Effects of tunnel structures of two termite species on territorial competition and territory size

    Effects of Tunnel Structures of Two Termite Species on Territorial Competition and Territory Size Wonju Jeon, Sang-Hee Lee PII: DOI: Reference:

S1226-8615(14)00006-5 doi: 10.1016/j.aspen.2014.01.004 ASPEN 485

To appear in:

Journal of Asia-Pacific Entomology

Received date: Revised date: Accepted date:

11 June 2013 20 November 2013 8 January 2014

Please cite this article as: Jeon, Wonju, Lee, Sang-Hee, Effects of Tunnel Structures of Two Termite Species on Territorial Competition and Territory Size, Journal of Asia-Pacific Entomology (2014), doi: 10.1016/j.aspen.2014.01.004

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Effects of Tunnel Structures of Two Termite Species on Territorial

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Wonju Jeon and Sang-Hee Lee*

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Competition and Territory Size

Division of Fusion Convergence of Mathematical Sciences, National Institute for

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Mathematical Sciences, Republic of Korea

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Abstract

The foraging territories of 2 subterranean termites, Coptotermes formosanus Shiraki and

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Reticulitermes flavipes (Kollar), were simulated using a model to explore how territorial

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intraspecific competition changes with 4 variables characterizing the formation of territory:

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the number of primary tunnels, N0; the branching probability, Pbranch; the number of territories, N; and the blocking probability, Pblock. The blocking probability Pblock quantitatively describes the probability that a tunnel will be terminated when another tunnel is encountered; higher Pblock values indicate more likely termination. Higher tunnel-tunnel encounters led to denser tunnel networks. We defined a territory as a convex polygon containing a tunnel pattern and explored the effects of competition among termite colonies on territory size distribution at steady state attained after sufficient simulation time. At the beginning of the simulation, N = 10, 20, …, 100 initial territory seeds were randomly distributed within a square area. In our previous study, we introduced an interference coefficient γ to characterize territorial competition. Higher γ values imply higher limitations on network growth. We theoretically derived γ as a function of Pblock and N. In this study, we considered the constants in γ as functions of N0 and Pbranch so as to quantitatively examine the effect of tunnel structure on 1

ACCEPTED MANUSCRIPT territorial competition. By applying statistical regression to the simulation data, we determined the generalized γ functions for both species. Under competitive conditions,

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territory size is most strongly affected by N0, while the outcome of territorial competition is

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most strongly affected by N, followed by Pblock and N0.

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Keywords: Termite territorial competition; Tunnel network; Territory simulation model;

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foraging efficiency

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Correspondence: Sang-Hee Lee ([email protected]; [email protected])

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Tel. +82-42-717-5736 / Fax. +82-42-717-5758

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ACCEPTED MANUSCRIPT Introduction

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Subterranean termites are colony-forming social insects whose colonies may contain

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hundreds of thousands to millions of individuals. Colony members disperse throughout the soil, constructing underground tunnel networks spanning tens to hundreds of meters for

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foraging (Su et al. 1984). The manner in which the tunnel networks grow determines how

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territories are formed under the influence of competition. Territory size and shape reflects a compromise between foraging activity and other biological and/or ecological constraints,

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such as the number of foragers, soil density, and food availability (Adams and Levings 1987; Ganeshaiah and Veena 1991; Buhl et al. 2006; Lee and Su 2011). Thus, information on

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territory distribution is important not only to comprehend the stability and functioning of the

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(Forschler 1994).

