Territory size and shape in fire ants: a model based on neighborhood interactions.
Animals that defend foraging territories often show considerable intraspecific variation in territory area. Ecologists have long been interested in explaining this variation and in predicting how the sizes or shapes of territories differ among individuals (Covich 1976, Carpenter 1987). Within a population, variation in territory area contributes to the unequal division of limiting resources, which may in turn affect the dynamics or stability of animal numbers (e.g., Lomnicki 1988). Furthermore, the processes determining territory area may shed light on decision making by competing animals.
In many territorial species, territories are molded by boundary disputes among neighboring residents. Mosaics of contiguous territories have been observed in diverse animals (e.g., birds: Watson and Miller 1971, Woolfenden and Fitzpatrick 1984; fish: Kodric-Brown 1978, McNichol and Noakes 1981; mammals: Koford 1957, Kitchen 1974; insects: Mabelis 1979, Adams and Levings 1987). Numerous experiments have shown that neighbors restrict one another's territory areas by fighting and display (e.g., Watson 1967, Krebs 1971, Welsh 1975, Nursall 1977, Norman and Jones 1984, Adams 1990, Gordon 1992). One approach to modeling territory size is to predict the positions of boundaries formed by these neighborhood interactions.
Although many models of territory size have been proposed, there has been little effort to predict the geometric consequences of boundary conflicts. Various optimality models consider how the costs and benefits of territory defense change with the area or perimeter of the defended space (reviewed by Holldobler and Lumsden 1980, Schoener 1983, Davies and Houston 1984, Carpenter 1987). While these models successfully predict some aspects of territorial behavior, their application to territory mosaics is limited because most consider decision making by only a single resident (but see Tullock 1983, Maynard Smith 1982, Jones and Krummel 1985). When territories are contiguous, the size of any particular territory depends not only on the decisions of the resident, but also on actions of each of its neighbors, which may in turn be affected by the behavior of more remote residents. Therefore, to predict territory boundaries, it may be necessary to consider the simultaneous actions of groups of interacting residents.
An alternative to the standard optimality models is to apply geometric procedures that divide the habitat into cells. For example, territories have been approximated by Dirichlet tessellation, in which straight line segments are placed midway between neighbors (Hasegawa and Tanemura 1976, Tanemura and Hasegawa 1980, McCleery and Perrins 1985). The resulting cells, known as Thiessen or Voronoi polygons, among other names (Weaire and Rivier 1984), have also been used in plant ecology to provide indices of neighborhood competition (Czaran and Bartha 1992). Similar results are obtained by drawing circular territories and dividing equally the areas of overlap (Grant 1968, Maynard Smith 1974, Covich 1976, Stamps and Krishnan 1990). While this approach yields quantitative predictions of territory size and shape, the methods applied to date have been incomplete, usually assuming that the positions of boundary segments depend only on the distances to the two closest residents. Territory partitioning in ant populations has been represented by nonoverlapping circles or rectangles (Korzukhin and Porter 1994, Stoker et al. 1994); however, these models assume a strict size hierarchy such that any particular colony is unaffected by the presence of smaller colonies, regardless of the degree of size difference.
Territory boundary positions may also be predicted from the behavioral mechanisms underlying the movement and interaction of residents. The spatial consequences of such mechanisms have been modelled for wolf packs by the use of partial differential equations (Lewis and Murray 1993, White et al. 1996). These models predict the distribution of wolf density for isolated or interacting packs based on movement of the wolves relative to the den, and on the distribution of and reaction to scent marks. This approach is similar in intent to the one developed in this paper in that both consider the relationship between individual behaviors and the partitioning of space. However, the computer algorithm described below is easier to modify or extend, particularly when the behavior of a resident towards one of its neighbors depends upon the outcome of its interactions with other neighbors.
This paper describes a model that predicts the equilibrial positions of boundaries molded by interactions among a set of competing animals. I illustrate this approach by its application to the fire ant Solenopsis invicta Buren, in which colonies defend exclusive foraging territories (Wilson et al. 1971, Tschinkel et al. 1995). I then briefly discuss ways to extend and test the model for other organisms and ecological circumstances.
