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Territory size distribution of Formosan subterranean termites in urban landscape: Comparison between experimental and simulated results
Journal of Asia-Pacific Entomology 14 (2011) 1–6
Contents lists available at ScienceDirect
Journal of Asia-Pacific Entomology j o u r n a l h o m e p a g e : w w w. e l s e v i e r. c o m / l o c a t e / j a p e
Territory size distribution of Formosan subterranean termites in urban landscape: Comparison between experimental and simulated results Sang-Hee Lee a,⁎, Nan-Yao Su b a b
Division of Fusion Convergence of Mathematical Sciences, National Institute for Mathematical Sciences, Daejeon, Republic of Korea Department of Entomology and Nematology, Ft. Lauderdale Research and Education Center, University of Florida, Ft. Lauderdale, FL, USA
a r t i c l e
i n f o
Article history: Received 21 August 2010 Revised 4 November 2010 Accepted 5 November 2010 Available online 11 November 2010 Keywords: Territory size distribution Termite Lattice model Urban landscape Territory competition
a b s t r a c t The territory size distribution of Formosan subterranean termites in an urban landscape was studied by using a two-dimensional lattice model with minimized local rules on the basis of empirical data to determine the development of a territory. An urban landscape including components such as complex man-made structures, trees, and lakes was constructed on a lattice space. Each component was described by assigning values ranging from 0.0 to 1.0. These values were interpreted as transition probabilities, Ptrans, which represented the spatially distributed properties of the components. The higher the Ptrans values, better were the environmental conditions for territory growth. We applied this model to the termite colonies in Louis Armstrong Park in New Orleans, LA. A comparison of the model prediction and the observed territory size distribution showed very similar trends in maximum territory on a size-normalized scale. The trends emerged from two effects: first, when a founding pair was surrounded by lattice cells with low Ptrans values, its growth was strongly inhibited, and second, when territories competed with each other at shared boundaries, their growth was restrained. In addition, we investigated the effect of the presence/absence of landscape components on the territory size distribution in the simulation. © Korean Society of Applied Entomology, Taiwan Entomological Society and Malaysian Plant Protection Society, 2010. Published by Elsevier B.V. All rights reserved.
Introduction Territoriality can exert strong effects on the population dynamics of aggressive animals (Patterson, 1980; Davies and Houston, 1984; Lomnicki, 1988; Newton, 1992; Sutherland, 1996). The nature of these effects depends largely on how the territory size is adjusted according to ecological circumstances. When the territory size is inflexible, local populations may be regulated at stable densities. In contrast, when the territory size is readily altered according to the changes in the abundance of food or the density of competitors, population densities also vary. Therefore, understanding the territory size distribution in territorial competition is not only important for demography and population regulation but also has applications in spatial ecology and conservation biology (Lee et al., 2007; Korb and Lenz, 2004). Thus far, researches have been conducted on the various forms of territorial behavior showed by different animals (Nursall, 1977; Norman and Jones, 1984; Watson, 1967; Watson and Miller, 1971). To account for the diverse pattern of territorial behavior, several models have been developed using different approaches. The first economic model for the study of the territorial behavior of animals was
⁎ Corresponding author. Tel.: + 82 42 717 5736; fax: + 82 42 717 5758. E-mail address: [email protected] (S.-H. Lee).
proposed by Brown (1964). This model was based on the hypothesis that the benefits of exclusive access to a resource should exceed the costs of territory defense. Following Brown's model, three additional models were developed: an optimality model, a geometrical model, and a partial differential equation model. In the optimality model, the marginal costs and utility of territorial defenses were considered with respect to the area and perimeter of the space defended (Carpenter 1987). In the geometrical model, territorial boundaries placed midway between neighbors and division of territory on the basis of the distance between the residents’ centers of control were addressed (McCleery and Perrins, 1985). In the partial differential equation model, the spatial distribution resulting from the interplay of behavioral mechanisms between residents, especially those that underlay movement and interaction, were underlined (White et al., 1996). Although the predictive accuracy of these territory models has been high with respect to certain characteristic features of territorial behavior, factors influencing the territory size and cost, especially the environment faced by underground animals, have not been taken into account in these models. This is, in large part, because of the difficulties involved in determining interaction functions between the environment and animals and observing underground animals. To overcome these difficulties, we formulated a lattice-based territory model based on empirical data to simulate territory growth dynamics. The empirical data was obtained by observing the territory of
1226-8615/$ – see front matter © Korean Society of Applied Entomology, Taiwan Entomological Society and Malaysian Plant Protection Society, 2010. Published by Elsevier B.V. All rights reserved. doi:10.1016/j.aspen.2010.11.002
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the Formosan subterranean termite, Coptotermes formosanus Shiraki, in the urban landscape of Louis Armstrong Park in New Orleans, LA. Because determining time-varying territories in the field is difficult and data are scarce, in the present study, we explored the territory size distribution at a stable state using a model simulation. A stable state indicates that the territory distribution does not change with time.