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termite ecosystem but also to determine the efficacy of control tactics for termite infestations

To date, researchers have studied the territory size and shape of subterranean termites using a variety of methods. In field studies, foraging territories have been mapped by direct excavation (Ratcliffe and Greaves 1940; King and Spink 1969), by tracing radioisotopelabeled foragers in their galleries (Li et al. 1976; Spragg and Paton 1980), and by examining whether wooden stakes driven into soil in lawns and planters were infested by the foragers (La Fage et al. 1973; Su and Scheffrahn 1986). These studies were aimed at estimating population size in a territory and developing termite control technologies. Thus, these studies did not provide territorial information pertaining to the growth and structure of tunnel networks. In our previous studies (Lee and Su 2008, 2009a,b; Lee et al. 2007a), we explored this aspect and suggested a lattice model to simulate territory growth of subterranean termites in a 3

ACCEPTED MANUSCRIPT natural landscape. In this model, a territory was defined as a convex polygon containing the entire tunnel pattern of a colony (Su and Scheffrahn 1986; Lee and Su 2006; Jeon and Lee

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2011). This model successfully explained the territory size distribution of mangrove termites

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on the Atlantic coast of Panama that had been reported by Adams and Levings (1987). However, this lattice model had a limitation in that territorial competition was simulated

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through a territory-territory bordering process. For mangrove termites, this process can be

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justified because they move from tree to tree via dense and complicated pathways provided by the prop roots of fallen trees. However, subterranean termites construct their foraging

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tunnels with tree-like branching patterns; thus, territory-territory bordering is not valid for such species. In territory-territory bordering process of the previous lattice model, the

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bordering zone between two adjacent territories consists of a series of consecutive lattice

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cells because the territories was formed by lattice cells occupied by termites. On the other

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hand, in the present model, the tunnel-tunnel encountering occurs at the intersections between two tunnels. These two different processes lead to the different competing dynamics. Tunnels of a colony can advance through the empty spaces between tunnels radiating from the nests of other colonies, implying that territories can overlap. Therefore, competition should be considered a tunnel-tunnel encountering process (Jost et al., 2012). To incorporate this phenomenon into a revised model and better understand territorial competition, we developed a new “continuous territory model” based on the tunnel pattern and introduced the interference coefficient, γ, as a function of the number of territories N and the blocking probability Pblock

(Jeon and Lee 2011).

The interference coefficient is defined as in Eq. 1 from Jeon and Lee (2011).

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ACCEPTED MANUSCRIPT ∑ (A N

γ=

n =1

Pblock = 0 n N

− AnPblock

) (1)

Pblock =0 n

∑A

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n =1

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where N is the total number of territories. AnPblock represents the nth territory size for a given Pblock value. By calculating differences between the territory size for Pblock = 0 and Pblock

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≠ 0, the effect of Pblock on the territory size distribution was quantified. The denominator was

among territories in a given space.

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introduced to normalize the γ value. Consequently, γ captures the degree of interactions

The blocking probability Pblock quantitatively describes the probability that a tunnel will

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be terminated when another tunnel is encountered; higher Pblock values indicate greater

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probability of termination. The formula is as follows.

N Pblock a( N ) Pblock γ ( N , Pblock ) = = N +c Pblock + b( N ) P + d block N +c

(2)

In the above formula, c and d were considered constant when characterizing territorial competition. However, these 2 values may vary according to changes in environmental conditions, such as heterogeneity of soil type, because the conditions play a significant role in territory shape and size (Lee et al. 2008a; Su and Lee 2009). Changes in the tunnel pattern are likely to cause some degree of territorial competition. The major difference between the territory-territory bordering and tunnel-tunnel encountering 5

ACCEPTED MANUSCRIPT processes is in the compactness of tunnels. When the tunnels are very compact, territorial competition follows the territory-territory bordering process, while in the case of lower tunnel

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compactness, competition is expected to be of the tunnel-tunnel encountering type. The

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number of primary tunnels and the branching probability are closely related to tunnel compactness. Ecologically, these 2 variables also play important roles in controlling foraging

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strategy to increase foraging efficiency (Lee and Su 2008; Lee et al. 2009c). For this reason,

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as the objective of this study, we formulated c and d as functions of the number of primary tunnels (N0) and the branching probability (Pbranch), which together represent how densely the

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territory is filled with primary and secondary tunnels. In addition, we carried out a sensitivity analysis to quantify the effect size of each variable on the γ function and briefly discuss

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Model description

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termite foraging efficiency in relation to territorial competition among neighboring territories.