A TERRITORY MODEL based ON NEIGHBORHOOD INTERACTIONS
Although there have been few experimental studies on the processes by which territory boundaries are formed or shifted (Stamps 1994), it may be hypothesized that boundaries develop where the aggressive tendencies of residents match those of adjacent neighbors. The degree of aggression by a resident towards intruders often declines as it moves farther from its nest or core area (e.g., Patterson 1980, McNichol and Noakes 1981). At greater distances from its nest, a resident may flee from a rival that it would otherwise attack (e.g., Van den Assem 1967). At intermediate distances, the tendencies to attack and to flee are balanced, producing a boundary that is respected by each each neighbor (e.g., Baerends and Baerends-Van Roon 1950, Nursall 1977). When the ability or motivation of a resident to fight is experimentally manipulated, this balance is disrupted, causing shifts in boundary positions (e.g., Peek 1972, Dill et al. 1981, Adams 1990).
The partitioning of habitat into territories can be modeled by predicting where the aggressive tendencies of adjacent residents will reach such a balance (see also Maynard Smith 1974, Patterson 1980). This requires a quantitative description of the way that defensive behaviors vary spatially and among residents. I use the term "aggressive pressure" to indicate the intensity and persistence of territorial behaviors directed towards an intruding neighbor at a particular location. The important elements of territorial behavior are those that are effective in deterring competitors or allowing a resident to expand into a neighbor's territory. The appropriate operational definition of aggressive pressure will vary from species to species. In many ants, a primary determinant of pressure is the rate at which workers are recruited to contested areas (Holldobler 1981, Adams 1990). In other taxa, the most important components contributing to aggressive pressure may include the resident's fighting ability, or resource-holding power (Parker 1974), its persistence in chasing intruders or in continuing to intrude upon a defended area (e.g., Stamps 1994), or displays, including scent marks (e.g., Gosling and McKay 1990). Field data can be used to ascertain which of these variables are the most important determinants of boundary positions and to begin to develop quantitative descriptions of aggressive pressure. This is illustrated for S. invicta below.
The initial models assume that: (1) territories completely fill available space; (2) each resident has a nest site, or some other fixed position, from which it centers its activities; (3) the residents do not show strategic responses in the level of aggressive pressure. However, these assumptions, or others, can be modified easily within this modeling framework. Residents may be solitary animals, such as male birds, or cooperating groups of animals, such as colonies of social insects. Prediction of boundary positions is a two-step process. First, by observing how the fighting behavior of territorial residents varies with fighting ability, ecological circumstances, and spatial position, one can develop a mathematical description of the rules describing spatial variation in aggressive pressure. Second, an algorithm is applied that locates the curves along which the aggressive pressure applied by adjacent neighbors is equal.
The rules describing territorial fighting for a particular species and environment may be very complex. The approach taken here is to begin with simple hypotheses concerning aggressive pressure, based upon field observations, and to evaluate the predictive ability of models incorporating these hypotheses. Even a crude description of territorial behavior may predict many of the most important aspects of size and shape variation; it remains to be seen how much the fit can be improved by adding detail to the model. I begin with three aspects of territorial fighting that appear to be especially important predictors of territory boundary positions in uniform habitats.
Distance from the nest site. - Many territory residents conduct defense from nest sites or other fixed positions in the interior of their territories. It is sometimes observed that the probability or intensity of attack against intruders declines with distance from these central positions (e.g., Patterson 1980, McNichol and Noakes 1981, Gordon and Kulig 1996). Therefore, one can hypothesize that the aggressive pressure brought to bear against an adjacent resident is a decreasing function of distance.
Fighting ability. - In some species, there is considerable variation in fighting ability among interacting residents due to differences in size, health, weaponry, or experience. Animals with greater fighting ability can exert greater pressure, which tends to push boundaries away from stronger residents towards their weaker neighbors. Territory area is often positively correlated with resident size (e.g., Adams and Levings 1987, Grant et al. 1989). However, one should not expect a fixed relationship between fighting ability and territory area, especially if relative fighting ability determines where the boundary is formed. For example, Petrie (1984) found that territory size in the moorhen, Gallinula chloropus, was tightly correlated with the relative masses of neighboring males, but was only weakly correlated with resident mass per se.