Model description A termite foraging territory was simulated on a two-dimensional lattice space composed of L × L cells, where L (=200) is the system size. Each cell was assigned one of three possible states: occupied by active termites (active termite cell), occupied by inactive termites (inactive termite cell), and empty (empty cell). A site occupied by a termite cell corresponded to a place where a termite tunnel network existed. All of the cells had values ranging from 0.0 to 1.0, which represented the components of an urban landscape such as buildings, asphalt pavement, and trees. These values can be understood as the degree of ease in constructing tunnels. Higher values corresponded to easier environmental conditions for termite tunneling. Tunneling was more active at the ends of tunnels with better environmental conditions, while the behavior was less active at tunnel tips with worse environmental conditions (Su et al., 1991; Su and Puche, 2003). Thus, the value can also be interpreted as a transition probability, Ptrans, for an active termite cell to grow into its neighboring cell. In order to simulate our model in the urban landscape of Louis Armstrong Park, each region occupied by different components of the landscape was extracted from a color image of the park (Fig. 1). A transition probability, Ptrans, was assigned to each region (tree and grass: 0.9 (green), lake: 0.0 (blue), building: 0.05 (red), asphalt pavement: 0.1 (white)) (Fig. 1). In addition, seasonal cycles (summer and winter) were incorporated. It was assumed that one season changed into the next season according to a step function with two time scales, summer and winter time, for the sake of theoretical simplification. The simulated territory grew in the summer and shrank in the winter. At the beginning of the simulation run, N active termite cells, referred to as seed cells representing founding pairs, were randomly distributed on the lattice space. In the simulation study, we used the case where all of the seed cells were located on tree and grass sites. The following rules determined the seed cells' growth from one generation to the next in the framework of the territory dynamics.
Growth rules (summer season) • When an active termite cell was surrounded by empty neighbor cells, its growth was determined by the Ptrans values of its neighbors. To accomplish this, a random process was used to generate a probability from the active termite cell's position. When the value of the probability was smaller than the Ptrans value of a neighbor cell, the active termite cell grew into the neighbor cell (Fig. 2 (a)). • When two active termite cells met each other, they could not share the same site (Fig. 2 (b)). • When two active termite cells competed with each other to occupy an empty site, the occupant of the site was determined by a coin toss (Fig. 2 (c)). State-changing rules (from an active termite cell to an inactive termite cell) • When one active termite cell was surrounded by 7 obstacle cells, the active termite cell was changed to an inactive termite cell (Fig. 3 (a)). • When one active termite cell was surrounded by less than 7 obstacle cells, the termite cell could grow into empty cells or stop probabilistically. Once it stopped, it was changed to an inactive termite cell (Fig. 3 (b)). Shrinkage rules (winter season) • Messenger and Su (2005) reported that C. formosanus colony territories contracted to ≈80% of their summer season size during the winter. Based on this observation, we made a simple rule mimicking territorial shrinkage: active and inactive termite cells were removed in distance order, with distal cells from the seed cell removed first (Fig. 4(a)). The remaining 20% of the termite cells became inactive cells to prevent the territory from growing during the winter. The Ptrans values of the sites of the removed cells increased at a certain rate, β: t + 1 t Ptrans i′ ; j′ = Ptrans i′ ; j′ ð1 + βÞ; where i′ and j′ represent the sites of the removed cells. This rule was based on the observation that when a shrunken territory grows again in the following summer season, the territory tends to grow in accordance with the territory site established in the previous year (Messenger et al., 2005). This may be explained by termites using
Fig. 1. Regional characteristic extraction from image of Louis Armstrong Park.
S.-H. Lee, N.-Y. Su / Journal of Asia-Pacific Entomology 14 (2011) 1–6
Active termite cell
a
b 0.0
c
3
Coin toss
0.2
0.0
0.0
0.3
0.9 0.5
0.0
Fig. 2. All possible configurations that active termite cell could encounter. The arrow signs indicate the growing direction of each active termite cell. The cell values represent the transition probabilities. The square with termites is an active termite cell.
year corresponded to T = 37. The update was synchronous for all cells. The simulation results were statistically averaged over 30 runs.
previously constructed tunnels for expansion rather than spending energy to construct new tunnels. The value of β was set to appropriately 0.3. • After the territory size shrank, several cells among the 20% inactivated cells were chosen by a probability function, being proportional to the distance from the seed cell, and changed into active termite cells to initiate territory growth (Fig. 4 (b)). The preferential weighting of the distal cells as starting sites for new tunnel growth stemmed from the assumption that growth was most likely at the colony's periphery.