Tunnel patterns of foraging termites were simulated within a continuous two-dimensional area. The simulated tunnels were classified into primary and secondary tunnels; those originating from the nest were classified as primary, while those branching from the primary tunnel were classified as secondary (Selkirk 1982). Tertiary and quaternary tunnels were excluded because they were rarely formed during the test period (Su et al. 2004). In this study, based on the study of Su et al. (2004) and the Puche and Su (2001), we only considered the primary and secondary tunnels. In the statistical viewpoint, the simulated tunnel patterns from our model are same with those obtained from the consideration of tertiary and quaternary tunnels because the value of the thirdly branching probability is negligible. As well, the

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ACCEPTED MANUSCRIPT exclusion of the branching of tertiary and quaternary tunnels decreased the computational burden.

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Simulated tunnel patterns were constructed for 2 species, C. formosanus and R. flavipes.

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For all tunneling activities, the model presented in this study is based on probabilistic decisions. The simulation model has 2 main procedures. One describes tunnel network

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growth and the other represents territorial competition based on tunnel-tunnel interactions

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(see Appendix).

In the model, 8 variables are used to simulate the tunnel network pattern. The variables

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are as follows: (1) number of primary tunnels, (2) linear length of primary tunnel segments, (3) turning angle of each linear segment of the primary tunnel, (4) branching angles of

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secondary tunnels, (5) probability of branching per linear segment (defined as the linear line

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connecting two closest points of a tunnel that did not deviate off the tunnel path) of a primary

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tunnel, (6) termination probability for primary tunnels, (7) termination probability for secondary tunnels, and (8) probability of continuing through an intersection of 2 tunnels. The values for these 8 variables for the 2 species simulated were obtained from Su et al. (2004) (Table 1). A single termite tunnel pattern was generated by the addition of linear segments with lengths and turning angles statistically obtained from the empirical tunnel patterns. The values for the variables determined whether a tunnel terminated, branched, or extended beyond any intersecting tunnels. To simulate territorial competition, 2 additional variables were used: number of colonies (N) and blocking probability (Pblock). At the beginning of the simulation, N (= 10, 20, …, 100) territory seeds were used as initial pairs and uniform randomly distributed in a given L × Lsized area, where L = 56cm. The blocking probability was introduced to describe the extent to which an advancing tunnel stops when it encounters another tunnel such that higher Pblock 7

ACCEPTED MANUSCRIPT values indicate that it is easier for a tunnel to stop advancing. According to Lee et al. (2008b), Pblock should be a function of the intersection angle between two encountering tunnels and

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tunnel widths. However, for the purpose of the model simplification, we merely described the

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encounter phenomenon as the blocking probability.

When the growth of all of the tunnels was stopped by termination of the primary and

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branching tunnels, the simulation ended. Each territory was defined as a convex polygon

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containing its corresponding tunnel pattern (Fig. 1). When the territories grown from the

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seeds located near the space boundary can exceed the space with L×L area, we considered the exceeded area in the analysis of the simulation. In this study, the simulation results were

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Results

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statistically averaged over 200 runs.

The tunnel pattern of a colony territory is directly related to the degree of interference between territories, because the 2 variables that characterize the tunnel pattern, the number of primary tunnels N0 and the branching probability Pbranch, determine the degree of interaction between territories. When N0 and Pbranch increased, the tunnel pattern became more compact. Figure 2 shows the simulated tunnel patterns of C. formosanus. The tunnel pattern is sparser when N0 = 3 and Pbranch = 0, whereas the network is densest for higher values, such as N0 = 11 and Pbranch = 0.2. Denser tunnel networks are more likely to increase tunnel-tunnel encounters between different colonies, which results in an increase in the interference coefficient characterizing territorial competition. Figure 3 shows territory size distributions for C. formosanus for N = 60 and Pblock = 0.5 in size-descending order by changing N0 and Pbranch in the domain of 3≤N0≤11 and 0.0≤ Pbranch 8

ACCEPTED MANUSCRIPT ≤0.2. Comparing the decreasing tendencies of Fig. 3 (a) and Fig. 3 (b), the territory size distribution is much more influenced by the number of primary tunnels than the branching

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probability.

