Food encounter rate of simulated termite tunnels in heterogeneous landscapes

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1 BioSystems 90 (2007) Food encounter rate of simulated termite tunnels in heterogeneous landscapes S.-H. Lee, N.-Y. Su, P. Bardunias, H.-F. Li Department of Entomology and Nematology, Ft. Lauderdale Research and Education Center, University of Florida, Ft. Lauderdale, FL 33314, United States Received 2 June 2006; received in revised form 15 September 2006; accepted 16 September 2006 Abstract The aim of this study was to explore how a heterogeneous landscape affects food encounter rate in the Formosan subterranean termite, Coptotermes formosanus Shiraki. To do this, a lattice model was formulated to simulate the tunneling structure of the termite. The model made use of minimized local rules derived from empirical data. In addition, a landscape structure was generated on a lattice space by using a neutral landscape model. Each lattice cell has a value h, representing spatially distributed property of the landscape (e.g., temperature or moisture). The heterogeneity of the landscape was characterized by a parameter, H controlling aggregation of lattice cells with higher values of h. Higher H values correspond to higher aggregation levels. The effect of the landscape heterogeneity on the encounter rate was clear in the presence of higher food density than in lower density. The effect was also enhanced by the increase of the number of primary tunnels Elsevier Ireland Ltd. All rights reserved. Keywords: Food encounter rate; Neural landscape model; Termite; Lattice model; Heterogeneous landscape 1. Introduction Landscape structure produces environmental or spatial heterogeneity (Rohani et al., 1997), which may act as a template in generating complexity at other levels of ecological organization (e.g., Bonabeau, 1998). Theory (De Roos et al., 1991; Wiens et al., 1993) and empirical observations (Levin et al., 1971; Kareiva, 1985; Crist et al., 1992; Cartar and Real, 1997) indicate that the foraging behavior of an animal is influenced by the heterogeneity of the landscapes they occupy. The influence is closely connected with animal extinction or colonization (Gustafson and Gardner, 1996; King and With, 2002), and an understanding of how the landscape heterogene- Corresponding author. Tel.: ; fax: address: sunchaos@pusan.ac.kr (S.-H. Lee). ity affects foraging behavior is an essential component in developing a mechanistic foundation for landscape ecology (Merriam, 1988; Ims, 1995; Wiens, 1995). The influence of landscape heterogeneity on foraging behavior has been investigated (King and With, 2002; McIntyre and Wiens, 1999; Crist et al., 1992; Szacki and Liro, 1991; Etzenhouser et al., 1998; Johnson et al., 1992), but most of these studies have been limited to terrestrial animals because little is known about the behavior of below-ground foragers such as subterranean termites (Capowiez et al., 1998; Su, 2001). Although subterranean termite foraging behavior has been studied through either direct gallery excavation (Greaves, 1962; King and Spink, 1969; Ratcliffe and Greaves, 1940) or studies on baited plots by using spatio-temporal and mark-release-recapture methods (Grace et al., 1989; LaFage et al., 1973; Su et al., 1984), these methods offered no direct information on foraging behavior in /$ see front matter 2006 Elsevier Ireland Ltd. All rights reserved. doi: /j.biosystems

2 relation to landscape structure. Our limited understanding of termite social interaction and how the efforts of individual subterranean termites are coordinated during excavation further obscures the issue. In the present study, we dealt with the Formosan subterranean termite, Coptotermes formosanus Shiraki and carried out an experiment to observe its foraging galleries in a two-dimensional arena. Based on the data obtained from the experiment, we formulated a lattice model mimicking termite foraging pattern, that is, tunneling structure on the heterogeneous landscape. Using the model, we explored the effect of the heterogeneous landscape on termite foraging pattern in relation to food encounter rate. 2. Empirical data S.-H. Lee et al. / BioSystems 90 (2007) The empirical data on termite tunnel geometry were obtained from the two-dimensional foraging arena study as reported by Su et al. (2004). The arena was composed of two sheets of transparent Plexiglas (105 cm 105 cm and 0.6 cm thick) separated from each other by four Plexiglas laminates (105 cm 2.5 cm and 0.2 cm thick) placed between the outer margins. The 0.2 cm gaps Fig. 1. A TVC by a cadre of termite workers. Fig. 2. Schematic representation of the rules defining TVCs movement constructing primary tunnels with a transition probability from one position to its nearest neighboring position.