Simulation results Fig. 5 shows typical territory distribution patterns according to time in the urban landscape representing Louise Armstrong Park. Initially, 30 seed cells were randomly distributed. After 1 year, small and isolated territories were formed because of the weak competition among territories, or the worse environmental conditions surrounding the seed cells (Fig. 5 (a)). As time passed, territories grew and the competition among them became stronger at their shared boundaries, which led to more complicated patterns (see inside of dotted circles in Fig. 5 (b) and (c)). Consequently, after 20 simulation years, the
In the present study, we set the duration of the summer season to the iteration time of T = 36. The processes in the winter season were fulfilled in one iteration time (T = 1) (Lee et al., 2007; Lee and Su, 2009). Therefore, in the model, the unit for time t was the “year” and 1
a 0.0
0.0
0.0 0.0
0.0
0.0
0.0
becomes t
t+1
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0.0 0.0
0.0
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0.0
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b 0.0 0.0 0.5 0.0 0.0
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becomes t+1 t 0.0
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Fig. 3. Multiplication process (state-changing rule) for each active termite cell: (a) surrounded by seven cells with Ptrans = 0.0 or (b) toward next-neighbor cells with Ptrans N 0.0 during one discrete time step t → t + 1.
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a 2 7
1
5 13
4
11 12 8
15 20
3 9
17 16 18 19
10 6 14
Summer season
Winter season
b Seed cell Active termite cell Inactive termite cell
Fig. 4. Shrinkage process of territory during winter season: (a) territory size decreased to ~ 80%. Termite cells were removed according to the rule of “farther cells removed earlier,” and (b) after shrinkage, new active termite cells were chosen to initiate territory growth.
Fig. 5. Typical patterns of territory distribution at t = 1, 2, 5, and 20.
S.-H. Lee, N.-Y. Su / Journal of Asia-Pacific Entomology 14 (2011) 1–6
Discussion In order to better understand the territory size distribution of Formosan subterranean termites in the urban landscape of Louise Armstrong Park, New Orleans, LA, we used a two-dimensional lattice model to simulate the growth dynamics of the termite territory and compared the simulated and experimental results. In the simulation, the urban landscape was characterized by Ptrans values ranging from 0.0 to 1.0. These values represented the degree of ease in constructing tunnels. The higher the Ptrans values, better are the environmental conditions for termite tunneling. As components of the urban landscape, trees and grass, lakes, buildings, and asphalt
N=13 (observation) N=10 N=20 N=30 N=40
Normalized territory size
1.0 0.8 0.6 0.4 0.2 0.0 1
10
100
n Fig. 6. Plots of territory size distributions normalized by largest territory size at stable state in which territory pattern is not changed with seasonal time against rank, n, for simulated and empirical data.
N=13 Observation data Simulated in ideal landscape Simulated in urban landscape
1.0
Normalized territory size
territory pattern reached a stable state in which the territory distribution shape was not changed with time (Fig. 5 (d)). In order to analyze the territory size distribution in the stable state, we ranked the order of territory size, n. For instance, the smallest territory had a value of n = 1 and the next smallest territory had a value of n = 2. The territory size was normalized by the maximum territory size at the state. Fig. 6 shows the plots of territory size distributions against the rank, n, for the different N. In the distribution, two different regions partitioned by a break point, τ, were observed. Where n b τ, smallersized territories arose from seed cells surrounded by neighbors with low values of Ptrans, as seen in Fig. 5 (a), or were loser territories in the territorial competition. On the other hand, where n N τ, larger-sized territories were winner territories from strong competition among the territories, or were developed from seed cells surrounded by neighbors with high values of Ptrans, at shared boundaries. In Armstrong Park, there were 13 territories. The territory size distribution lay between the simulated distributions for N = 10 and 20, which means the empirical data was in very good agreement with the simulated results. This reflects that termite territory formation is strongly governed by two effects: a founding pair location-related effect in an early stage and a territory competition-related effect in a late stage. In addition, we investigated the effect of the presence/absence of the landscape components on the territory size distribution. Fig. 7 shows territory size distributions normalized by the maximum territory size in the urban landscape and the ideal landscape. The ideal landscape was generated by assigning the value of Ptrans = 1.0 to all of the lattice cells. The territory size distribution curve in the urban landscape was closer to the empirical data than that in the ideal landscape. This means that the urban landscape significantly affected the territory size distribution.
