An empirically based simulation of group foraging in the harvesting ant, Messor pergandei

Journal of Theoretical Biology 340 (2014) 186–198

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An empirically based simulation of group foraging in the harvesting ant, Messor pergandei Nicola J.R. Plowes a,n, Kai Ramsch b, Martin Middendorf b, Bert Hölldobler a a b

School of Life Sciences, Arizona State University, P.O. Box 874501, Tempe 85287, AZ, USA Universität Leipzig, Fakultät für Mathematik und Informatik, Institut für Informatik, PF 100920, D-04009 Leipzig, Germany

H I G H L I G H T S

G R A P H I C A L

A B S T R A C T


Empirically based group model of foraging interactions in column foraging ant.
Messor pergandei ants balance column length with costs of aggressive interactions.
Influence of behavioral strategy, nest distribution and density on foraging dynamics.

art ic l e i nf o

a b s t r a c t

Article history: Received 16 November 2012 Received in revised form 12 July 2013 Accepted 17 July 2013 Available online 24 August 2013

We present an empirically based group model of foraging interactions in Messor pergandei, the Sonoran desert harvesting ant. M. pergandei colonies send out daily foraging columns consisting of tens of thousands of individual ants. Each day, the directions of the columns may change depending on the resource availability and the neighbor interactions. If neighboring columns meet, ants fight, and subsequent foraging is suppressed. M. pergandei colonies face a general problem which is present in many systems: dynamic spatial partitioning in a constantly changing environment, while simultaneously minimizing negative competitive interactions with multiple neighbors. Our simulation model of a population of column foragers is spatially explicit and includes neighbor interactions. We study how different behavioral strategies influence resource exploitation and space use for different nest distributions and densities. Column foraging in M. pergandei is adapted to the spatial and temporal properties of their natural habitat. Resource and space use is maximized both at the colony and the population level by a model with a behavioral strategy including learning and fast forgetting rates. & 2013 Elsevier Ltd. All rights reserved.

Keywords: Group model Self-organized conflict Territoriality Animal contests Resource exploitation

1. Introduction Many animals defend territories, including social insects such as bees (Wainwright, 1978; Breed et al., 2002), ants (Hölldobler and Wilson, 1990, 2009), and termites (Adams and Levings, 1987). n

Corresponding author. Tel.: þ 1 480 727 9434; fax: þ 1 480 965 6899. E-mail addresses: [email protected] (N.J.R. Plowes), [email protected] (K. Ramsch), [email protected] (M. Middendorf), [email protected] (B. Hölldobler). 0022-5193/$ - see front matter & 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.jtbi.2013.07.014

Social insects are particularly interesting because territory boundaries arise from distributed decision making processes. In addition, partitioning of space between neighbors emerges from decisions made by multiple groups. Classification of territoriality in ants is based on temporal and spatial components (Hölldobler and Lumsden, 1980). When a territory persists over space and time, it is referred to as “absolute,” e.g., Adams (1998, 1990), Brian et al. (1967), Brian and Elmes (1974), Plowes and Adams (in preparation) but when the territory is transient, it is referred to as “spatiotemporal.” Spatio-temporal territories are those in which only the