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To quantify the degree of territory interaction, the interference coefficient γ was calculated for various values of N0 and Pbranch. Figure 4 (a) shows γ vs. Pblock with Pbranch = 0.0, 0.05, 0.1,

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…, 0.20 and N0 held constant at 7. γ values were close together, indicating that the influence

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of Pbranch was relatively small. Figure 4 (b) shows γ vs. Pblock with N0 = 3, 5, …, 11 and Pbranch held constant at 0.1. γ values were farther apart in Figure 4 (b) than Figure 4 (a), indicating

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that variations in Pbranch have less influence on territorial interaction than variations in N0; thus, N0 is a significant factor in territorial competition.

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Considering the effects of N0 and Pbranch on γ values in Eq. (2), we generalized the form of

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Eq. (2) to include N0 and Pbranch as follows.

N Pblock N + f ( N 0 , Pbranch ) γ ( N 0 , Pbranch , N , Pblock ) = g ( N 0 , Pbranch ) Pblock + N + f ( N 0 , Pbranch )

(3)

We calculated γ values for C. formosanus in territory simulations in which N0 = 3, 5, 7, 9, and 11 and Pbranch = 0.00, 0.05, 0.10, 0.15, and 0.20; and for R. flavipes in which N0 = 4, 6, 8, 10, and 12 and Pbranch was as above (Table 2). As shown in Table 2, the degree of interference between territories increased as each tunneling parameter increased. We recast Eq. (3) as following form by inversion and performed the regression for the acquisition of more concise formula.

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(4)

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( N + f ( N 0 , Pbranch )) Pblock + g ( N 0 , Pbranch ) 1 = γ ( N , Pblock ; N 0 , Pbranch ) NPblock

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To determine the 2 unknown functions f(N0, Pbranch) and g(N0, Pbranch), we used a set of 2

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equations for arbitrarily selected N = 60 and Pblock = 0.5 and 1.0. Using the surface-fitting toolbox of MATLAB Version 7.10 (2010a), we approximated a linear combination of a first-

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order polynomial and a rational function.

for C. formosanus

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1.43N 0 + 10.0 1  2  γ ( P = 0.5) = N + 0.73 − 1.28Pbranch (r = 0.9906)  block 0  1 . 44 N 0 + 5.43 1  = − 0.68Pbranch (r 2 = 0.9906) N 0 + 0.73  γ ( Pblock = 1.0)

(5) for R. flavipes

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1.64 N 0 + 6.73 1  2  γ ( P = 0.5) = N + 0.06 − 0.98Pbranch (r = 0.9834)  block 0  1 . 48 N 0 + 3.65 1  = − 0.29 Pbranch (r 2 = 0.9921)  γ ( Pblock = 1.0) N 0 + 0.06

From Eqs. (4) and (5), we obtained the following set of equations.

 (60 + f ( N 0 , Pbranch )) ⋅ 0.5 + g ( N 0 , Pbranch ) 1.43N 0 + 10.0 − 1.28Pbranch =  60 ⋅ 0.5 N 0 + 0.73  for C. formosanus   (60 + f ( N 0 , Pbranch )) ⋅ 1.0 + g ( N 0 , Pbranch ) = 1.44 N 0 + 5.43 − 0.68P branch  60 ⋅ 1.0 N 0 + 0.73   (60 + f ( N 0 , Pbranch )) ⋅ 0.5 + g ( N 0 , Pbranch ) 1.64 N 0 + 6.73 = − 0.98Pbranch  60 ⋅ 0.5 N 0 + 0.06  for R. flavipes   (60 + f ( N 0 , Pbranch )) ⋅ 1.0 + g ( N 0 , Pbranch ) = 1.48 N 0 + 3.65 − 0.29 P branch  60 ⋅ 1.0 N 0 + 0.06 