Spatial reallocation of effort. - The behavior of a resident towards a particular neighbor may depend upon its degree of success in contests with other neighbors. If the total time or energy available for territory defense is limited, then effort expended in one region of the territory subtracts from the defense of other regions. Furthermore, when a resident loses territory on one front, it may partially compensate by increasing its activity elsewhere. This supposition is similar to the hypothesis of compensatory growth proposed for competing plants (Brisson and Reynolds 1994). An important consequence is that the positions of the various segments that comprise territory boundaries are interrelated. When this is true, one must consider all neighborhood interactions simultaneously in order to model territory size and shape for even a single resident.
A simple way to incorporate such interdependence of boundaries is to assume that aggressive pressure varies inversely with territory area. When territory size increases, the resident must patrol and defend a greater area and may therefore be less able to vigorously defend any particular point. This assumption links the positions of boundary segments: when a resident expands its territory in one direction, territory area is enlarged, and pressure is reduced along other fronts. This possibility has received little attention, but is suggested by field observations on birds and fish (e.g., Krebs 1971; Sale 1975).
An additional example of boundary interdependence is shown in Fig. 1. In 1991, several colonies of S. invicta were removed during an experimental study on competition among colonies (Adams and Tschinkel, unpublished data). Changes in territory boundaries were followed for a single set of four colonies on the periphery of an area from which colonies were removed and for a set of four colonies on an undisturbed control plot (Fig. 1) The study site and methods of mapping are described below. Colonies were removed from a circular area 36 m in diameter by application of the pesticide Amdro (American Cyanamid, Princeton, New Jersey), which leaves no detectable residue after several days (Apperson et al. 1984). Colonies surviving application of pesticide were killed by opening holes deep into the nest mound with a metal stake, then filling the nest with boiling water. Within days, untreated colonies adjacent to the removals expanded their territories into the areas formerly occupied by their neighbors (Fig. lb). These expanded territories subsequently shrank along borders with surviving neighbors (Fig. l c). This suggests that when a colony increases its foraging and defensive efforts in one direction, it tends to lose territory in other directions. These cases give only anecdotal support for the hypothesis that positions of different boundary segments are interrelated. However, I will show below that the ability of a model to predict territory shapes is substantially improved by including this feature of territory aggression.
Prediction of boundary positions. - The effects of these three variables - fighting ability, distance from the nest site, and territory area - can be described by simple formulae for aggressive pressure. Given a map of the positions of residents and a mathematical description of aggressive pressure, one can solve for the expected positions of the territory boundaries. For simple cases, this can be done analytically. However, as the number of residents rises and the complexity of the rules describing aggressive pressure increases, mathematical solutions become cumbersome or infeasible. Instead, I describe an algorithm for producing a map of predicted boundary positions that can be implemented on a desktop computer. Because the equation describing aggressive pressure requires only a single line of code, the program can be modified easily to implement alternative rules of territorial fighting.
To illustrate the algorithm, consider a species in which aggressive pressure is a function of the resident's mass, of distance from the nest site to the boundary, and of territory area. Thus, the aggressive pressure, P, applied by resident R at point X is given by
[P.sub.R, X] = C[M.sub.R]/[A.sub.R][[d.sub.R].sup.Y] (1)
where C is a constant, [M.sub.R] is the biomass of resident R, [A.sub.R] is the area of the resident's territory, d[.sub.R] is the distance between point X and R's nest site, and Y is an exponent that determines how rapidly aggressive pressure declines with distance. At the boundary between residents 1 and 2, the following condition is met:
[M.sub.1]/[A.sub.1][[d.sub.1].sup.Y] = [M.sub.2]/][A.sub.2][[d.sub.2].sup.Y].
Notice that the constant, C, cancels; thus, its value need not be estimated.
The assumption that aggressive pressure depends upon territory area complicates efforts to solve for boundary positions. The goal of the analysis is to predict territory boundaries, from which the total area can be calculated. However, to apply Eq. 1, the territory areas must already be known. To remedy this problem, an iterative approach can be applied that initially assumes each resident holds the same territory area, then repeatedly improves the estimates of areas and boundary positions until each boundary segment describes a line of equal pressure. In all cases tested, the outcome of this procedure was not changed by varying the initial estimates of territory area, suggesting that there is a unique solution. The algorithm is described briefly here; a more complete account is given in the Appendix.