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actively foraged area is defended against conspecifics e.g., Hölldobler (1974), Whitehouse and Jaffe(1996), Hölldobler (1981). The large majority of models of territoriality in social insects have described absolute territoriality e.g., Hölldobler and Lumsden (1980), Adams (1998, 1990). In this paper we describe a group level simulation model of spatio-temporal territoriality in Messor pergandei, a common harvesting ant in the Sonoran desert (Tevis, 1958; Gordon, 1978; Wheeler and Rissing, 1975; Rissing and Wheeler, 1976; Rissing and Pollock, 1989). Colonies of M. pergandei send out daily foraging columns consisting of tens of thousands of ants. Each day, colonies can change the direction of columns e.g., Wheeler and Rissing (1975), Rissing and Wheeler (1976), Rissing and Pollock (1989), and when neighboring columns intersect, fights break out between colonies (Wheeler and Rissing, 1975; Went et al., 1972) (Fig. 1). A simple self-organized, individual behavior based model describing the formation and movement of columns in M. pergandei was developed by Goss and Deneubourg (1989). The major premise behind this model was that foraging columns ‘rotated’, like the hands of a clock, on subsequent foraging bouts (Wheeler and Rissing, 1975; Rissing and Wheeler, 1976; Rissing and Pollock, 1989; Went et al., 1972; Bernstein, 1975). In this model, individual behaviors incorporated include foragers laying trails when returning to the nest, and foragers following a simple rule of choosing the most heavily marked trail when leaving the nest. The foraging area is divided into sectors which start out with similar abundances of food. Positive feedback via differences in pheromone concentration results in increased recruitment to one sector. As a sector′s food supply diminishes through foraging, the relative abundance of preferred food types is higher in neighboring sectors, and foraging shifts to the adjacent sector. Thus the foraging trail shifts incrementally, sweeping across the foraging area as food supplies decrease, like the hands of a

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clock. The model does not include the effect of neighbor presence, nor can it explain how or why changes in column direction do not follow a predictable clockwise pattern of movement (Plowes et al., 2013) or what causes colonies show preferences for certain foraging locations (Gordon, 1978; Plowes et al., 2013). Columns in natural colonies extend and retract at each foraging bout (once or twice a day depending on the season) (Plowes et al., 2013). Alternative hypotheses for the function of foraging columns include both food acquisition and defense of resources (Ryti and Case, 1984, 1986, 1998). In many territorial ants, aggressive encounters have a role in establishment and maintenance of foraging areas (Adams, 1990; Hölldobler, 1974; Mabelis, 1979). Previous studies in M. pergandei suggest that colonies may establish trails in directions away from interactions with neighbors (Ryti and Case, 1998). In a study of a related group forager, Messor andrei, foraging direction does not appear to be affected strongly by the distribution of food sources (Brown and Gordon, 2000). Experimental data suggested that foragers were more likely to return to locations where aggressive encounters with neighbors occurred in preceding days (Brown and Gordon, 2000). In contrast, previous studies suggest that M. pergandei colonies switch from group foraging along a column with rich, dense resources, to individual foraging when resource density is low (Rissing and Wheeler, 1976; Bernstein, 1975, reviewed in Plowes et al. (2013)). There is some evidence that incorporating information about the behavior of neighboring colonies is important, including that nearest neighbors tend to forage in opposite directions, especially when resources are low (Ryti and Case 1986, 1998). Our preliminary observations in M. pergandei suggest that after two colonies have an aggressive interaction (Fig. 1), one of the colonies is likely to refrain from long trails, and to have individual foraging o4 m from the nest, or to establish trails in directions away from the interaction (Ryti and Case, 1986). The system which we model is novel for a number of reasons: first, it is derived from the behavioral repertoire of a species which shows highly dynamic spatio-temporal territories; second, our model describes the behavior of a group foraging social animal, and third, it is spatially explicit, incorporating neighbor interactions. We use our simulation model to test different foraging strategies, some of which might not be possible in ants or which have not been seen in ants. For instance, we explore how changing the nest density, nest distribution, or behavioral strategies can change the occurrence of aggressive encounters and foraging patterns at a population level. We can also study the spatial distribution of foraging from a different perspective: How intense is the resource exploitation regarding the total space occupied by a population of colonies, and what is the spatial distribution of exploitation? In addition, our simulations can run for long periods, providing more data than is possible in field observations.

2. Empirical data 2.1. Methods

Fig. 1. We used the presence of aggressive behavior between individual ants to ascertain whether columns had intersected or not and were interacting with each other. (a) Individual ants from different M. pergandei colonies can be induced to display aggressive behavior by placing individuals from neighboring nests in a small arena. (b) Naturally observed aggressive behavior includes immobilization followed by biting.