(6) By solving the above equations, we determined the functions for each species as follows. 10

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27.3N 0 + 7.69   f ( N 0 , Pbranch ) = N + 0.73 − 4.68Pbranch  0 for C. formosanus  0 . 66 N 274 − + 0 g(N , P − 35.9 Pbranch 0 branch ) =  N 0 + 0.73

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19.1N 0 + 28.5   f ( N 0 , Pbranch ) = N − 0.06 + 23.7 Pbranch  0 for R. flavipes  9.48 N 0 + 183 g(N , P − 41.2 Pbranch 0 branch ) =  N 0 − 0.06 (7)

Eqs. (7) for the 2 species as below.

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NPblock    274 − 0.66 N 0  27.3 N 0 + 7.69  N + − 4.68Pbranch  Pblock +  − 35.9 Pbranch  N 0 + 0.73   N 0 + 0.73  

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γ ( N , Pblock ; N 0 , Pbranch ) =

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For C. formosanus,

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The γ function taking into account N, Pblock, N0, and Pbranch can be determined based on

For R. flavipes,

γ ( N , Pblock ; N 0 , Pbranch ) =

NPblock    183 + 9.48 N 0  19.1N 0 + 28.5  N + + 23.7 Pbranch  Pblock +  − 41.2 Pbranch  N 0 − 0.06    N 0 − 0.06 

(8) To verify the accuracy of the final formulas, we statistically compared the γ values obtained from the model simulation and the γ values calculated using the formula by calculating the goodness of fit (Table 4). When N = 20, the goodness of fit was relatively low, because 20 territories did not sufficiently fill the allotted space and thus competition among the territories was weak. For larger values of N, the generalized formula had a good fit to the γ values obtained from model simulation. 11

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Discussion and conclusions

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In this study, we mathematically generalized the γ formula, characterizing territorial

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competition as a function of the number of territories, N; the number of primary tunnels, N0; the tunnel branching probability, Pbranch; and the blocking probability, Pblock. The formula was

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verified by comparing the simulation results obtained from a two-dimensional territory

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competition model to predictions obtained from the formula. This formula can be used to explore termite foraging strategies to optimize foraging efficiency under territorial competition, because the formula includes the key variables that play an important role in

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tunnel and territory formation.

An approach to understanding the foraging strategy is to explore the degree of the

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influence among the variables. Thus, with the generalized γ formula, we investigated which variables most strongly affected territorial competition. The level of influence is important because the stability of a termite colony is strongly affected by the degree of territorial competition, which determines “winner” and “loser” territories (Lee and Su 2008). We performed a sensitivity analysis to determine variations in γ with each variable (Cacuci et al. 2005; Grievank and Walther 2008). The average local sensitivity is calculated in discrete form as shown below.

I =11 J =11 K =11R =11

∑∑ ∑∑

SX =

i =1 j =1 k =1 r =1

 ∂   ∂X γ ( N , Pblock ; N 0 , Pbranch ) ⋅ X (i) N ( i ), Pblock ( j ), N 0 ( k ), Pbranch ( r ) I ⋅J ⋅K ⋅R (9)

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ACCEPTED MANUSCRIPT where X = N, Pblock, N0, or Pbranch; N(i) = 60(0.5 + 0.1(i – 1)); Pblock(j) = 0.5(0.5 + 0.1(j – 1)) for (i = 1, 2, …, I; j = 1, 2, …, J). For C. formosanus, N0(k) = 7(0.5 + 0.1(k – 1)) and

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Pbranch(r) = 0.1(0.5 + 0.1(r − 1)) for k = 1, 2, …, K; r =1, 2, …, R. For R. flavipes, N0(k) =