5
0.8 0.6 0.4 0.2 0.0 0
2
4
6
8
10
12
14
n Fig. 7. Plots of territory size distributions normalized by largest territory size at stable state against rank, n, for simulation of ideal and urban landscapes, and using empirical data.
pavements were considered. The Ptrans values were 0.9 (for trees and grass), 0.0 (for lakes), 0.05 (for buildings), and 0.1 (asphalt pavements). In a preliminary simulation test, the results, as shown in Figs. 6 and 7, the Ptrans values showed slight changes when the components were changed in the following domains—trees and grass: 0.85–0.95, buildings: 0.01–0.1, and asphalt pavements: 0.5–0.3. Thus, although the simulation results (Figs. 6 and 7) were based on Ptrans values that were not directly obtained from experimental data, the results can be considered to be reliable because the Ptrans values obtained under field conditions are likely to stay within the domains. The colony territories of C. Formosanus in Armstrong Park were determined by conducting a stake survey and they were found to be located 10–60 m away from each other (Messenger and Su, 2005), while the simulated territories shared boundaries. This discrepancy can be explained by the fact that the territory was considered to be a minimized polygon, including the termite tunnel web, with a spokeand-wheel pattern. Although the territories were close-packed near the boundaries, the areas between the tunnels in the soil remained unexplored. Further, as the distance between the tunnels and their seed cell increased, the distance between the tunnels also increased, because the tunnels tended to grow outward from the origin (Su et al., 2004; Lee et al., 2006). Thus, the probability of finding termites near the border of a territory was much lower than that of finding them near the origin. In our model, the parameter values were based on observational data that justified their application. However, in the field, the parameters pertaining to the transition probability Ptrans might be different because of termite individual state (e.g., age, health, and nutrition). Moreover, the seasonal cycle was treated as a binary switch for the purpose of theoretical simplification, and seasonal variations such as fluctuations in temperature and moisture were not included. Nevertheless, the simulation results provide a possible explanation for the territory size distribution of subterranean termites in an urban landscape. The simulation model can be used as a tool to explore termite territorial behavior in the viewpoint of cost-benefit. Termites should protect their territories from predators located outside of the territories. Thus, the cost for the protection could be quantified as the length of territory boundaries. Benefit can be related to the area of the territory because the area has strongly something to do with the probability of obtaining food resources. This study is also valuable in that the simulation results provide a basis for future work on termite territory formation in consideration of territory–landscape interactions in territorial dynamics.
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Acknowledgments This study was carried out with the support of the National Institute for Mathematical Sciences, Republic of Korea. References Brown, J.L., 1964. The evolution of diversity in avian territorial systems. Wilson Bulletin 76, 160–169. Carpenter, F.L., 1987. Food abundance and territoriality: to defend or not to defend? American Zoologist 27, 387–400. Davies, N.B., Houston, A.I., 1984. Territory economics, In: Krebs, J.R., Davies, N.B. (Eds.), Behavioural ecology: an evolutionary approach, 2nd ed. Blackwell Sci, Oxford, pp. 148–169. Korb, J., Lenz, M., 2004. Reproductive decision-making in the termite, Crptotermes secundus (Kalotermitidae), under variable food conditions. Behav. Ecol. 15, 390–395. Lee, S.-H., Bardunias, P., Su, N.-Y., 2006. Food encounter rates of simulated termite tunnels with variable food size/ distribution pattern and tunnel branch length. J. Theor. Biolo. 243, 493–500. Lee, S.-H., Su, N.-Y., 2009. A simulation study of subterranean termite's territory formation. Ecol. Infor. 4, 111–116. Lee, S.-H., Su, N.-Y., Bardunias, P., 2007. Exploring landscape structure effect on termite territory size using a model approach. Biosystems 90, 890–896. Lomnicki, A., 1988. Population ecology of individuals. Princeton Univ. Press, Princeton, NJ. McCleery, R.H., Perrins, C.M., 1985. Territory size, tit, Parus major. In: Sibly, R.M., Smith, R.H. (Eds.), Behavioral ecology: ecological consequences of adaptive behavior. Blackwell Scientific Publications, Oxford, England, pp. 353–373.
