) if in rangeR6.To simplify computation, it is assumed in the simula-tions that the dimensionless relaxation timeη = 1; thatis, the time step size and the relaxation time are thesame so thatP∗

i (x, t) = Peqi (x, t).

The convergence of the above LBE model for indi-vidual larvae to the continuum form DCE as given inEq. (5) in the limit is easy to prove. Suppose that thenumber of larvae at positionx and timet is n; from Eq.(13) it is known that the number of larvae moving withvelocity ζi is

nPi(x + ζi t1, t + t1)

= nPi(x, t) + nPeqi (x, t) − nPi(x, t)

η. (17)

Dt ipb

n

p

i

∂z≈

2x, (15)

here the subscriptsj, k andl are indexes numberinhe cubes in thex, y andz directions, respectively;|c|s estimated by

∇c| ≈ 1

2x

√(cj+1,k,l − cj−1,k,l)2 + (cj,k+1,l − cj,k−

mplementation of the above LBE model for individuarvae is similar to that for the chemical signal, consng of a collision step and a streaming step to advane time step. The collision step is to calculate the pbilities of a larva, which is at positionx at timet, mov-

ng in each of the seven directions; the streamings to move the larva to a new position based on theulated probabilities. Using the data available at timt,he seven probabilities,P∗

i (i = 0, 1, 2, 3, 4, 5, 6), are ca

(cj,k,l+1 − cj,k,l−1)2. (16)

enoting the number of larvae at positionx and timemoving with velocityζi by pi , then the relationshetweenn andpi is

=6∑

i=0

pi(x, t) =6∑

i=0

peqi (x, t),

eqi = n

7

(ε + 3.5χ x

t1ζi

∇c

|∇c|)

= 1, 2, 3, 4, 5, 6, peq0 = n −

6∑i=1

peqi . (18)


As for the LBE model for the chemical signal, it can beproven using the Chapman–Enskog’s multiple expan-sion that Eqs.(17) and (18)converge in the limit toEq. (5) with the random foraging coefficient given by(Zhang et al., 2002a)

DL = 2ε

7

x2

t1

(η − 1

2

). (19)

6. Parameter determination

6.1. Parameters for larval movement

The parameters required for simulating the move-ment of neonatal larvae are the random foraging coef-ficient DL and the chemotactic coefficientχ. Theirexperimental measurement is nontrivial because of thedifficulty of observing the tiny neonatal larvae in anopaque soil matrix. In this paper we have estimatedthe two parameters based on the data obtained fromthe above X-ray tomographic scanning. Although thesample number was insufficient to give a robust esti-mation, these values are used because they are the bestavailable information.

If the distance a larva travels from time zero to timet is denoted byl, thenl consists of a random movementand a directional movement, and its kinetic equation is

w rec-t icals va’sr medt t is,〈 ver-afr

T -t

Fig. 5. Calculated variance of the travel distance〈l′(t)l′(t)〉 vs. timefor larval movement in soil cores.

follows (Dgan, 1989):

χ = U,

DL = 1

2

∂ < l′(t)l′(t) >

∂t

= 1

2

∂

∂t

∫ t

0

∫ t

0u′(t)u′(t′′) dt′ dt′′. (22)

The average speedU is given in Fig. 2 and Fig. 5shows the variation of〈l′(t)l′(t)〉 with timet, from whichthe random forage coefficient was calculated to beDL = 2.25 mm2 h−1.

6.2. Parameters for the chemical signal

S. lepidus larvae are attracted to white clover rootsin soil from up to 5 mm away, so a gaseous signallingmechanism must underpin root location (Johnson etal., 2004a). The diffusion coefficient of CO2 was usedas an approximate diffusion coefficient of the uniden-tified gaseous signal, which is 6.5× 104 mm2 h−1 inair at temperature 25C. The diffusion of the chem-ical signal in soil, however, is much less than thatbecause of the tortuosity of the air-filled pores, andis affected by soil moisture content. Dependence ofthe diffusion coefficient of the chemical signal on soilmoisture is described by Eq.(4). A numerical anal-ysis of the diffusion of the chemical signal in soilunder various soil water contents demonstrated thatu d oft ve-m tent

∂l

∂t= U + u′, (20)

hereU is the average speed of the larva moving diionally towards the nodules guided by the chemignal, andu′ is a speed deviation caused by the larandom movement. The speed deviation is assuo be a zero-mean normal random process, thau′〉 = 0, where the angle bracket means taking age. Because of the randomness ofu′, l is also a random

unction. Splittingl into a determinant part,X, and aandom part,l′, leads to

∂X

∂t= U,

∂l′

∂t= u′. (21)

he random foraging coefficientDL and the chemoactic coefficientχ in Eq.(5) are related to Eq.(21) as

nless the soil is extremely wet, the diffusive speehe chemical signal is much faster than larval moent and thus the impact of soil moisture con


on the directional response of the neonatal larvae isinsignificant. We therefore used a constant diffusioncoefficient, 6.5× 102 mm2 h−1, for the chemical signalin all the simulations. It is possible that soil mois-ture content might affect the motility of the larvaeand this is a well-documented effects in nematodes(Caenorhabditis elegans) movement in sand as affectedby particle size, moisture and the presence of bacte-ria (Escherichia coli) (Young et al., 1998). However,we have no knowledge of the impact on the move-ment ofS. lepidus larvae, so at this stage we assumethat the speed of the movement is not affected by soilmoisture.