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12(0.5 + 0.1(k – 1)) and Pbranch(r) = 0.2(0.5 + 0.1(r − 1)) for k = 1, 2, …, K; r = 1, 2, …, R. Figure 5 shows the average sensitivity for the 4 variables. N is the most important factor

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in determining the level of territorial competition, followed by Pblock and N0. Pbranch had the

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smallest effect on γ among the variables. In our previous study (Lee et al. 2007b), Pbranch was an important factor to increase the foraging efficiency of a single colony. However, under

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territorial competition conditions, Pbranch is less important. This reflects that the foraging strategy is strongly subject to the degree of territorial competition. In other words, the

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branching tunnels play more important role in non-competitive condition than highly

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competitive condition. Based on the sensitivity analysis, we can speculate that termites have

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three types of strategies according to the time from the settlement of the founding pairs to territorial competition. One is the manipulation of the number of primary tunnels, N0, which is the strategy for quick occupation of space in the early stage. The second is the controlling the branching probability, Pbranch. This is a kind of defense strategy. The branching tunnels, formed between two primary tunnels, of a colony can play a role in stopping the advancing tunnels from other colonies towards the empty space between the primary tunnels. This strategy could be effective in the middle stage. The third is to regulate the blocking probability, Pblock. This seems to be a sort of aggressive strategy. What a colony takes a low value of Pblock means that the tunnels of the colony strongly invade other colony territory. To be winner territory through the competition, termites should be optimally driven by a combination or mix of the three strategies according to the stage of the territory growth. The combination is more likely to be different according to the complicated environmental factors, 13

ACCEPTED MANUSCRIPT such as heterogeneous landscape. In addition, by combining the size distribution information for the combination and the food encounter rate for the randomly distributed food resources,

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we may conjecture which foraging strategy is optimal over the long term. Further

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investigation would be interesting and required to understand the combinational mechanism. In this study, the simulation model has a conceptual problem related to the tunnel-tunnel

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encountering process. Termite tunnel networks represent 3-dimensional structures, so that

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tunnels meeting other tunnels below the ground are likely to be rare when considering the narrow tunnel width (~3 mm). Nevertheless, in the field, we can observe that when 2

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territories meet, their growth tends to slow down and stop (Messenger and Su 2005), as though tunnel-tunnel interaction occur. This could be understood by the study of Evans et al

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(2009). The authors reported that termites communicate vibro-acoustically and, as these

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signals can travel over long distances, they are vulnerable to eavesdropping.

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Our simulation study was based on experimental data obtained from homogeneous sand substrates (Su et al. 2004). However, under field conditions, the γ function is likely to be affected by heterogeneity due to numerous abiotic factors such as moisture, soil particle size, temperature, and seasonal cycles (Messenger and Su 2005). Nevertheless, this study is valuable in that the proposed formula enables prediction of territory size distributions resulting from territorial competition.

Acknowledgment This research was supported by the National Institute for Mathematical Sciences (NIMS).

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Cambridge University Press, New York Spragg, W.T., Paton, R., 1980. Tracing, trophallaxis and population measurement of colonies of subterranean termites (Isoptera) using a radioactive tracer. Ann. Entomol. Soc. Amer. 73, 708-714

Su, N.Y., Lee, S.H., 2009. Tunnel volume regulation and group size of subterranean termites (Isoptera: Rhinotermitidae). Ann. Entomol. Soc. Amer. 102, 1158-1164 Su, N.Y., Scheffrahn, R.H., 1986. A method to access, trap, and monitor field populations of the Formosan subterranean termite (Isoptera: Rhinotermitidae) in the urban environment. Sociobiology 12, 299-304 Su, N.Y., Tamashiro, M., Yates, J.R., Havety, M.I., 1984. Foraging behavior of the Formosan subterranean termite (Isoptera: Rhinotermitidae). Environ. Entomol. 13, 1466-1470 Su, N.Y., Stith, B.M., Puche, H., Bardunias, P., 2004. Characterization of tunneling geometry 17

ACCEPTED MANUSCRIPT of subterranean termites (Isoptera: Rhinotermitidae) by computer simulation. Sociobiology 44, 471-483

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Figure captions

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Figure 1

Simulated tunnel networks (solid lines) and territory borders (dashed lines) of single-colony

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termites for Coptotermes formosanus (left) and Reticulitermes flavipes (right). The filled

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circles represent the nests of each colony.