We collected field data in Sonoran Desert Scrub, on Arizona State Trust land (321 56'N, 1111 41'W), just north of Casa Grande, Arizona. A detailed habitat description can be found in Ode and Rissing (2002). Data used in this study was acquired from June 2008 through August 2009. Two dominant ant species are found at the site: M. pergandei and P. rugosus. Nests of P. rugosus were primarily located outside the study area. In 2009, every colony of M. pergandei found in a 1 ha plot was mapped and marked, allowing us to calculate the density and distribution of nests (Fig. 2). M. pergandei is a monodomous species: the entire colony uses a single nest entrance. Monodomy was confirmed using behavioral tests where workers from

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Table 1 The following data includes all pairs of observations where we could evaluate a colony′s behavior on two sequential days. Behavioral transition is a description of the foraging behavior observed on the consecutive days: SF ¼ column(s), AB¼ fighting recorded, NC¼no foraging/fighting (at most there was foraging which was classified as diffuse/individual foragers to 2 m from the nest). For example, the percentage of transitions from a column to no foraging was 7%, whereas the percentage of transitions from fighting to no foraging was 34%.

Fig. 2. The location of M. pergandei nests (black circles) at the Casa Grande field site during observations made in 2009. The corners of the study area are marked in open squares. The line between open squares shows the original western boundary of the field site. The line between open triangles indicates the adjusted boundary, which removes space unoccupied by M. pergandei nest.

neighboring nests were placed into an arena- in all cases workers behaved aggressively to workers from neighboring nest entrances. In June 2009, we followed the foraging behavior of every colony in a 1 ha plot for 18 days. In addition to the compass direction of the column, the length of the column, and overall column topology, we searched for instances of aggressive behavior at nests, along columns and within foraging areas.

Behavioral transition

Number of observations

% Of transitions

SF-SF SF-AB SF-NC AB-SF AB-AB AB-NC NC-SF NC-AB NC-NC

436 27 35 24 9 17 39 9 43

88 5 7 48 18 34 43 10 47

mortality and dismemberment of 100′s–1000′s of individuals, and the interactions affected colony behavior on subsequent days (Table 1). The colonies at Casa Grande which experienced aggressive encounters with neighbors were not located closer together than colonies which did not interact (when the size of Voronoi cells generated by the nest locations is compared)(T-test, T ¼1.553, p¼ 0.126). The colony observation data was used to generate a matrix of behavioral transitions that colonies make from one day to another (Table 1). For example, 88% of colonies which are successfully foraging along a column will forage the next day, whereas colonies which are not foraging are about equally likely to continue to not forage or to form a column the next day. The behavioral transitions were used in the implementation of the model (Section 3.1.1).

2.2. Field results In June 2009, the mean nearest neighbor distance (NND) (Clark and Evans 1954) between nests in the study plot was 8.9 m, and the mean distance to the 3 closest neighbors between nests was 12.7 m. The mean trail length, for all trails, was 11.8 m (n ¼558 trails), indicating potential overlap of foraging areas. Trails were variable in length and shape, but are consistently straight between the nest and foraging areas. The end of a trail is easily recognized by the behavior of individual ants- instead of racing forward in a straight path, they veer off and begin searching for seed. Trails were categorized as: individual, narrow, medium or broad, based on the width of the path of worker ants 2 m away from the nest entrance. Individual foraging is when individuals forage randomly in all directions around the nest (n ¼ 79, 13% of observations). Narrow trails had ants walking in a pathway 10–30 cm wide (n¼ 232, 39% of observations), whereas medium trails were 30–100 cm wide (n ¼190, 32% of observations). Broad trails were more than 1 m wide at 2 m from the nest entrance, with ants fanning out rapidly (n ¼89, 15% of observations). Colonies switched from individual foraging to broad fan-like trails to narrow trails during different foraging bouts. There is a trade-off between length and shape, in that longer trails are more likely to be narrow (n¼ 523, Kruskall–Wallis: H¼88.04, d.(f).¼ 3, po0.001). Of all observations (total n ¼899 over 90 colonies), 7.7% included observations of overt aggression. Overt aggression fell into three categories: dyads/groups, tussles, or posturing. Dyads/ groups involve grappling and biting between 2 or more ants and last10 s to several minutes. Tussles are shorter bouts of 1–2 s where two ants circle each other, snapping and biting, but breaking away. Posturing ants stand still and very alert, and are easily aroused by passing antennation to spin and snap their mandibles. Approximately 39% of the colonies were involved in aggressive encounters during the observation period. Aggression resulted in

