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Journal of Theoretical Biology 243 (2006) 493–500

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Food encounter rates of simulated termite tunnels with variable foodsize/distribution pattern and tunnel branch length

S.-H. Lee�, P. Bardunias, N.-Y. Su

Department of Entomology and Nematology, Ft. Lauderdale Research and Education Center, University of Florida, Ft. Lauderdale, Florida 33314

Received 23 January 2006; received in revised form 24 May 2006; accepted 20 July 2006

Available online 27 July 2006

Abstract

Subterranean termites excavate tunnels in a search pattern to encounter food in soil. To investigate the effect of food size, food

distribution and the branch length of tunnels on food encounter rate we used a lattice gas model to simulate tunnels of the Formosan

subterranean termite, Coptotermes formosanus Shiraki. The model made use of minimized local rules derived from empirical data to

simulate termite tunnel patterns in featureless soil. Food distributions with three types (uniform, random, and clumped) were defined by

using an I-index proposed by Zimmer and Johnson (1985). The food encounter rate was higher in a clumped than in non-clumped

(uniform and random) distribution of food particles. When food particle size was varied in random distributions of food particles a

maximum encounter rate was found, with particles of larger or smaller size being encountered less frequently. We also discussed the

relationship between the branch tunnel length and the tunnel search pattern in minimizing the redundancy of overlapping branches.

r 2006 Elsevier Ltd. All rights reserved.

Keywords: Encounter rate; Termite; Lattice gas model; Clumped distribution

1. Introduction

Animals must find food efficiently or they risk a loss offitness because energy that could be allocated to reproduc-tion is instead needed to survive. Foraging models areplentiful and diverse, ranging from static optimal foragingtheory (Stephens and Krebs, 1986) to deterministic partialdifferential equations (Okubo, 1980; Shlesinger et al., 1995)or stochastic simulations (Siniff and Jensen, 1969; Durrett,1999). Particularly in the theory of insect movement anddispersal, the stochastic simulation has received muchattention in recent years (DeAngellis and Gross, 1992;Czaran, 1998; Diekmann et al., 2000). In these models,search mechanisms were addressed by explicitly describingthe movement tracks of individuals in two or three-dimensional space to evaluate the adequacy of behaviorsin finding food resources (Harkness and Maroudas, 1985;Bell, 1991). Along with the development of theoreticalmodels, empirical data have been accumulated for thespecial case of animal foraging, in which an animal

e front matter r 2006 Elsevier Ltd. All rights reserved.

i.2006.07.026

ing author. Tel.: +1954 577 6326.

ess: sunchaos@pusan.ac.kr (S.-H. Lee).

optimizes its search for food (Cain et al., 1985; Focardiet al., 1996; Gillingham and Bunnell, 1989).In contrast to terrestrial animals, little is known for

foraging behavior of below-ground foragers such as earthworms (Capowiez et al., 1998) or subterranean termites.This is because most foraging galleries of subterraneananimals remain a ‘‘black box’’ that eluded direct observa-tion (Su, 2001). Subterranean termite foraging behaviorhas been studied through either direct gallery excavation(Greaves, 1962; King and Spink, 1969; Ratcliffe andGreaves, 1940) or studies on baited plots by using spatio-temporal and mark–release–recapture methods (Graceet al., 1989; LaFage et al., 1973; Su et al., 1984). Thesemethods offered no direct information on foragingefficiency in relation to food resource and most recentstudies of termite foraging investigated only the effect ofresource presence or quality (Gallagher and Jones, 2005;Su, 2005).We developed a lattice gas model to simulate termite

tunneling pattern in order to test the efficiency of the tunnelcomplex in encountering food particles at differentdistributions. Lattice gas models, a form of discretestochastic spatial population models, have been used by

ARTICLE IN PRESSS.-H. Lee et al. / Journal of Theoretical Biology 243 (2006) 493–500494

theoretical biologists to provide a simple, flexible frame-work for modeling spatial dynamics that arise from eventsat the individual level (Harada and Iwasa, 1994). In thisstudy, the food encounter rate was defined by counting thenumber of food encountered by simulated tunnel in searchtime per lattice site used in the place of the foragingefficiency. We examined the food encounter rate for threepatterns, namely, uniform, random, and clumped distribu-tion. We also investigated the effect of food size onencounter rate in a random distribution pattern and theeffect of the secondary branch tunnel length on foodencounter rate in relation to travel cost.