Messenger, M.T., Su, N.-Y., 2005. Colony characteristics and seasonal activity of the Formosan subterranean termite (Isoptera: Rhinotermitidae) in Louis Armstrong Park, New Orleans, Louisiana. J. Entomol. Sci. 40, 268–279. Messenger, M.T., Su, N.-Y., Husseneder, H., Grace, J.K., 2005. Elimination and reinvasion studies with Coptotermes formosanus (Isoptera: Rhinotermitidae) in Louisiana. J. Econ. Entomol. 98, 916–929. Newton, I., 1992. Experiments on the limitation of bird numbers by territorial behaviour. Biol. Rev. 67, 129–173. Norman, M.D., Jones, G.P., 1984. Determinants of territory size in the pomacentrid reef fish, Parma victoriae. Oecologia 61, 60–69. Nursall, J.R., 1977. Territoriality in redlip blennies (Ophiolennius atlanticus-Pisces: Blennidae). J. Zool. 182, 205–223. Patterson, I.J., 1980. Territorial behaviour and the limitation of population density. Ardea 68, 53–62. Su, N.-Y., Puche, H., 2003. Tunneling activity of subterranean termites (Isoptera: Rhinotermitidae) in sand with moisture gradients. J. Econ. Entomol. 96, 88–93. Su, N.-Y., Scheffrahn, R.H., Ban, P.M., 1991. Uniform size particle barrier: a physical exclusion device against subterranean termites (Isoptera: Rhinotermitidae). J. Econ. Entomol. 84, 912–916. Su, N.-Y., Stith, B.M., Puche, H., Brdunias, P., 2004. Characterization of tunneling geometry of subterranean termites (Isoptera: Rhinotermitidae) by computer simulation. Sociobiology 44, 471–483. Sutherland, W.J., 1996. From individual behaviour to population ecology. Oxford Univ. Press, Oxford. Watson, A., 1967. Population control by territorial behavior in red grouse. Nature 215, 1274–1275. Watson, A., Miller, G. R., 1971. Territory size and aggression in a fluctuating red. White, K.A.J., Lewis, M.A., Murray, J.D., 1996. A model for wolf-pack territory formation and maintenance. J. Theor. Biol. 178, 29–43.
Contents lists available at ScienceDirect
Journal of Asia-Pacific Entomology j o u r n a l h o m e p a g e : w w w. e l s e v i e r. c o m / l o c a t e / j a p e
Territory size distribution of Formosan subterranean termites in urban landscape: Comparison between experimental and simulated results Sang-Hee Lee a,⁎, Nan-Yao Su b a b
Division of Fusion Convergence of Mathematical Sciences, National Institute for Mathematical Sciences, Daejeon, Republic of Korea Department of Entomology and Nematology, Ft. Lauderdale Research and Education Center, University of Florida, Ft. Lauderdale, FL, USA
a r t i c l e
i n f o
Article history: Received 21 August 2010 Revised 4 November 2010 Accepted 5 November 2010 Available online 11 November 2010 Keywords: Territory size distribution Termite Lattice model Urban landscape Territory competition
a b s t r a c t The territory size distribution of Formosan subterranean termites in an urban landscape was studied by using a two-dimensional lattice model with minimized local rules on the basis of empirical data to determine the development of a territory. An urban landscape including components such as complex man-made structures, trees, and lakes was constructed on a lattice space. Each component was described by assigning values ranging from 0.0 to 1.0. These values were interpreted as transition probabilities, Ptrans, which represented the spatially distributed properties of the components. The higher the Ptrans values, better were the environmental conditions for territory growth. We applied this model to the termite colonies in Louis Armstrong Park in New Orleans, LA. A comparison of the model prediction and the observed territory size distribution showed very similar trends in maximum territory on a size-normalized scale. The trends emerged from two effects: first, when a founding pair was surrounded by lattice cells with low Ptrans values, its growth was strongly inhibited, and second, when territories competed with each other at shared boundaries, their growth was restrained. In addition, we investigated the effect of the presence/absence of landscape components on the territory size distribution in the simulation. © Korean Society of Applied Entomology, Taiwan Entomological Society and Malaysian Plant Protection Society, 2010. Published by Elsevier B.V. All rights reserved.