3 316 S.-H. Lee et al. / BioSystems 90 (2007) Fig. 3. 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 θ 0. between the Plexiglas sheets were filled with moistened sand, into which 1000 workers (plus 100 soldiers) of C. formosanus were introduced. The arena was placed in the horizontal position in a dark room at 25 ± 2 C. Digital images of the arena were taken daily to record tunnel development until one tunnel reached the arena edge. The experiments were replicated by using nine colonies. 3. Model description of continuing through an intersection used in this model was obtained from the empirical data of the 2D arena as described by Su et al. (2004). Each lattice cell can either be empty or occupied by exactly one TVC. Each TVC can advance only one lattice per time-step. In the present study, tunnels were designated as two classes: primary or secondary. Tunnels originating from the release position were classified as primary. Tunnels branching 3.1. Termite tunnel We used the stochastic lattice model to describe termite tunneling pattern under a closed boundary condition. A tunnel was mimicked by the movement of a tunnel vector cell (TVC) with transition probabilities from one site to its nearest neighboring site. A TVC in reality is a discrete unit of tunnel excavated by a cadre of termite workers (Fig. 1). In the figure, the dotted box indicates the TVC. This model was defined on the square lattice of L L sites (L = 300). In this model, the movement trace of TVCs corresponded to the tunneling pattern. Other parameters such as average branching angle, probability of branching, branching length exponent and probability Fig. 4. Normalized frequency distribution of secondary tunnel length.

4 S.-H. Lee et al. / BioSystems 90 (2007) from the primary tunnel were classified as secondary (Selkirk, 1982). The tertiary and quaternary tunnels were excluded because they rarely occurred under the test period. Initially, TVCs were positioned on the center cell of the lattice space. The TVCs advanced to form primary tunnels. Each TVCs advances to the preferential direction with no back step because termite tunnels did not loop back to the origin released (Su et al., 2004). We set the preferential direction as outward direction from the release position. We applied the conservation law of probability P(i, j) to the TVC existing on a site (i, j). We denoted the transition probabilities from site (i, j) to the nearest neighbors as P up (i, j + 1); P down (i, j 1); P left (i 1, j); P right (i +1, j) (Fig. 2). In order to determine the probabilities, we extracted primary tunnel shapes from empirical images and then performed linear fitting as shown in Fig. 3(a). From nine empirical tunneling patterns, the angle between primary tunnels ( φ) and the number of primary tunnels (N) were determined as 53.1 ± 8.5 and 6.78 ± 1.01, respectively. The data indicated the lack of spatial correlation among tunnels, namely, angle interval between tunnels was nearly equal thus, φ = 360 /N and the directional angle of TVCs is determined as θ k = k φ, where the subscript k was a label assigned to primary tunnel ranging from 1 to N. Based on the directional angle θ k, the transition probabilities were determined: P up = P down = c sin(θ k ) and P left = P right = c cos(θ k ), where c is a proportional constant. This form can be changed under the condition, c sin(θ k ) + c cos(θ k ) = 1 as follows: P up = P down = tan(θ k ) /(1 + tan(θ k ) ) and P left = P right = 1/(1 + tan(θ k ) ). To construct secondary branches of tunnels, the probability was generated by random process at a TVC s position. When the value of probability was larger than 0.9, the TVC site was chosen as the branching node where a new TVC was introduced (Su et al., 2004). The new TVCs at branching node begin advancing with branching angle, θ 0 =50 as reported previously (Fig. 3(b)) (Su et al., 2004). To describe the movement of new TVCs, a new transition probability P (i, j) was introduced: P up = P down = tan(θ k ± θ 0 ) /(1 + tan(qk ± q0) ) and P left = P right = 1/(1 + tan(θ k ± θ 0 ) ). Upper- or lower-side branching was determined by coin toss rule. The sign ± indicates upper or lower branch, respectively. The length of branch tunnels L was characterized by the frequency distribution of branch length P(L) exp( αl) with a branch length exponent, α = 0.15, which was obtained from nine empirical images (Fig. 4). Fig. 5. All possible configurations that up TVC may encounter.