3. Simulation model The following questions were the focus of the simulations: Do natural M. pergandei colonies experience higher or lower levels of interactions with neighboring colonies than expected? In other words, do colonies actively avoid aggressive confrontations with neighboring colonies? We compared the frequency of aggressive interactions between observed levels at Casa Grande, and simulated scenarios with different types of learning behavior. Simulated nest distributions either had the same density and Clark–Evans ratio (Clark and Evans, 1954) as the Casa Grande site, or the same density, but randomly distributed nests. How does the distribution of nests and foraging behavior of M. pergandei compare with an optimizing strategy? We asked if the distribution and behavior of Casa Grande nests minimized variance in patch use and maximized the average length of those foraging columns that did not include aggressive encounters. We hypothesized that M. pergandei colonies tradeoff the costs of establishing longer foraging columns with the benefits gained by that effort. We compared the effectiveness of learnt behavior and random behavior while varying both density and distribution of nests. We hypothesized that simulations most closely matching the natural field conditions would more thoroughly exploit resources and have higher mean lengths of successful foraging columns. 3.1. Setup We simulated M. pergandei colonies by generating nest locations and foraging columns in the event based simulation framework Repast 3 (North et al., 2006) (see Table 2 for parameters and

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Table 2 Parameters and variables used in the simulations. lcg loff l

S

Length of foraging column (m), generated from empirical data Length (m) added to mean and mode of the distribution of column lengths Mean length (m) of successful foraging columns of all nests during a time period

f

Sum of all lengths of successful foraging columns of all nests during a time period divided by the number of days in the time period

l s ρ η ξ ξN G μ α αpref

Side length (m) of the square simulated area Density of nests in the simulated area, normalized by the area of Casa Grande. The observation site in Casa Grande contained 125 nests. Parameter generating nest distributions with particular Clark–Evans ratios via algorithm by Johnson et al. (1994). Casa Grande had a Clark–Evans ratio of 0.96. Exploitation, i.e., the number of visits per cell over a time period. The area (s2 ) was divided into 90,000 cells. Exploitation deviation (standard deviation of number of visits of a cell divided by the average number of visits in the cells of the grid) Forgetting rate. From one step to another, it is how fast the directional preference is relaxed (i.e., how fast ϕ approaches 0), as a fraction, e.g., G ¼ 0.1 is a faster rate of forgetting than 0.01. Memory variable, initially set at 0 (no preference). When an interaction occurs with another nest, and the colony establishes a preference for a new direction, μ is set to 1. Direction taken by a foraging column during a foraging bout. The direction which is 1801 from the most recent interaction with another colony.