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Figure 2

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Typical tunnel networks for C. formosanus for various N0 and Pbranch.

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Figure 3

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Territory size distributions for C. formosanus: (a) N = 60, Pblock = 0.5, N0 = 7, Pbranch = 0, 0.05, …, 0.2; and (b) N = 60, Pblock = 0.5, Pbranch = 0.1, and N0 = 3, 6, …, 11. An (y-axis) indicates the average territory size and n (x-axis) is the rank of the size in descending order.

Figure 4

Interference coefficient for C. formosanus: (a) N0 = 7 with various Pbranch values, and (b) Pbranch = 0.1 with various N0 values.

Figure 5 Sensitivity (S) of γ to 4 variables: number of territories (N), blocking probability (Pblock), number of primary tunnels (N0), and branching probability (Pbranch). Error bars indicate the standard deviations. 18

ACCEPTED MANUSCRIPT Appendix

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Pseudo-code for termite territory simulation model

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Main Program Parameters:

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Total number of primary and secondary tunnels (Tmax), Number of Primary Tunnels from

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the Nest (jmax), Length of Tunnel Segment,

Turning angle, Branching Angle, Termination Probability of Primary Tunneling

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(P_primary_term), Total Discrete Time Step (kmax), Number of Colony (imax), Blocking Probability (P_block), Branching Probability (P_branching), Termination Probability of

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Branching (P_branch_term)

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while k=1,kmax (discrete time loop) % until all tunnels in all colonies stop growing do i=1,imax (colony loop) do j=1,jmax (tunnel loop)

Call random number r1, r2

if j=P_primary_term, then make a new tunnel with the Turning angle call subroutine INTERSECTION if r2
ACCEPTED MANUSCRIPT endif elseif j>jmax

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if r1>P_branch_term, then make a new tunnel with the Turning angle

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call INTERSECTION

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endif

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end do (tunnel loop) end do (colony loop)

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end while (discrete time loop)

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descending order

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make a convex polygon for each colony, calculate the territory area, and order it in

End Program

Subroutine INTERSECTION call random number r3

if the new tunnel intersects with an existing tunnel & r3
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Figure 1

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Figure 2

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Figure 3a

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Figure 3b

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Figure 4a

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Figure 4b

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Figure 5

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ACCEPTED MANUSCRIPT Table 1 The values (mean ± SD) of the variables used to generate tunnel network patterns in the simulation

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model.

Tunneling Parameter

R. flavipes

6.78 ± 1.01

11.6 ± 0.69

1.03 ± 0.04

0.87 ± 0.02

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Number of primary tunnels per arena

C. formosanus

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Length of a linear segment (cm) Turn angle of the linear segment (deg)

19.69 ± 0.68

22.72 ± 0.36

51.14 ± 2.83

60.55 ± 1.87

10.52 ± 0.97

19.42 ± 3.46

18.68 ± 1.33

14.73 ± 1.88

Termination probability for a secondary tunnel (%)

41.17 ± 3.94

45.55 ± 5.01

Probability of continuing through an intersection (%)

50.0 ± 16.69

42.71 ± 6.15

Probability of branching (deg)

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Termination probability for a primary tunnel (%)

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Branching angle (deg)

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γ for C. formosanus Pbranch=0.00

0.05

0.10

0.15

0.20

3

0.253

0.257

0.278

0.276

0.283

5

0.335

0.332

0.359

0.364

0.369

7

0.387

0.395

0.414

0.419

0.421

9

0.437

0.440

0.448

0.456

0.462

11

0.465

0.473

0.479

0.492

0.498

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N0

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of primary tunnels (N0) and branching probabilities (Pbranch).