Space is represented as a grid of rectangular cells [ILLUSTRATION FOR FIGURE 2 OMITTED]. Each resident is initially assumed to have 1/n of the available area, where n = the number of residents within the mapped area. Each cell is assigned to the territory of the resident applying the greatest pressure at the center of the cell, using Eq. 1 and the initial estimates of territory area. When all cells have been assigned, territory areas are calculated based on the numbers of cells acquired. The ownership of each cell is then determined again, inserting the new estimates of territory area into Eq. 1. This process is repeated until cell ownership is stable. If the area term is removed, the iterative approach and initial estimates of territory area are not needed, and the assignment of cells to territories can be completed in a single pass.
When the area term is included, the iterations can be time-consuming and steps should be taken to ensure an efficient approach to the solution. In the examples illustrated, the model began with a coarse grid of cells and successively reduced cell size three times by a factor of four until each cell was 0.32 [m.sup.2]. Iterations at each scale ceased when the number of cells changing ownership dropped below a critical value [approximately] 1% of cells at the largest scale and [approximately] 0.05% of cells at the smallest scale. Finally, a curve was smoothed along the boundary by searching along lines perpendicular to the faces of the cells for positions where the pressure applied by adjacent residents was equal [ILLUSTRATION FOR FIGURE 2 OMITTED]. To prevent diverging oscillations in estimates of territory area, the changes in territory area from one iteration to the next were dampened. The result is an approximate solution, but because the final cell size is small, the magnitude of errors is well below the measurement error of territories mapped in the field.
Application of the model to an imaginary population of 10 residents is illustrated in Fig. 3. Notice that the shapes and relative sizes of territories vary with the exponent Y (Eq. 1), which describes the dependence of aggressive pressure on distance to the nest site. When the value of Y is high, implying that aggressive pressure declines rapidly with distance, the solution resembles Dirichlet tessellation (Weaire and Rivier 1984), and the correlation between resident mass and territory area is weak [ILLUSTRATION FOR FIGURE 3A OMITTED]. By contrast, when the value of Y is low, territory size depends strongly on the mass of the resident, and boundaries are often highly curved [ILLUSTRATION FOR FIGURE 3B OMITTED]. Boundaries are typically convex from the point of view of smaller residents. The effects of including territory area in the formula for aggressive pressure can be seen by contrasting Fig. 3b to Fig. 3c. When the area term is included, territories of crowded residents tend to take on more eccentric shapes, bulging away from strong neighbors.
Fig. 3d illustrates a modification of the model that allows free movement by the residents, rather than assuming fixed nest sites. After initial prediction of boundary positions, residents relocate to the center of gravity of their territories (see also Hasegawa and Tanemura 1976, Tanemura and Hasegawa 1980). Following relocation by each resident, the boundaries are solved anew. This is repeated until resident movement ceases. This modified boundary model shows how territory disputes may affect the spatial distribution of competitors.
APPLICATION OF THE MODEL TO FIRE ANTS
Study site and species. - Field studies were conducted in pasture on gently sloping land 1 km southeast of Tallahassee, Florida, in June and July 1993, and from June through August, 1994. For further description of the study site, see Tschinkel et al. (1995).
Colonies of the monogyne, or single-queen, form (Ross and Fletcher 1985) of the fire ant S. invicta defend exclusive foraging territories (Tschinkel et al. 1995). A second social form, characterized by multiple queens in mature colonies (Ross and Fletcher 1985), is not territorial and did not occur at our study sites. Fire ant colonies construct nest mounds that are clearly visible, allowing rapid location of all but the smallest colonies (Tschinkel 1993). Although colonies of S. invicta occasionally relocate (Hays et al. 1982), they typically remain fixed in position for months or years. In July 1994, we searched thoroughly for nest mounds in 12 circular plots, which were 72-90 m in diameter. The position of each mound was mapped from the center of the plot using a compass and tape measure. The biomass of each colony was estimated from the length, width, and height of the nest mound (Tschinkel 1993, Tschinkel et al. 1995).
Territories were mapped by placing baits in transects extending outward from the nest sites. Assays of aggression among the ants attracted to the baits were used to identify areas within each colony's territory (Tschinkel et al. 1995). When adjacent baits were found to attract ants from different colonies, additional baits were placed between them to pinpoint the location of the territory boundary.