variables used). The framework provides basic components used in the simulation: data storage, scheduling, and visualization. In our simulations, we define a colony as the queen, brood, alates, and worker ants found in a single nest. Simulated nests are modeled by describing the location of nest entrances and the formation of foraging columns. Each simulation step corresponds to one foraging bout or day. The simulated nests were placed in a square region of area s2 , which corresponds to the size of the area of the sampling plot in Casa Grande (Fig. 2). We varied the nest density by adjusting the number of simulated nests in a virtual region with an area of the same dimensions as the study site at Casa Grande. We used the algorithm developed by Wu et al. (1987) (see Appendix A) to generate nest distributions in our simulations having the Clark–Evans ratios with the following values: 2.0 (regular), 0.96 (Poisson distributed, the distribution of nests found at Casa Grande study site) and 0.82 (clumped). At every simulation step, each simulated nest is able to send out a column of ants to forage in the area around the nest. This is modeled by a straight line (foraging column) emerging from the nest and leading to a fan-shaped zone at the end of the line (foraging fan). While there is considerable variation in the size and shapes of foraging fans in nature, for the purposes of simplicity, we model the foraging fan as a circle segment with a radius of 1.5 m and an opening angle of 1201. The lengths of the foraging columns (lcg ) are distributed according to a gamma distribution fitted to the observed frequency of foraging columns lengths for 27 nests at Casa Grande which never interacted with other nests or columns during the 18 day study period (lcg A Γðshape ¼ k, scale ¼ ΘÞ with k :¼ 4:11 and Θ :¼ 2:85) (Fig. 3). In order to study the influence of the length of the columns, an offset parameter, lof f , is added simultaneously to the mean and the mode of the gamma distribution resulting in the parameters shape ¼ knðlof f =ðknΘÞ þ 1Þ and scale ¼ Θ for the gamma distribution. Every colony in the natural field site was classified in one of three classes each day: either no column was formed (NC), a column resulting in successful foraging was formed (SF), or a column was formed but aggressive behavior was observed (AB). To model days without foraging columns, we calculated the transition frequencies, i.e., the frequency to belong to one of these classes depending on the class of the previous day. In our simulations, aggressive behavior is a result of colony interactions; therefore we pooled the transitions from each class in any of the classes SF and AB (Table 1). Simulated colonies follow these same rules: at each simulation step a colony either forms a foraging column or does not forage, depending on whether it was classified as SF, AB or NC at the previous time step.

Fig. 3. The frequency of lengths of columns observed at the 27 nests at Casa Grande which did not have aggressive interactions during 18 days. The gamma distribution (shape¼ 4.11, scale ¼ 2.85) fitting generated from this data was used to determine column lengths used in the simulations.

3.1.1. Evaluating intersecting columns As M. pergandei individuals react very aggressively toward nonnestmates, the model assumes that when foraging columns intersect, fights immediately break out, and therefore at least one foraging column ends prematurely at the point of intersection. To simulate foraging columns intersecting, the following algorithm is used (see Appendix A for pseudocode). First we assume that the length of a foraging column scales with the time the ants need to reach the end of the column after emerging from the nest, that foraging columns elongate at the same rate, and that all colonies begin foraging at the same time. When two foraging columns intersect, the colony which has the shorter distance from the intersection to the nest is called the ‘defender’, the other colony is referred to as the ‘attacker’. The algorithm orders all column intersection events chronologically, by evaluating the distances of the attackers from their corresponding intersection. Beginning with the earliest intersection event, the algorithm determines the attacker in each remaining crossing event and truncates the attacker′s foraging column length at the intersection. Both the attacker and the defender are marked as being in a fight at that simulation step. That means later ‘events’ of the attacker′s foraging column are removed from the ordered list of crossing events because they do not happen (see Appendix A). This algorithm is designed to produce a deterministic and consistent solution of all crossing events with only pairwise interactions in each simulation

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step. All foraging columns which are not part of an intersection are regarded as ‘successful foraging columns’ at that simulation step. 3.1.2. Behavioral rules In the simulation model, two behavioral rules are used to determine the direction of foraging columns: random behavior and learned behavior. The column direction determined by “Random behavior” (R) is made by picking a compass direction randomly from a uniform distribution. With “Learned behavior” (L), if there has been no interaction with a neighbor, i.e., without previous intersections, the column direction is formed as in the random behavior. If there is an interaction with a neighbor a simulated nest will remember the direction of that interaction and will form the next foraging columns in opposite direction αpref , thus evading continued conflict. Directly after the interaction the next foraging columns will be close to that desired direction αpref but as time elapses the column directions increasingly deviate from the desired direction, and thus simulated nests forget the experienced interaction. This is controlled by a forgetting rate (G A ½0; 1). Formally the direction of foraging columns follows the dynamics: αðt þ1Þ ¼ αpref ðtÞ þ Uðð1μðtÞÞ; 1μðtÞÞπ