5 318 S.-H. Lee et al. / BioSystems 90 (2007) Fig. 5 shows all the possible configurations that the up TVC may encounter. If more than one TVC shares the same target lattice cell, one is chosen randomly, with equal probability. This TVC moves while its rivals for the same target remain their position (Fig. 5(a) (c)). When up TVC encounters other tunnel, passing or stopping movement was also chosen randomly, with equal probability which was based on empirical data (Fig. 5(d) (f)) (Su et al., 2004). In the similar way, we can define the similar treatment for the possible configurations that the down/left/right walkers, may encounter. In our model, when any TVC reaches boundary wall, the simulation is terminated Food resource The distribution of food was expressed by the field n(i, j) which was the number of foods at the lattice site (i, j). We allowed at most one food at any lattice site so n(i, j) was a binary field, i.e. n(i, j) = 0 or 1. The occupancy of the food in each lattice was given by thresholding based on the probability F, which represents density: { 1 when rand(i, j) <F n(i, j) = 0 otherwise where rand(i, j) represents the randomly generated number at the site (i, j). Here, F denotes the food density. In the present study, the results of simulations were statistically averaged over 300 runs Heterogeneous landscapes We used the neutral landscape model (Gardner et al., 1987; With, 1997) to create spatially complex patterns on lattice space (Fig. 6). In the figure, each lattice cell has a value representing the spatially distributed property of the landscape (e.g., soil particle size, temperature, and moisture). As details of complex effects of the properties Fig. 6. Neutral landscape model. Heterogeneous landscapes vary in the aggregation control parameter, H, which produces a gradient of fragmentation.

6 S.-H. Lee et al. / BioSystems 90 (2007) on termite foraging behavior is not known, the environmental properties were, for simplicity, represented as a height, h, ranging from 0 to 1. The high value of h represents a bad environmental condition. Here, an internal parameter, H, controlling aggregation of lattice cells with higher value of h, was assigned from 0.0 to 1.0. Higher H values correspond to higher aggregation levels. In order to represent interaction between TVC and landscape, the probability determining whether the TVC advances or not arises from random process at their position at each time step. When the value of the probability is larger than that of the landscape height, h, the TVC can advance according to transition rule mentioned in Section Simulation results and discussions Fig. 7 shows typical termite tunneling patterns for the four different values of H = 0.0, 0.4, 0.7, and 1.0 where the number of primary tunnels, N = 6. In the figure, landscapes were displayed using the pixels that have gray color to represent the value of h. The probability of a TVC advancing through a cell in each time step is reduced as h increases (bright color), while the probability becomes more likely with a lower h (dark color) until at h = 0 there is no impediment to TVC advancement. The variation in response to h represents a differential in the labor allotted to tunnel excavation based on the environmental conditions encountered. In case of H = 0.0, the lengths of all primary tunnels constructed by TVCs were similar because cells with higher h were strongly fragmented, leading to a more uniform distribution (Fig. 7(a)). As H increased, the cells were more likely to form clusters of high or low h. When H = 0.4, several primary tunnel lengths became shortened due to aggregations of cells of high h acting to reduce the probability of their advancement (Fig. 7(b)). As the primary tunnel length became shorter, the tunnel area, defined as the number of cells occupied by the Fig. 7. Typical termite tunneling patterns for four different values of H, H=0.0, 0.4, 0.7, and 1.0 where the number of primary tunnel N =6.