7. Simulation and analysis of results

The above model was used to simulate the move-ment of neonatal larvae within the root-zone of whiteclover in order to investigate the factors that are mostlikely to affect the survival of newly hatched larvae.The factors considered included nodule density, the rateof release of the chemical signal, and the sensitivityof neonatal larvae to the chemical signal. The spatiallocation of the eggs was assumed to be uniformly dis-tributed near soil surface (seeFig. 7A below), and thespatial location of the nodules on the white clover rootwas assumed to be randomly distributed around a singleroot located vertically at the centre of the domain (seeFig. 7below). For each experimental set-up, 10 simu-l erea dl thes larvafi tor ule( hatn ur-r medt adyb ouldc piedn tol im-ua eret icals par-

ticles (or neonatal larvae) leaving the domain from oneside will re-enter the domain from its opposite side. Thebottom boundary was treated as a no-escape boundaryfor both the chemical signal and neonatal larvae, so thatall the chemical particles (or neonatal larvae) hitting thebottom boundary were bounced back towards the direc-tion they came from. The top boundary was treated as ano-escape boundary for the neonatal larvae, and larvaewere bounced back in the directions they came from.For the chemical signal, the top boundary was treatedas a free boundary and the movement of the chemi-cal particles across it was solved by the mirror-imagemethod given inZhang et al. (2002b). Refer toFig. 6where the line AB is the top boundary. In the stream-ing step, the chemical particle moving with velocityξ6towards the boundary comes from outside the domainand does not physically exist; that is, the link for thisparticle is broken by the boundary. In the mirror-imagemethod, the boundary is seen as a mirror and all the

F ove-m be.

ations were run. In all the simulations, the eggs wssumed to hatch at timet = 0 and the newly hatche

arvae then foraged to find the nodules guided byignal released by the nodules. Once a neonatalnds a nodule, it will stay there and is assumedemain alive whilst it feeds on the nitrogen-rich nodGerard, 2001). Experimental observations show to more than two larvae live in the same nodule (May, personal communication). We therefore assuhat if a larva encountered a nodule which had alreeen occupied by another larva, then this larva wontinue to forage until it encountered an unoccuodule. The larva was assumed to die if it failed

ocate a nodule within 24 h of hatching. In all the slations, the domain size was 10 cm× 10 cm× 10 cm,nd the four vertical boundaries of the domain w

reated as periodic boundaries for both the chemignal and neonatal larvae, in which the chemical

ig. 6. Mirror-image method to solve the top boundary for the ment of chemical signal; the shadowed area is the imagery cu


cubes inside the domain see their images in the mirror.The imaginary cubes are also defined with particlesthat provide the link that is broken by the boundaryduring the streaming step. For the example shown inFig. 6, the boundary cubeo has an imaginary cubeo′inside the mirror, and the particle distribution functionsdefined at cubeo′, including their direction and magni-tude, are the same as those defined at the real cubeo, i.e.fi(o′,t) = fi(o′,t) (i = 0, 1, 2, 3, 4, 5, 6). So the particle,f6, moving towards to boundary cubeo from outsidethe domain is given byf6(o, t + t) = f ∗

6 (o, t), wheref ∗

6 (o, t) is the pro-collision and pre-streaming particledistribution function at cubeo shown inFig. 6.

7.1. Impact of root nodule density on the survivalof neonatal larvae

All the simulations in this investigation started with122 larvae, randomly distributed near the soil surfaceas shown inFig. 7A. The number of nodules variedfrom a very few to several hundreds, and each nodulereleased the chemical signal at a constant rate. Sincethe neonatal larvae respond only to the direction andnot the magnitude of the concentration gradient of thechemical signal, the exact value of the rate at which thenodules release the chemical signal does not affect theresult. The rate of the release used in all the simulationswassa = 0.001.Fig. 7B shows a simulated isosurface ofthe chemical signal and locations of the neonatal larvaeat the end of one simulation. The larval survival rate wasc ingt 22l sust ellfiw s,a e is5

7r

thate samec od-u bea e byp wn

Fig. 7. (A) Initial positions of the neonatal larvae, (B) a simulatedisosurface of the chemical signal and locations of the larvae at theend of one simulation.

about how these factors might affect signalling. Herewe make a simple assumption that the variation of therate that nodules release the chemical signal is log-normally distributed within the nodule population, thatis, ln(Sa) ∼ N(Sa,σ2), whereN(Sa,σ2) is a normal dis-tribution with a mean ofSa and variance ofσ2. Thesesimulations assumed a constantSa =−3, and that thevariance varied from 10−6 to 100; other parameterswere the same as in Section7.1. Fig. 8B shows thedependence of the survival rate on the number of nod-

alculated as the ratio of the number of larvae findhe nodules within 24 h after hatching to the initial 1arvae.Fig. 8A shows the averaged survival rate verhe number of nodules in the domain, which is wtted by the Michealis–Menten kinetics,r = M/(M + k),herer is the survival rate,M the number of nodulendk the number of nodules when the survival rat0% (M = 270 for this example).