ð1Þ

μðt þ 1Þ ¼ GμðtÞ

ð2Þ

where αðtÞ is the direction of a foraging column in radians and μðtÞ A ½0; 1 models the memory for the direction of the last intersection at simulation step t. Uðf rom; toÞ is the uniform distribution over the interval ðf rom; toÞ. The initial condition is μð0Þ ¼ 0 and when a new desired direction αpref is set at simulation step t, the memory variable is set to μðtÞ ¼ 1. We assume that when ants respond to local rules, group level behaviors emerge as a result, and that the group level behavior is predictable given the individual behavior. Thus in our model individual level behaviors are collapsed into the group level behavior. This approach does not incorporate how individual level decisions translate to group level decisions, for example behavioral thresholds (Greene and Gordon, 2007), quorum sensing (Pratt, 2005; Pratt and Sumpter, 2006), counting number of opponents or strength including head counting, caste polling, and queue flooding (Hölldobler and Lumsden, 1980; Adams, 1990; Lumsden and Hölldobler, 1983). To simplify our approach, we assume that all colonies are equivalently matched opponents, i.e., in our simulations behavioral rules are independent from the result of an encounter in terms of who ‘wins’ the fight. 3.1.3. Use of space, and measures of successful foraging To quantify how colonies use space, l f (the sum of lengths of all foraging columns which were not part of an intersection divided by the number of days in the time period) was calculated for each simulated colony. The variable l f is the length of the successful foraging columns of a nest per day and thus measures the size of uninterrupted flow of resource acquisition and the spatial influence of the colony simultaneously. We hypothesized that the length of foraging columns chosen by ant colonies is fairly constant, and yields primarily from geometric properties of the distribution of simulated nest locations. To test this, we measured the mean length l s of the successful foraging columns of all evaluated nests in a time period. That is the sum of lengths of all foraging columns which were not part of an aggressive interaction, divided by the number of successful foraging columns in that time period. Our hypothesis is supported if l s does not deviate from the mean length of foraging columns measured in the colonies of Casa Grande which had no interactions. The food exploitation of the area surrounding the nest might also be an important criterion when colonies live together with high density in a desert with few food sources. Therefore we

divided the foraging region (s2 Þinto a grid containing 90,000 cells (dividing s by 300) and counted how often a specific cell would have been visited by foraging ants. At each simulation step, for all foraging columns we tested if a cell was lying in the foraging fan of that foraging column. If it was, the counter for that cell was increased by 1. For a time step we denote the number of visits averaged over the cells of the grid by AVðξÞ and its standard deviation by SDðξÞ. The standard deviation of visits divided by the average number of visits of the grid is called normalized exploitation deviation ξN ¼ SDðξÞ=AVðξÞ and is used to measure the spatial food exploitation. The three measures: l s , l f and ξN , are scale free, i.e., values of each measure obtained from simulations with different length offsets, nest densities or nest distributions can be compared to each other. 3.1.4. Adjustments to the model In 2009, there was a portion of the sampling region in the Casa Grande field site where there were no M. pergandei nests (western border region in Fig. 2). This decreased the measured value for the density but had no effect on the Clark–Evans ratio as only internest distances are used for the Clark–Evans ratio. In order to get a realistic measure of density for the model, we moved the border of the region sampled in the field to remove unoccupied space without excluding any nests (see Appendix A). This approach is more biologically realistic than keeping the unoccupied space in the Casa Grande distribution which would result in fewer simulated intersections in the western border region, because that area contained P. rugosus colonies, which may interact with M. pergandei colonies competitively. We only evaluated nests in an area of size s2 , but ran the simulations on an area large enough to have the same degree of interactions at the border of the evaluated region as inside the evaluated region (see Appendix A). To measure the distribution of the spatial exploitation, we ran our simulations until the normalized exploitation ξN had converged to a stable state, i.e., a state where longer simulation with the same set of colonies will not change the distribution of visits in space (see Appendix A).