Introduction Territoriality can exert strong effects on the population dynamics of aggressive animals (Patterson, 1980; Davies and Houston, 1984; Lomnicki, 1988; Newton, 1992; Sutherland, 1996). The nature of these effects depends largely on how the territory size is adjusted according to ecological circumstances. When the territory size is inflexible, local populations may be regulated at stable densities. In contrast, when the territory size is readily altered according to the changes in the abundance of food or the density of competitors, population densities also vary. Therefore, understanding the territory size distribution in territorial competition is not only important for demography and population regulation but also has applications in spatial ecology and conservation biology (Lee et al., 2007; Korb and Lenz, 2004). Thus far, researches have been conducted on the various forms of territorial behavior showed by different animals (Nursall, 1977; Norman and Jones, 1984; Watson, 1967; Watson and Miller, 1971). To account for the diverse pattern of territorial behavior, several models have been developed using different approaches. The first economic model for the study of the territorial behavior of animals was
⁎ Corresponding author. Tel.: + 82 42 717 5736; fax: + 82 42 717 5758. E-mail address: [email protected] (S.-H. Lee).
proposed by Brown (1964). This model was based on the hypothesis that the benefits of exclusive access to a resource should exceed the costs of territory defense. Following Brown's model, three additional models were developed: an optimality model, a geometrical model, and a partial differential equation model. In the optimality model, the marginal costs and utility of territorial defenses were considered with respect to the area and perimeter of the space defended (Carpenter 1987). In the geometrical model, territorial boundaries placed midway between neighbors and division of territory on the basis of the distance between the residents’ centers of control were addressed (McCleery and Perrins, 1985). In the partial differential equation model, the spatial distribution resulting from the interplay of behavioral mechanisms between residents, especially those that underlay movement and interaction, were underlined (White et al., 1996). Although the predictive accuracy of these territory models has been high with respect to certain characteristic features of territorial behavior, factors influencing the territory size and cost, especially the environment faced by underground animals, have not been taken into account in these models. This is, in large part, because of the difficulties involved in determining interaction functions between the environment and animals and observing underground animals. To overcome these difficulties, we formulated a lattice-based territory model based on empirical data to simulate territory growth dynamics. The empirical data was obtained by observing the territory of
1226-8615/$ – see front matter © Korean Society of Applied Entomology, Taiwan Entomological Society and Malaysian Plant Protection Society, 2010. Published by Elsevier B.V. All rights reserved. doi:10.1016/j.aspen.2010.11.002
2
S.-H. Lee, N.-Y. Su / Journal of Asia-Pacific Entomology 14 (2011) 1–6
the Formosan subterranean termite, Coptotermes formosanus Shiraki, in the urban landscape of Louis Armstrong Park in New Orleans, LA. Because determining time-varying territories in the field is difficult and data are scarce, in the present study, we explored the territory size distribution at a stable state using a model simulation. A stable state indicates that the territory distribution does not change with time.
Model description A termite foraging territory was simulated on a two-dimensional lattice space composed of L × L cells, where L (=200) is the system size. Each cell was assigned one of three possible states: occupied by active termites (active termite cell), occupied by inactive termites (inactive termite cell), and empty (empty cell). A site occupied by a termite cell corresponded to a place where a termite tunnel network existed. All of the cells had values ranging from 0.0 to 1.0, which represented the components of an urban landscape such as buildings, asphalt pavement, and trees. These values can be understood as the degree of ease in constructing tunnels. Higher values corresponded to easier environmental conditions for termite tunneling. Tunneling was more active at the ends of tunnels with better environmental conditions, while the behavior was less active at tunnel tips with worse environmental conditions (Su et al., 1991; Su and Puche, 2003). Thus, the value can also be interpreted as a transition probability, Ptrans, for an active termite cell to grow into its neighboring cell. In order to simulate our model in the urban landscape of Louis Armstrong Park, each region occupied by different components of the landscape was extracted from a color image of the park (Fig. 1). A transition probability, Ptrans, was assigned to each region (tree and grass: 0.9 (green), lake: 0.0 (blue), building: 0.05 (red), asphalt pavement: 0.1 (white)) (Fig. 1). In addition, seasonal cycles (summer and winter) were incorporated. It was assumed that one season changed into the next season according to a step function with two time scales, summer and winter time, for the sake of theoretical simplification. The simulated territory grew in the summer and shrank in the winter. At the beginning of the simulation run, N active termite cells, referred to as seed cells representing founding pairs, were randomly distributed on the lattice space. In the simulation study, we used the case where all of the seed cells were located on tree and grass sites. The following rules determined the seed cells' growth from one generation to the next in the framework of the territory dynamics.