Development of equations for aggressive pressure.Three alternative descriptions of aggressive pressure were developed using information gathered from field populations. (1) In the simplest model, aggressive pressure depends only on distance to the nest site. The aggressive pressure, P, applied by resident R at point X is given by
[P.sub.R, X] = [C.sub.1] /[d.sub.R]. (3)
As before, [d.sub.R] is the distance to the nest of resident R and [C.sub.1] is a constant. This version of the model is equivalent to Dirichlet tessellation. (2) In the second model, aggressive pressure depends upon the competitive ability of the resident, as well as the distance to the nest site. Previous studies have shown that territory area is linearly related to the biomass of the colony during the season in which this study was performed (Tschinkel et al. 1995). A preliminary analysis of 16 colonies mapped in 1993 showed that the best fit between predicted and observed territory areas was achieved when aggressive pressure was assumed to decline inversely with the cube of the distance:
[P.sub.R, X] = [C.sub.2][M.sub.R] /[[d.sub.R].sup.3] (4)
where [M.sub.R] is the mass of resident R. (3) In the third model, aggressive pressure is affected by territory area. Analysis of the preliminary data showed that best fit was achieved when aggressive pressure was assumed to decline inversely with the square of distance:
[P.sub.R, X] = [C.sub.3][M.sub.R]/[A.sub.R][[d.sub.R].sup.2]. (5)
As in Eq. 2, the constants [C.sub.n] cancel; thus, their values need not be estimated.
The alternative models (Eqs. 3, 4, and 5) are listed in order of increasing complexity (number of parameters). However, the parameters are not estimated from the same data used to test goodness-of-fit; therefore, the more complex model (Eq. 5) is not necessarily the most successful. Data from territories mapped in 1993 were used to estimate the value of the exponent Y, the exponent to which the distance to the nest site is raised. The model was then tested by application to an independent set of data, consisting of territories mapped in the same habitat in 1994. No colony was included in both data sets.
Descriptions of territories. - Observed and predicted territories were represented as polygons [ILLUSTRATION FOR FIGURE 4 OMITTED]. The curved sides of predicted boundaries were represented by a large number of short, straight line segments. For each polygon, the total area and the center of mass were calculated. Territory displacement was measured by the angle and length of a vector extending from the nest site to the center of mass.
Territory shapes were quantified by Mead's indices of eccircularity and abcentricity, which are independent of polygon size (Mead 1966). Eccircularity ranges from 1.0 for circles to higher values for increasingly elongate polygons (1.25 for the territory in Fig. 4a). Abcentricity ranges from 0 when the nest mound is at the center of the territory to higher values when the nest mound is off center (0.88 for the territory in Fig. 4a).
Selection of territories. - The models were applied to each of the 12 plots mapped in July 1994 to yield predictions of territory boundaries. The territories of 27 non-neighboring colonies, scattered across the study plots and selected to represent a range of colony sizes, were mapped. To minimize edge effects, only colonies at least 18 m from the perimeter of the plot were included. For the angles of territory displacement, the rank correlation for circular data was calculated (Batschelet 1981); the Spearman rank correlation is given for the other statistics.
[TABULAR DATA FOR TABLE 2 OMITTED]
TABLE 1. Rank correlations between observed and predicted measures of territory area and shape for three functions describing aggressive pressure (Eqs. 3-5). When aggressive pressure varies with distance only, the result is equivalent to Dirichlet tessellation. Aggressive pressure varies with: Distance, biomass, Distance and Distance and territory Measure only biomass area Area ([m.sup.2]) 0.20 NS 0.68(***) 0.73(***) Eccircularity 0.36 NS 0.45(*) 0.62(***) Abcentricity 0.47(*) 0.54(**) 0.68(***) Distance of territory displacement (m) 0.15 NS 0.60(***) 0.65(***) Angle of territory displacement 0.52(**) 0.59(***) 0.63(***) * P [less than] 0.05; ** P [less than] 0.01; *** P [less than] 0.001; NS: not significant.