Fig. 4. The percentage of colonies which showed preferences for foraging in particular sectors (Rayleigh test with po0.05) for natural colonies at Casa Grande (C.G.), or simulated colonies with the same density and distribution as Casa Grande: random behavior (R), learned behavior (L) with values of G ranging from 0.1 to 0.001. For each parameter combination, 30 repetitions were run. Each run started with a 365 day period to stabilize foraging patterns, followed by an 18 day period during which data was gathered for comparison with the Casa Grande colonies.

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3.2. Simulation results and analysis In the field observations at Casa Grande, approximately 25% of colonies showed a statistically significant preference to forage in particular sectors around their nest (i.e., a Rayleigh test performed on the compass direction of columns showed rejection probabilities of p o0.05). Simulated colonies (with the same density and distribution as the Casa Grande site) were followed for an equivalent 18 days (Fig. 4). Simulations run with random behavior had an average of 7% of colonies showing directionality, well below the range for natural colonies. Simulations run with learned behavior had higher proportions of colonies showing significant preference for direction- with the proportion increasing as G (the forgetting rate) decreased. Increasing the value of the forgetting rate (G) resulted in increases in all measures of interactions: the proportion of days with no columns averaged over colonies, the number of encounters (intersections between colonies) divided by the number of days with columns averaged over colonies, and the number of encounters divided by the length of the 18 day period averaged over colonies (Fig. 5). There is a consistent decrease in the measures of interactions with learned behavior runs compared to random behavior, in that the more colonies ‘learn’ or can respond to their neighbors, the more successfully intersections, i.e., conflicts, are avoided. The effect of nest distribution on

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interactions is negligible, which suggests that the differences in behavioral mode are robust. The field data does not match simulated data under a behavioral rule of learning, with the same density and distribution as Casa Grande. There are several explanations for showing higher rates of encounter in the simulations. First, the simulations used empirically derived behavioral transition frequencies (Table 1), which, from a mathematical standpoint, do not produce the same behavior as observed in real ants, because they are local probabilities sampled from within a limited time frame. Second, we used a simple model of interaction (suppressing successive columns in both colonies), but there are other biologically reasonable possibilities, such as only the ‘late’ colonies experiencing column suppression, or that M. pergandei colonies use different strategies to change column direction. To explore the effects of spatial parameters on the model, we first ran simulations representing a “null” model: random behavior with randomly distributed nests, with varying nest densities in an area of size s2 . With increasing offsets (lof f ), the mean length S of successful foraging columns (l Þ becomes increasingly shorter than the ‘ideal length’ (represented by function f ðxÞ ¼ lCG þ x), i.e., the length in the absence of intersections (Fig. 6). As the offset value increases, the probability of hitting another column increases, which subsequently results in shorter successful (non interaction) columns. This was true in all parameter configurations we tested (see also Fig. 7).

Fig. 5. The following figures compare measures of interaction between random behavior (R) and learning behavior (L, values of G¼0.001, 0.01, 0.1) with natural field data (O). Simulated nest distributions with exactly distribution as the Casa Grande site have the subscript “cg”, whereas simulations with the same nest density, but randomly distribution, have the subscript “r”. The smaller the value of G (forgetting rate), the lower the rates of interactions, for both simulations run with nest distributions matching Casa Grande (Clark–Evans ratio¼0.96) and random distribution. High values of G correspond with a faster rate of forgetting the previous encounter. (a) ■¼proportion of days where there were no columns averaged over colonies. (b) ●¼the number of encounters, divided by the number of days with columns averaged over colonies, over 18 day period (c) ○¼the number of encounters, divided by length of the 18 day period averaged over colonies. For all simulations, lof f ¼0, N¼ Casa Grande, 30 repetitions were run. Each run started with a 365 day period, followed by an 18 day observation period. Mean and SE bars shown.