2. Empirical data

The empirical data on termite tunnel geometry wereobtained from the two-dimensional foraging arena study asreported by Su et al. (2004). The arena was composed oftwo sheets of transparent Plexiglas (105� 105 and 0.6 cmthick) separated from each other by four Plexiglaslaminates (105� 2.5 and 0.2 cm thick) placed between theouter margins. The 0.2 cm gaps between the Plexiglassheets were filled with moistened sand, into which 1000workers (plus 100 soldiers) of C. formosanus wereintroduced. The arena was placed in the horizontalposition in a dark room at 2572 1C. Digital images ofthe arena were taken daily to record tunnel developmentuntil one tunnel reached the arena edge. The experimentswere replicated nine times by using termite collected fromnine colonies.

Pright

Pdown

Pright

Pup

Pleft

Pdown

Pleft

Pup

Tunnel Vector Cell(TVC)

�i (i=1, 2,…,6)

Fig. 1. Schematic representation of the rules defining an individual

movement constructing primary tunnels with a transition probability from

one position to its nearest neighboring position.

3. Model description

3.1. Termite tunnel behavior on lattice space

We used the lattice gas model to describe termitetunneling pattern under a closed boundary condition.A tunnel was mimicked by the movement of a tunnel vectorcell (TVC) with transition probabilities from one cell to itsnearest neighboring cell. A TVC in reality is a discrete unitof tunnel excavated by a cadre of termite workers. Thismodel was defined on the square lattice of L�L sites(L ¼ 200). In this model, the movement trace of TVCscorresponded to tunneling pattern. Other parameters suchas average branching angle, probability of branching, andprobability of continuing through an intersection used inthis model was obtained from the empirical data of Su et al.(2004). Each lattice cell can either be empty or occupied byexactly one TVC. Each TVC can advance only one latticeper time step. In the present study, tunnels were designatedas two classes: primary or secondary. Tunnels originatingfrom the release position were classified as primary.Tunnels branching from the primary tunnel were classifiedas secondary (Selkirk, 1982). The tertiary and quaternarytunnels were excluded because they rarely occurred underthe test period (Su et al., 2004).

Initially, TVCs were positioned on the center cell of thelattice space. The TVCs advanced to form primary tunnels.Each TVC advances to the preferential direction with noback step because termite tunnels did not loop back to theorigin of release (Su et al., 2004). We set the preferentialdirection as outward direction from the release position.We applied the conservation law of probability P(i, j) to theTVC existing on a site (i, j). We denoted the transitionprobabilities from site (i, j) to the nearest neighborsas Pup(i, j+1); Pdown(i, j�1); Pleft(i�1, j) and Pright(i+1, j)(Fig. 1).In order to determine the transition probabilities of

initially introduced TVCs, we extracted primary tunnelshapes from empirical images with 0.5� 0.5 cm resolutionand then measured angles between primary tunnels underlinear fitting (Fig. 2(a)). From nine empirical tunnelingpatterns, the angle between primary tunnels (Df) and thenumber of primary tunnels (N) were determined as53.178.51 and 6.7871.011, respectively. The data indi-cated the lack of spatial correlation among tunnels,namely, angle interval between tunnels was nearly equalthus, Df ¼ 3601/N and the directional angle of TVCs yi isdetermined as yi ¼ iDf (i ¼ 1, 2, 3, y , 6).The transition probabilities were set by the directional

angle of TVC, yi as follows: Pup ( ¼ Pdown) ¼ |tan(yi)|/(1+|tan(yi)|)+b and Pleft ( ¼ Pright) ¼ 1/(1+|tan(yi)|)+b0,where the subscript i was a label assigned to primary tunnelranging from 1 to N. b and b0 were introduced formimicking the curvy shape of tunnel. When Pup (or Pdown)has the addition term b ( ¼ 0.1), the value of b0 isappropriately changed under the condition that thesummation of transition probabilities should become 1.The opposite case is in the same manner. The priority

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j

i

(a)

Primary

Lowersecondary

Uppersecondary

Branchingnode

(b)

��i j

�i (i=1, 2,…,N)

�0 = -50°

�0 = +50°

Fig. 2. Determination of transition probability of initially introduced TVCs: (a) the calculation of angle through the linear fitting of primary tunneling

shape extracted from real images, (b) the calculation of transition probability of secondary tunneling with branching angle y0.

P(L) ~ e-0.15 L

Nor

mal

ized

freq

uenc

y

Branch length [cm]

1.0

0.8

0.6

0.4

0.2

-0.2

0.0

0 10 20 30

Fig. 3. Branching length exponent a determining the length of secondary

tunnel.

S.-H. Lee et al. / Journal of Theoretical Biology 243 (2006) 493–500 495

between b and b0 is determined by coin toss occurred atTVC’s position at each time step.