Results. - Correlations between observed and predicted measures of size and shape are summarized in Table 1. By all criteria, the greatest predictive ability was achieved when aggressive pressure was assumed to vary with distance, resident biomass, and territory area (Eq. 5). Correlations were lower when area was omitted (Eq. 4), especially for predictions of shape. Dirichlet tessellation (Eq. 3) produced poor predictions of territory area and only crude predictions of shapes. Furthermore, territory areas were more tightly correlated with the predictions of the model of boundary formation than with nest mound volumes (Spearman rank correlation coefficient between territory area and mound volume = 0.46, P [less than] 0.05). Thus, although colony size explains much of the variation in territory area (Tschinkel et al. 1995), better predictions are achieved by also considering the colony's competitive neighborhood.
Although predicted and observed territory measurements were correlated, natural fire ant territories tended to deviate more from circular shapes and to be centered farther from the nest site than predicted. Observed measures of eccentricity, abcentricity, and distance of territory displacement were greater than those produced by any of the three models (Table 2). This may be because of habitat heterogeneity or because aggressive pressure is not exerted equally in all directions. In nature, many ant colonies allocate more of their worker force in some directions than in others due to branching trail systems (Holldobler and Wilson 1990). Furthermore, in both insects and vertebrates, residents may be favored to allocate defensive effort into sectors where they achieve the greatest gains. This will tend to produce more eccentric territory shapes. In essence, the dependence of aggressive pressure upon territory area assumed in this model provides a simple way to account for shifts of territory defense into regions with relatively low competition. More realistic representations are possible, but will be computationally more demanding.
TESTING AND EXTENDING BOUNDARY MODELS
The modeling approach outlined in this paper allows prediction of the spatial consequences of competitive interactions among neighbors. Although the version developed for S. invicta predicts much of the variation in areas and shapes observed in the field, it is unlikely that this model will apply to other taxa without considerable modification. There are at least three ways to develop and test descriptions of spatial variation in aggressive pressure. First, one can construct alternative mathematical descriptions of territorial behavior and accept the version that most accurately predicts territory sizes and shapes. This approach was illustrated above for S. invicta. Phenomenological models of this kind are useful for prediction, but potentially misrepresent mechanisms, since alternative equations may predict similar outcomes.
A second approach is to test experimentally whether particular attributes of the animals or the environment affect boundary position. For example, one can measure how territories change when the fighting or signaling ability of the resident is experimentally altered (e.g., Peek 1972, Adams 1990) or when resource distribution is manipulated (e.g., Ebersole 1980, Hixon et al. 1983). These experiments should be designed so that the quantitative relationship between territory area and the manipulated variable can be determined. I know of no experimental tests of the hypothesis that the behavior of a resident towards one of its neighbors depends on interactions with other neighbors. This could be tested by monitoring shifts in boundary positions following removal of residents. If boundary positions are interdependent, then when neighbors expand their territories into the vacated region, they should also lose territory along boundaries with surviving neighbors.
A third approach is to build predictions of spatial variation in fighting behavior from details of the behavioral mechanisms, rather than by describing gross characteristics mathematically. This approach is exemplified by White et al.'s (1996) mechanistic model of wolf pack territories. For ants, one could employ individual-based models of pheromone trail systems to produce branching patterns that change with the distribution of resources and competitors (e.g., Crist and Haefner 1994, Haefner and Crist 1994). Thus, the decline of aggressive pressure with distance from the nest can arise from a description of the movement and density of the ants, rather than being assumed to fit a simple functional form.
Neighborhood models of territory size can be extended readily to incorporate assumptions other than those adopted for S. invicta. For example, the model can be modified to include resident movement (e.g., [ILLUSTRATION FOR FIGURE 3D OMITTED]), mosaics that do not completely fill the available habitat, overlapping home ranges, heterogeneous environments, time lags in responses to neighbors, and constrained patterns of movement, such as branching trail systems. Since the habitat is represented as a fine grid of cells, it is easy to represent heterogenous resources or topographical features that may influence territory shapes (Reid and Weatherhead 1988). Finally, neighborhood and optimality approaches may be combined by assuming that a resident within a mosaic of contiguous territories adjusts its fighting effort to achieve the maximum net benefit of territory defense. It remains to be determined whether including strategic decisions will improve the predictive ability of the model. The success of simpler rules for S. invicta shows that much of the variation in territories can be predicted using information that is easily acquired in natural conditions.