Growth rules (summer season) • When an active termite cell was surrounded by empty neighbor cells, its growth was determined by the Ptrans values of its neighbors. To accomplish this, a random process was used to generate a probability from the active termite cell's position. When the value of the probability was smaller than the Ptrans value of a neighbor cell, the active termite cell grew into the neighbor cell (Fig. 2 (a)). • When two active termite cells met each other, they could not share the same site (Fig. 2 (b)). • When two active termite cells competed with each other to occupy an empty site, the occupant of the site was determined by a coin toss (Fig. 2 (c)). State-changing rules (from an active termite cell to an inactive termite cell) • When one active termite cell was surrounded by 7 obstacle cells, the active termite cell was changed to an inactive termite cell (Fig. 3 (a)). • When one active termite cell was surrounded by less than 7 obstacle cells, the termite cell could grow into empty cells or stop probabilistically. Once it stopped, it was changed to an inactive termite cell (Fig. 3 (b)). Shrinkage rules (winter season) • Messenger and Su (2005) reported that C. formosanus colony territories contracted to ≈80% of their summer season size during the winter. Based on this observation, we made a simple rule mimicking territorial shrinkage: active and inactive termite cells were removed in distance order, with distal cells from the seed cell removed first (Fig. 4(a)). The remaining 20% of the termite cells became inactive cells to prevent the territory from growing during the winter. The Ptrans values of the sites of the removed cells increased at a certain rate, β: t + 1 t Ptrans i′ ; j′ = Ptrans i′ ; j′ ð1 + βÞ; where i′ and j′ represent the sites of the removed cells. This rule was based on the observation that when a shrunken territory grows again in the following summer season, the territory tends to grow in accordance with the territory site established in the previous year (Messenger et al., 2005). This may be explained by termites using
Fig. 1. Regional characteristic extraction from image of Louis Armstrong Park.
S.-H. Lee, N.-Y. Su / Journal of Asia-Pacific Entomology 14 (2011) 1–6
Active termite cell
a
b 0.0
c
3
Coin toss
0.2
0.0
0.0
0.3
0.9 0.5
0.0
Fig. 2. All possible configurations that active termite cell could encounter. The arrow signs indicate the growing direction of each active termite cell. The cell values represent the transition probabilities. The square with termites is an active termite cell.
year corresponded to T = 37. The update was synchronous for all cells. The simulation results were statistically averaged over 30 runs.
previously constructed tunnels for expansion rather than spending energy to construct new tunnels. The value of β was set to appropriately 0.3. • After the territory size shrank, several cells among the 20% inactivated cells were chosen by a probability function, being proportional to the distance from the seed cell, and changed into active termite cells to initiate territory growth (Fig. 4 (b)). The preferential weighting of the distal cells as starting sites for new tunnel growth stemmed from the assumption that growth was most likely at the colony's periphery.
Simulation results Fig. 5 shows typical territory distribution patterns according to time in the urban landscape representing Louise Armstrong Park. Initially, 30 seed cells were randomly distributed. After 1 year, small and isolated territories were formed because of the weak competition among territories, or the worse environmental conditions surrounding the seed cells (Fig. 5 (a)). As time passed, territories grew and the competition among them became stronger at their shared boundaries, which led to more complicated patterns (see inside of dotted circles in Fig. 5 (b) and (c)). Consequently, after 20 simulation years, the
In the present study, we set the duration of the summer season to the iteration time of T = 36. The processes in the winter season were fulfilled in one iteration time (T = 1) (Lee et al., 2007; Lee and Su, 2009). Therefore, in the model, the unit for time t was the “year” and 1
a 0.0
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becomes t
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0.0 0.0
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Inactive termite cell
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b 0.0 0.0 0.5 0.0 0.0
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becomes t+1 t 0.0
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Fig. 3. Multiplication process (state-changing rule) for each active termite cell: (a) surrounded by seven cells with Ptrans = 0.0 or (b) toward next-neighbor cells with Ptrans N 0.0 during one discrete time step t → t + 1.
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a 2 7
1
5 13
4
11 12 8
15 20
3 9
17 16 18 19
10 6 14
Summer season
Winter season
b Seed cell Active termite cell Inactive termite cell
Fig. 4. Shrinkage process of territory during winter season: (a) territory size decreased to ~ 80%. Termite cells were removed according to the rule of “farther cells removed earlier,” and (b) after shrinkage, new active termite cells were chosen to initiate territory growth.
Fig. 5. Typical patterns of territory distribution at t = 1, 2, 5, and 20.