7 320 S.-H. Lee et al. / BioSystems 90 (2007) Fig. 8. The relation between encounter rate ε and the heterogeneity of landscape H for different food densities F and the number of primary tunnels N. total complex of primary tunnels in an arena, became smaller. At H = 0.7, the combination of the size and distribution of clusters makes it likely that many primary tunnels will encounter areas of high h and be foreshortened, minimizing tunnel area at this value (Fig. 7(c), dotted circles indicate cell clusters with higher (h > 0.7)). When the degree of aggregation H was maximized (Fig. 7(d)), the larger, but less frequent clusters made it likely that some primary tunnels do not encounter zones of reduced tunnel propagation. These few TVCs that construct longer primary tunnels contributed to the increase of tunnel area in comparison with that of Fig. 7(c). Fig. 8 shows the relationship between the encounter rate ε and the landscape heterogeneity H for different food density F and the number of primary tunnel N. Regardless of H and N, when the value of F was low (F = and 0.002), the encounter rate ε was much less influenced by H. IfF is low, even though tunnel area significantly varies in different H, the probability that TVCs will encounter sparse food particles is little changed. On the other hand, in case of the high value of F (F > 0.003), H greatly impacted food encounter rate ε. As H increases, ε becomes similar to the distribution of total tunnel area, because each cell encountered is more likely to contain food. With high F, ε drops markedly at H = 0.7 (Fig. 8(a)), just as tunnel area dropped at this value (Fig. 7(c)). As N increased, the inhibitory effect of clusters on tunnel area extended over whole ranges of H (Fig. 8(b) (d)). At these values of H, an increase in the number of primary tunnels leads to a larger proportion of those tunnels impeded relative to either low H, that lacked clusters, or very high H, where fewer clusters were unlikely to enclose a majority of primary tunnels. In order to understand the relationships among H, N, F, and ε, we investigated encounter rate ε and food density F for different values of H (Fig. 9(a)). This result shows that there is a linear relationship between ε and F regardless of H. If food particles were evenly distributed on a resulting tunneling pattern, the number of particles intercepted by the tunnel pattern will linearly increase with the number of food particles distributed or the tunnel area. The degree of increasing proportion, the slope m (= ε/ F), depends on the tunnel area. Higher values

8 S.-H. Lee et al. / BioSystems 90 (2007) Fig. 9. (a) The linear relationship between encounter rate ε and food density F for different the landscape heterogeneities. (b) The changes of the slope φ of the linear line fitted in the plot of ε against F according to H. of m imply larger tunnel area. From this relationship, we can justify the use of m as criteria for characterizing the tunneling pattern. Fig. 9(b) shows the changes of m against H for different N ranging from 6 to 12. The broken line is intended as a visual aid to demonstrate the valley-shaped trend of the slope data. When N = 9, the valley-shaped trend was relatively diluted. When N > 9, a few tunnels will invariable escape the inhibitory effect of high h clusters and move quickly to the edge of the area. This made the running time of simulation short because the model has constraint condition that when a TVC reaches the system boundary, the simulation is terminated. Thus, except for a few TVCs reaching the boundary wall, most TVCs did not have sufficient time to construct longer tunnel, decreasing of the encounter rate ε. In case of N < 9, most TVCs were interrupted by the clusters with higher h, so that even if the experiment was not terminated early, the tunnels propagated so slowly that tunnel area and encounter rate ε were still low. Consequently, at N = 9, the dilution of valley-shaped trend was attributed to the optimization of trade-off between the two variables, TVC s travel time and the asymmetric growth rate of primary tunnels. The results of this study showed that in field, subterranean termites may employ a foraging strategy controlling the number of primary tunnels to counter the decreased food encounter rate resulting from the effect of heterogeneous landscape. 5. Conclusions We demonstrated the effects of landscape heterogeneity on the termite tunnel pattern in relation to food encounter rate. To do this, we proposed a model mimicking termite tunnel pattern in heterogeneous landscapes. Most parameter values used in our model have been based on empirical data, which should justify their application. However, in field, the parameters pertaining to the transition probability might be different due to individual state (e.g. age, health and nutrition). Nevertheless, our model provides insight into the effect of the heterogeneity of landscape H on the encounter rate. When food density F was high (F > 0.003), the effect of H on ε appeared obviously, the valley-shaped trend, while in case of low F, the encounter rate ε was much less influenced by H (Fig. 8). In addition, we also showed the relationship among N, F, H, and ε (Fig. 9). Experimentation in field has not determined the relationship between termite tunnel pattern and the heterogeneity of landscape, but our simulation results provided not only possible prediction to the influence of the heterogeneous landscape but also a baseline for future empirical-work on the foraging pattern of subterranean termite. Acknowledgements We would like to thank P. Ban and R. Pepin (University of Florida) and for technical assistance. This research was supported by the Florida Agricultural Experiment Station and a grant from USDA-ARS under the grant agreement No References Bonabeau, E., Social insect colonies as complex adaptive systems. Ecosystems 1, Capowiez, Y., Pierret, A., Daniel, O., Monestiez, P., Kretzschmar, A., D skeleton reconstruction of natural earthworm burrow sys-

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