I thank W. R. Tschinkel for advice and commentary; P. Dietz for help with computer algorithms; M. Balas, J. Jaenike, D. McInnes and H. A. Orr for critically reading the manuscript; D. Cassill, T. Foster, B. Geary, T. Macom, D. McInnes, G. T Marks, and B. Womble for assistance in the field. I thank the St. Joe Land and Development Company's Southwood Plantation and Florida State University for the opportunity to conduct field work. This work was supported by a Fellowship in Science and Engineering from the David and Lucile Packard Foundation to E. Adams and by NSF grant BSR-8920710 to W. R. Tschinkel and E. Adams.
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AN ALGORITHM FOR PREDICTING TERRITORIAL BOUNDARY POSITIONS
Space is treated as a two-dimensional array of rectangular cells (Fig. 2). Each cell is assigned to the territorial resident that applies the greatest aggressive pressure at the center of the cell. When aggressive pressure depends upon territory area (e.g., Eq. 5), one must first estimate territory area before pressures can be calculated. The approach adopted here is to begin with initial estimates of territory area (Step 1), then to successively improve these through a series of passes over the entire array of cells. Inputs to the model include the position and size of each resident.
Step 1. Since the formula for aggressive pressure includes a term for territory area (Eq. 5), an initial estimate of each resident's area must be provided. The simplest rule is to assign each resident (l/n) x (the area of the habitat), where n = the number of residents. Howe, a better initial approximation may be provided by
[Mathematical Expression Omitted] (A.1)
where [A.sub.i] is the area of the ith resident's territory, [M.sub.i] is the mass of the ith resident, and Z is a scaling factor which can be adjusted to improve the initial guess. By choosing an appropriate value of Z, the time needed to find territory boundaries can be substantially reduced. For many of the examples discussed in this paper, Z = 0.5 has provided reasonable initial estimates.
Step 2. Cycle through each row and column of cells. For each cell, calculate the coordinates of the center of the cell. Consider each resident in turn and calculate the aggressive pressure applied by that resident at the cell center (e.g., by Eq. 5). The cell is assigned to the resident applying the greatest aggressive pressure. On subsequent passes through this step, a count is made of the number of cells for which ownership changes.
Step 3. The area assigned to each resident is likely to differ from the initial estimate of area used to calculate aggressive pressure in Step 2. The territory area estimate for each resident is updated as follows:
New estimate of area for resident i
= [Lambda] (Number of cells assigned to resident i)
x (cell area)] + (1 - [Lambda])(Former estimate of area) (A.2)
where 0 [less than] [Lambda] [Lambda] 1. Thus, the new estimate is a weighted average of the former estimate of territory area and the newly measured territory area, based on assignment of cells. This procedure dampens changes in territory area from one iteration to the next, which prevents diverging oscillations in the estimates of territory size for particular residents. The appropriate value of [Lambda] depends upon other specifics of the model. In general, the lower the value of Y, the exponent in the equation for aggressive pressure to which distance is raised (e.g., Eq. 5), the lower the value that [Lambda] must take in order to prevent oscillations. As an example, [Lambda] = 0.5 was used for Fig. 3c.
Step 4. Repeat steps 2 and 3 until the number of cells changing ownership drops below a critical value representing a small fraction of the total habitat. After the first pass through the array, cells in the interiors of territories (those for which all neighboring cells have the same owner) can be assumed to retain the same owner. Cells on the margins of territories (those for which at least one of the adjacent cells has a different owner) are examined to see if the owner has changed. This reduces the time needed for each iteration and dampens large shifts in territory areas.
Step 5. Repeat step 4 for reduced cell sizes. For each size reduction, each cell is divided into four smaller cells that are initially assigned the same resident as the parent cell from which they are derived. This procedure allows a crude but rapid assignment of territories at the largest scale, then successive refinement at smaller and smaller cell sizes.
Step 6. Smooth the territory boundaries. The territories originally have jagged edges, which run along the rows and columns of rectangular cells. Boundaries are smoothed by tracing along the perimeter of a focal resident's territory. At fixed intervals along each vertical or horizontal face, a number of closely spaced points are examined along a line perpendicular to that face. The resident applying the greatest pressure at each point is determined. The boundary is drawn at the outermost point owned by the focal resident. The territory is treated as a polygon connecting boundary points located in this fashion.
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|Author:||Adams, Eldridge S.|
|Date:||Jun 1, 1998|
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