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Discussion In order to better understand the territory size distribution of Formosan subterranean termites in the urban landscape of Louise Armstrong Park, New Orleans, LA, we used a two-dimensional lattice model to simulate the growth dynamics of the termite territory and compared the simulated and experimental results. In the simulation, the urban landscape was characterized by Ptrans values ranging from 0.0 to 1.0. These values represented the degree of ease in constructing tunnels. The higher the Ptrans values, better are the environmental conditions for termite tunneling. As components of the urban landscape, trees and grass, lakes, buildings, and asphalt
N=13 (observation) N=10 N=20 N=30 N=40
Normalized territory size
1.0 0.8 0.6 0.4 0.2 0.0 1
10
100
n Fig. 6. Plots of territory size distributions normalized by largest territory size at stable state in which territory pattern is not changed with seasonal time against rank, n, for simulated and empirical data.
N=13 Observation data Simulated in ideal landscape Simulated in urban landscape
1.0
Normalized territory size
territory pattern reached a stable state in which the territory distribution shape was not changed with time (Fig. 5 (d)). In order to analyze the territory size distribution in the stable state, we ranked the order of territory size, n. For instance, the smallest territory had a value of n = 1 and the next smallest territory had a value of n = 2. The territory size was normalized by the maximum territory size at the state. Fig. 6 shows the plots of territory size distributions against the rank, n, for the different N. In the distribution, two different regions partitioned by a break point, τ, were observed. Where n b τ, smallersized territories arose from seed cells surrounded by neighbors with low values of Ptrans, as seen in Fig. 5 (a), or were loser territories in the territorial competition. On the other hand, where n N τ, larger-sized territories were winner territories from strong competition among the territories, or were developed from seed cells surrounded by neighbors with high values of Ptrans, at shared boundaries. In Armstrong Park, there were 13 territories. The territory size distribution lay between the simulated distributions for N = 10 and 20, which means the empirical data was in very good agreement with the simulated results. This reflects that termite territory formation is strongly governed by two effects: a founding pair location-related effect in an early stage and a territory competition-related effect in a late stage. In addition, we investigated the effect of the presence/absence of the landscape components on the territory size distribution. Fig. 7 shows territory size distributions normalized by the maximum territory size in the urban landscape and the ideal landscape. The ideal landscape was generated by assigning the value of Ptrans = 1.0 to all of the lattice cells. The territory size distribution curve in the urban landscape was closer to the empirical data than that in the ideal landscape. This means that the urban landscape significantly affected the territory size distribution.
5
0.8 0.6 0.4 0.2 0.0 0
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6
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n Fig. 7. Plots of territory size distributions normalized by largest territory size at stable state against rank, n, for simulation of ideal and urban landscapes, and using empirical data.
pavements were considered. The Ptrans values were 0.9 (for trees and grass), 0.0 (for lakes), 0.05 (for buildings), and 0.1 (asphalt pavements). In a preliminary simulation test, the results, as shown in Figs. 6 and 7, the Ptrans values showed slight changes when the components were changed in the following domains—trees and grass: 0.85–0.95, buildings: 0.01–0.1, and asphalt pavements: 0.5–0.3. Thus, although the simulation results (Figs. 6 and 7) were based on Ptrans values that were not directly obtained from experimental data, the results can be considered to be reliable because the Ptrans values obtained under field conditions are likely to stay within the domains. The colony territories of C. Formosanus in Armstrong Park were determined by conducting a stake survey and they were found to be located 10–60 m away from each other (Messenger and Su, 2005), while the simulated territories shared boundaries. This discrepancy can be explained by the fact that the territory was considered to be a minimized polygon, including the termite tunnel web, with a spokeand-wheel pattern. Although the territories were close-packed near the boundaries, the areas between the tunnels in the soil remained unexplored. Further, as the distance between the tunnels and their seed cell increased, the distance between the tunnels also increased, because the tunnels tended to grow outward from the origin (Su et al., 2004; Lee et al., 2006). Thus, the probability of finding termites near the border of a territory was much lower than that of finding them near the origin. In our model, the parameter values were based on observational data that justified their application. However, in the field, the parameters pertaining to the transition probability Ptrans might be different because of termite individual state (e.g., age, health, and nutrition). Moreover, the seasonal cycle was treated as a binary switch for the purpose of theoretical simplification, and seasonal variations such as fluctuations in temperature and moisture were not included. Nevertheless, the simulation results provide a possible explanation for the territory size distribution of subterranean termites in an urban landscape. The simulation model can be used as a tool to explore termite territorial behavior in the viewpoint of cost-benefit. Termites should protect their territories from predators located outside of the territories. Thus, the cost for the protection could be quantified as the length of territory boundaries. Benefit can be related to the area of the territory because the area has strongly something to do with the probability of obtaining food resources. This study is also valuable in that the simulation results provide a basis for future work on termite territory formation in consideration of territory–landscape interactions in territorial dynamics.
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