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Table 2 γ values for Coptotermes formosanus and Reticulitermes flavipes for various numbers

N0

Pbranch=0.00

0.05

0.10

0.15

0.20

4

0.295

0.310

0.306

0.330

0.330

6

0.371

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γ for R. flavipes

0.389

0.393

0.401

8

0.408

0.430

0.440

0.436

0.437

10

0.447

0.455

0.456

0.474

0.474

12

0.470

0.475

0.485

0.492

0.499

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0.376

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ACCEPTED MANUSCRIPT Table 3 Relative error of γ values for Coptotermes formosanus (top) and Reticulitermes

N0=7

N0=11

N=40

N=60

N=80

N=100

N=20

N=40

N=60

N=80

N=100

N=20

N=40

N=60

N=80

N=100

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.2

2.8

0.5

0.6

3.1

2.9

12.9

0.8

1.0

4.5

4.8

1.5

0.7

0.4

0.5

0.4

7.2

3.2

2.6

2.8

0.9

0.1

4.1

1.7

2.9

3.0

5.2

1.2

0.2

0.2

1.0

0.6

5.0

1.4

0.4

3.7

1.6

8.8

0.2

1.2

2.0

2.6

1.6

0.5

0.1

0.4

0.3

0.8

0.9

2.7

3.0

0.3

1.3

6.1

1.0

6.5

1.6

4.4

0.5

2.3

1.1

2.2

1.6

1.9

1.5

2.0

5.9

1.4

0.2

0.7

0.0

0.1

1.0

2.1

1.2

4.9

0.3

0.9

0.5

0.3

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N0=8

Pblock

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N=20

0.0

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N0=3

Pblock

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flavipes (bottom). The dimension of the error is “%”.

N0=12

N0=16

N=40

N=60

N=80

N=100

N=20

N=40

N=60

N=80

N=100

N=20

N=40

N=60

N=80

N=100

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.2

7.5

1.1

1.5

1.0

1.3

13.9

0.3

1.8

2.3

2.2

3.1

4.6

1.9

3.1

3.0

0.4

3.2

0.0

0.8

0.4

1.0

8.6

1.1

1.6

1.2

1.5

1.4

3.9

2.1

1.7

1.6

0.6

2.4

1.7

0.3

0.1

1.0

3.5

0.5

0.4

1.2

0.8

0.4

0.6

0.6

0.3

0.8

0.8

5.7

0.9

0.1

0.0

1.1

6.6

0.4

0.6

0.2

0.7

3.1

0.5

0.8

0.2

0.0

1.0

3.0

1.7

0.3

0.3

0.7

3.8

1.3

0.3

0.1

0.0

1.2

1.2

1.1

0.9

0.6

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0.0

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ACCEPTED MANUSCRIPT Table 4 Goodness of the fitting for γ values against the values of Pblock for Coptotermes

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formosanus and Reticulitermes flavipes.

N0=16

0.9809

0.9980

0.9995

0.9974

0.9995

0.9990

0.9999

0.9993

0.9989

0.9994

0.9993

0.9990

For C. formosanus

For R. flavipes

N0=7

N0=11

N0=8

20

0.9900

0.9863

0.278

0.9916

40

0.9982

0.9985

0.359

0.9991

60

0.9957

0.9989

0.414

0.9998

80

0.9998

0.9972

0.448

100

0.9984

0.9979

0.479

N0=12

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Graphical abstract

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ACCEPTED MANUSCRIPT Highlights A formula for the degree of territorial competition was developed.

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Denser tunnels increase tunnel-tunnel encounters, increasing territorial competition.

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Territory size is most strongly affected by the number of initial tunnels.

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Territorial competition is the most strongly affected by the number of territories.

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