To construct secondary branches of tunnels, the prob-ability of a branching node was generated by randomprocess from a TVC’s position. When the value ofprobability was larger than 0.9, the position was chosenas branching node (Su et al., 2004). The new TVCs atbranching node begin advancing with branching angle,y0 ¼ 501 as reported previously (Fig. 2(b)) (Su et al., 2004).To describe the movement of new TVCs, we modified thetransition probability P(i, j) to P0(i, j); P0up ¼ P0down ¼

|tan(yi7y0)|/(1+|tan(yi7y0)|) and P0left ¼ P0right ¼ 1/(1+|tan(yi7y0)|). Upper- or lower-side branching wasdetermined by coin toss rule. The sign ‘7’ indicates upperor lower branch, respectively.

The length of secondary tunnels L can be characterizedby the frequency distribution of branch lengthP(L)� exp(�aL) with a branch length exponent, a ¼ 0.15,which was obtained from nine empirical tunnel patterns(Fig. 3).

Fig. 4 shows all possible configurations that the up-TVCs, those moving in the ‘up’ direction, may encounter.If more than one TVC shared the same target lattice, onewas chosen at random, with equal probability. This TVCmoved while its rivals for the same target maintained theirposition (Fig. 4(a)–(c)). When the up-TVC encountersother tunnel, ‘passing’ or ‘stopping’ movement was also

chosen randomly, with an equal probability as derivedfrom the empirical data (Fig. 4(d)–(f)).In the similar way, we defined the similar treatment for

the possible configurations that the down, left, and rightwalkers may encounter. When one of TVCs reached atboundary wall, the simulation was terminated. The results

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(a) (b) (c)

orbecomes

(d) (e) (f)

becomes or

stop

Tunnel

TVC

Fig. 4. All possible configurations that an up-TVC (the cell moving in the ‘up’ direction) may encounter.

S.-H. Lee et al. / Journal of Theoretical Biology 243 (2006) 493–500496

of simulations were statistically averaged after 30 differentruns.

3.2. Spatial distribution of food resource

Three patterns of food distribution, uniform, randomand clumped, were generated to investigate their effects offood encounter rate. The distribution of food wasexpressed by the field n(i, j) which was the number offoods at the lattice site (i, j). We allowed at most one foodparticle at any lattice site so n(i, j) was a binary field, i.e.n(i, j) ¼ 0 or 1.

The occupancy of foods in each lattice was given bythresholding based on the probability C, which representsdensity:

nði; jÞ ¼1 when rand ði; jÞC;

0 otherwise;

((1)

where rand(i, j) represents the random number at the site(i, j), as generated from a uniform random functionprovided by MATLABTM (Mathworks Inc., Natick,MA). The total number of food on the lattice N0 isX

i;j

nði; jÞ ¼ N0, (2)

where the sum extends over all lattice sites. This N0 isdetermined by C.

A uniform distribution of food particles with the totalnumber N0 was created by placing food particles regularlyon the lattice space (Fig. 5(a)). Using Eqs. (1) and (2), wealso created a random distribution of food particles(Fig. 5(b)). In addition to uniform and random, a clumpedfood distribution was generated. The 200� 200 latticespace was further divided into 40� 40 subspaces. Initially,

three or four subspaces chosen by coin toss rule were filledwith higher food density using Eq. (1), then othersubspaces were filled with lower food density. This processallowed us to obtain various clumped food distributions(Fig. 5(c) and (d)). When system size was expanded, thesubspace size increased proportionally to correct for biasdue to a limited system size.

3.3. Quantification of food distribution

I-index proposed by Johnson and Zimmer (1985) wasused to confirm uniform, random or clumped fooddistribution. While the index is based on point-to-individual distances, it is a useful index for testing spatialpatterns:

I ¼ ðN0 þ 1Þ

Pi x2

i

� �2P

i x2i

� �2h i2 ,where xi is a nearest neighbor distance between specimens.The index I has an expected value of approximately 2 for arandom pattern, less than 2 for uniform pattern, andgreater than 2 for clumped patterns. Food distributionpatches with different I values are shown in Fig. 5.

4. Simulation results and discussion

Fig. 6 shows the empirical tunneling pattern (Fig. 6(a))and the simulated tunneling pattern with the branch lengthexponents a ( ¼ 0.15). A fractal dimension (Mandelbrot,1983) was calculated to geometrically compare thesimulated and empirical tunnel patterns for their complex-ity. The fractal dimension, D was derived by dividing thelattice space into squares (pixels) of side r, and then

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Fig. 5. Three food distributions distinguished by I-index [(a) uniform; (b)

random; (c) and (d) clumped]. All have 400 food particles.