.2. Impact of varying the rate of chemical signalelease from nodules

It was assumed in the previous investigationach nodule released the chemical signal at theonstant rate. In reality, this rate might vary from nle to nodule because nodule quality is likely toffected by factors such as nodule age, damagests and genetic difference. However, little is kno


Fig. 8. Survival rate vs. the number of nodules for: (A) the rate ofnodules releasing chemical signal is constant, and (B) the rate variesfrom nodule to nodule; the symbols are simulation results and thesolid line is the fitting of the Michealis–Menten kinetics.

ules with differentσ2. Clearly, the impact of varyingthe rate of chemical release from the nodules on lar-val survival is small, and the relationship between thesurvival rate and the number of nodules for allσ2 iswell fitted by the same Michealis–Menten kinetics asin Section7.1.

7.3. Impact of the random foraging coefficient onlarval survival

The random foraging coefficient measures how farthe neonatal larvae might deviate from the directiongoverned by the gradient of the chemical signal. Sim-ulations in this investigation used different values ofrandom coefficientDL and the same chemotactic coef-ficient (χ = 1.8 mm h−1); the rate of chemical releasefrom the nodules was constant. The number of nodulesand the number of larvae was 238 and 122, respec-tively. Fig. 9A shows the variation of larval survival

Fig. 9. Survival rate vs.: (A) the larvae’s random forage coefficientand (B) larvae’s sensitivity to the chemical signal.

rate with the parameterDL; the survival rate increasedonly slightly asDL increased.

7.4. Impact of the chemotactic coefficient onlarval survival

The chemotactic coefficientχ measures the sensi-tivity of the larvae to the chemical signal released bythe nodules. Simulations in this example used differ-ent values ofχ and the same random foraging coeffi-cient (DL = 2.25 mm2 h−1); the rate of chemical signalrelease from the nodules was constant. The number ofnodules and the number of larvae was 236 and 122,respectively. The dependence of larval survival rate onthe parameterχ is shown inFig. 9B. Clearly, a posi-tive response to the chemical signal is crucial for theneonatal larvae to survive. For the example shown inFig. 9B, the survival rate increased from less than 5%when the larvae had no response to the signal to over40% as the sensitivity increased.


8. Discussion and conclusions

The work reported in this paper aimed to simu-late the movement and subsequent survival of newlyhatchedS. lepidus larvae in the root-zone of whiteclover. The movement of neonatal larvae in soil coresand in open arenas showed that the larvae move towardsthe root nodules, indicating the existence of a sig-nalling mechanism that allows the neonatalS. lepiduslarvae to locate white clover roots and the root nod-ules. The experimental results from the open arenasrevealed that in the absence of white clover roots androot nodules, the movement of neonatal larvae can bemodelled as a diffusive process if the sampling inter-val is long enough. The results from the observationsof larval movement in both soil cores and open arenasshowed that the average speed of neonatal larvae did notchange with time, indicating that the signal released bythe root nodules appears to be for guidance: it affectedthe direction but not the speed of larval movement.

An individual-based lattice Boltzmann method ispresented to simulate the responsive movement ofneonatal larvae to the chemical signal released by thenodules and used to investigate the factors that are mostlikely to affect the movement and survival of newlyhatched larvae in the root-zone of white clover. Thetwo parameters required in the model, the random for-aging coefficient and the chemotactic coefficient, areestimated from experimental data. The results of thesimulations indicate that among the factors being inves-t nsitya thatt duled et-i hicht d lit-t vali ienti nt isk

m-b odelt of ar ant.T andt ndi-t forma time

and the speed of larval movement is independent ofsoil moisture. Real situations are more complicated.For example, the soil density is heterogeneous, the spa-tial locations of the nodules change temporally becauseof root growth, and the time that the eggs take tohatch is affected by temperature, pH, soil moisture etc.(Johnson, personal communication). The model alsoassumes that the larvae do not follow each other duringtheir movement; this assumption needs further inves-tigation. However, the development of such models isimportant if a better understanding of how root-feedinginsects locate their host plants in the soil is to be gar-nered. This study is an attempt to begin to address thesecomplex issues.

Acknowledgements

This work was funded by the Biotechnology andBiological Science Research Council (BBSRC) of theUK (Project 45/D14536). The authors would like tothank Denise Headon of the Institute of Grassland andEnvironmental Research (IGER) for collecting insectsused in the experiments. We thank the two anonymousreviewers for their constructive comments to improvethe manuscript. IGER is supported by the BBSRC.

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Modelling the movement and survival of the root-feeding clover weevil, Sitona lepidus, in the root-zone of white clover