S.-H. Lee et al. / Journal of Theoretical Biology 243 (2006) 493–500 497

counting the number of the pixels including objects(tunnel), N(r). By repeating this process with increasing r

values, a range of N(r) values was obtained. Doublelogarithmic plots of N(r) against r generated a linear linefrom which the slope was estimated as the fractaldimension, i.e. N(r)�rD. The fractal dimension of empiricaltunneling pattern was 1.26670.0131 (Su et al., 2004). Thisvalue was in good agreement with our simulated tunnelingpattern (D ¼ 1.21270.012).

Fig. 7(a) shows the plot of encounter rate as a functionof I values when the numbers of food particles (N0) were225, 400, 625, or 900, where a ¼ 0.15 and L ¼ 200.Encounter rate increased with larger N0 as expected,because a greater number of food particles could coverlattice space more compactly. Non-clumped distributions(uniform and random) did not affect the encounter rate.Encounter rates were maximized at I ¼ 3, suggesting thatthe termite tunneling pattern is advantageous for an

appropriately clumped food resource with high density.At the smaller particle numbers (N0) of 225 and 400, thevalue of I corresponding to maximum encounter ratewas less clear than when N0 ¼ 625 or 900. A lowerfood density attributed to dilute the food distributioneffect. For food resource with highly clumped distributions(I43), we found a slow fall-off in encounter ratereaching a stable plateau. Except for some branchespassing through the area of clumped food, most branchesrarely encountered food particles because the increaseof I value led to the enlargement of lower food densityarea. Encounter rate was higher for clumped (I ¼ 3)than for highly clumped (I44) even as the total systemsize was increased. To confirm that the maximizedencounter rate at I ¼ 3 is not artifact (i.e. a bias due tosystem size), we performed the simulation in different sizes(Fig. 7(b)). This figure indicates that the loss of efficiencyfor highly clumped distributions was not simply an artifactof placing clumps in a system size for which they weretoo large.One possible explanation for the maximum encounter

rate at I ¼ 3 is that when a primary tunnel passes throughan area of clumped food particles, the branch tunnels playa major role in the increase of the food encounter rate. Thebranch tunnel effect that contributed to the increase of theencounter rate can be maximized at clumped fooddistribution of I ¼ 3.Fig. 7(c) shows the effect of food size on encounter rate

in random food distribution. In this study, different sizefoods were created by the combination of food particlesoccupying a lattice cell. Because the distance of foodparticles from the center was randomly distributed, a largefood particle close to the origin would be encounteredimmediately and by multiple tunnels.Encounter rate was higher at food size ¼ 9 than at other

size regardless of N0. When the food size was larger than 9,encounter rate decreased with food size. This can beexplained by the fact that the geometry of the distributionof large food is analogous to a highly clumped adistribution of smaller food particles with the same N0.Therefore, the regime (food size49) can be understood bythe same way as explained in Fig. 7(a).Simulations presented in this study were performed

under the assumption of the limited termite population size(�1100). Tunnel complex will change as population sizeincreases based on other networks found in nature. Livingorganisms are sustained by the transport of materialsthrough linear networks that branch to supply all parts ofan organism. In such cases, dissipated energy to transportmaterials needs to be minimized to improve systemefficiency (West et al., 1997). In our model, the dissipatedenergy to construct tunnels was interpreted as travel costbeing defined as the number of lattice cell occupied bytunnel. If termite tunnel complexes are subject to similarpressures, then as tunnels become longer due to largerpopulation size, branch tunnels should become longer toexplore the area between diverging primary tunnels. The

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Fig. 6. Tunneling pattern of observed termite (a) and simulated termite with a ¼ 0.15 (b).

Enc

ount

er r

ate

I

0 5 10 15 20 25

Food size

1 2 3 4 5 6

I

1 2 3 4 5 6

N0=225N0=400N0=625N0=900

(a) (b)

(c)

�=0.15, L=200 �=0.15

N0=200N0=400N0=1000

I-index=2.0�=0.15, L=200

L=100, N0=144L=200, N0=625L=300, N0=1444L=400, N0=2500

1.0

0.8

0.6

0.4

0.2

Enc

ount

er r

ate

0.9

0.6

0.3

0.0

Enc

ount

er r

ate

1.0

0.8

0.6

0.4

0.2

Fig. 7. (a) Encounter rate with respect to I values for four different N0, where a ¼ 0.15. (b) Encounter rate with respect to I values for four different system

size where a ¼ 0.15. (c) Encounter rate with respect to food size for three different N0, where I ¼ 2.0 and a ¼ 0.15.

S.-H. Lee et al. / Journal of Theoretical Biology 243 (2006) 493–500498

distribution of branch lengths, exponent a, should changeas population size changes in relation to energy cost.

In Fig. 8(a), the encounter rate increased with I valuewhen Io3. In case of a ¼ 1.0, the encounter rate was

significantly lower than a ¼ 0.15 and 0.05, while there isnot much difference between a ¼ 0.05 and 0.15. This resultpoints out the important role of branch tunnels in castingan effective search net. Long tunnels with fewer branches

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0.11 2 3 4 5 6

0.2

0.3

Enc

ount

er r

ate

I-index

N0=220

(a) (b)

0.1�

�=1.0

�=0.05�=0.15

1

0.5

0.5

Nor

mal

ized

cos

t

L=100L=200L=300L=400

Fig. 8. (a) Encounter rate against I-index for three different a (b). Normalized cost averaged over 30 runs for each branching length exponent a with four

different system sizes.

S.-H. Lee et al. / Journal of Theoretical Biology 243 (2006) 493–500 499

are much more likely to bypass a resource site in a clumpeddistribution without intercepting it than shorter, highlybranched tunnels are.

The coincidence of values for a ¼ 0.05 and 0.15 impliesthat as a decreased, the encounter rate reaches anasymptotic value and an optimized a might exist in relationto the travel cost. To confirm this, we further investigatedthe travel cost changes in response to a for different systemsizes in Fig. 8(b). The figure shows the plots of travel costnormalized by the maximum value against the branchlength exponent a where system size L ¼ 100, 200, 300, and400. The cost decreased substantially with a (ao0.15). Inthis situation, an inflection point occurred at a ¼ 0.15regardless of system size. The value of a derived directlyfrom empirical data coincided with this inflection point. Ifa40.15, the branch tunnel length is too short to the searcharea between diverging primary tunnels. If, on the otherhand, the value of a is defined by (D cos t/Da)o0, then thebranch lengths will be such that they overlap those ofadjacent primary tunnels. Thus, the value of a ¼ 0.15produces a branch length distribution for adjacent primarytunnels that allows an effective search of the area betweenthem as they diverge, while minimizing the redundancy ofoverlapping branches.

5. Conclusions

The search pattern of simulated tunnel complexesexcavated by C. formosanus is optimized to encounterresources that are clumped, as opposed to randomly oruniformly distributed. Studies on standing dead wood(snags) and fallen dead wood indicated that they were notto be found in a random or uniform distribution, but werepresent in clumped distributions on a variety of spatialscales due to substrate quality differences, windthrowevents and wildfire (Delaney et al., 1997; Harris, 1999;Pham et al., 2004). Because the tunnel patterns generatedreflect search patterns created in the absence of anyresources that might attract termite excavators or affect

tunnel propagation, we hypothesize that the characteristictunnel geometries arise from an evolved response to thedensities and distributions of dead wood in the wild.Under the same condition of N0, the food size effect is

analogous to the food distribution effect because large foodis composed of many small food particles. To understandthe effect of colony size on encounter rate in relation toenergy cost, a branch length exponent a was investigatedfor encounter rate and cost. In the graph of the relationshipbetween travel cost and branching length exponent a, aninflection point which divides a-space to two regimes wasfound. The value of a derived directly from empirical datalies near the inflection point where travel cost begins toplateau, displaying the efficiency of natural branchingpatterns.Using this model, we can extend possible application in a

number of ways. For instance, the framework in Section 3can accommodate a different algorithm for choosing thetransition probability P(i, j). This could involve modelingcombinations of stimulus gradients, such as soil hydrologyand soil particle size in the presence or absence of resource,and the manner in which their synergistic or competitiveinteractions affect tunnel search patterns. Termites mightreact to deterrent compounds by reducing the transitionalprobability of TVC advancement and thereby shortentunnels in an area treated with pesticides, which in turnincreases the volume of tunnels in other sectors. Such ashift can be simulated by our model and incorporated intotreatment schemes for a given area. The ability to modeltunnel patterns based on simulation allows for predictingtunnel morphology when direct characterization is im-possible.

Acknowledgments

We would like to thank P. Ban and R. Pepin (Univ. ofFlorida) and for technical assistance, and M.W. Crosland(Univ. of Florida) and T.-S. Chon (Pusan NationalUniversity) review of the manuscript. This research was

ARTICLE IN PRESSS.-H. Lee et al. / Journal of Theoretical Biology 243 (2006) 493–500500

supported by the Florida Agricultural Experiment Stationand a grant from USDA-ARS under the grant agreementno. 58-6435-3-0075.

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