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Fig. 6. Simulations run with Clark and Evan's ratio of 0.96, but different nest densities, following a strategy of random behavior. For the nest densities (a) N ¼40, (b) N ¼ 125 and (c) N ¼ 150. Phase I, i.e., the phase where longer foraging columns result in longer successful foraging columns and exploitation deviation is at the level of saturation, ends at (a) lof f Z 15 m, (b) lof f ¼ 0 m and (c) lof f ¼ 2:5 m, indicating that real ants at Casa Grande (N ¼125, lof f ¼ 0 m) are adapted to the spatial properties of their S environment. Solid black line ●: exploitation deviation ξN (right axis), solid black line ▲: Mean length of successful foraging columns l (m), solid black line ■: successful f foraging legth averaged over the days of successful foraging l (m), dashed black line: ideal length of foraging column ðy ¼ x þ lCG Þ. Mean and SE for 30 simulations shown.

As the offsets are increased, three phases with respect to our measures can be described qualitatively. The exact limits of these phases vary slightly for different parameter values so we will define them according to the limits in the models with parameters closest to the real ants (N ¼125 and Clark–Evans ratio 0.96): (1) Phase I (lof f A ½5; 0m): Increasing the length of foraging columns also increases l f ; the sum of all lengths of successful foraging columns of all nests over a time period. Both food income and spatial influence is therefore increased. This also results in a lower exploitation deviation ξN. (2) Phase II (lof f A ½2:5; 10 m): Increased frequency of intersections as the offset increases reduces the length of successful foraging columns per day. The exploitation deviation (ξN) drops to a level where every location is visited fairly often. Variation in exploitation deviation results from the subsequent changing of directions by foraging columns. (3) Phase III (lof f A ½15; 20 m): The high frequency of intersections resulting from longer foraging columns reduces the number of successful foraging columns. Therefore the length of successful foraging columns per simulation step decreases, and the potential increase in spatial influence does not compensate

for the lower number of successful foraging columns. An ever larger number of cells are not successfully visited over time, which leads to a higher exploitation deviation (ξN). Changing the density of nests does not change the general result, but merely scales the spatial properties of the setup. Nest density controls the column length offsets where qualitative changes in the measures of exploitation appear (Fig. 6). Thus increasing nest density or increasing the column length offsets have similar effects on the definitions of Phases I, II or III. Column lengths of real ant colonies appear to be adapted to the distribution of nests. With a lower nest density, colonies can form longer columns (lof f Z 5 m) without being interfered with by neighbors. With the density measured in Casa Grande this is not the case, the peak for l f is already at lof f ¼ 2:5 m. Even under a model of random behavior, the colonies operate at the limits of the spatial properties generated by the distribution of competing colonies. Extending the length of foraging columns will not increase the potential food income because interference competition becomes more frequent. The exploitation of the space is similar to the minimum values with higher offsets which can be explained by the fact that the ants do not have the time (12,000

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Fig. 7. Simulations run with 125 nests, but different Clark and Evan′s ratios, following a strategy of random behavior. The nest distributions have Clark and Evan′s ratio of (a) 2.0, S (b) 0.96 and (c) 0.82. Solid black line ●: exploitation deviation ξN (right axis), solid black line ▲: Mean length of successful foraging columns l (m), solid black line ■: successful f foraging length averaged over the days of successful foraging l (m), dashed black line: ideal length of foraging column ðy ¼ x þ lCG Þ. Mean and SE for 30 simulations shown.

days) to homogeneously explore the space but need to settle for a suboptimal solution. However, since the distribution of one of M. pergandei′s primary food resources, creosote (Larrea tridentata), is not homogeneous as assumed in our model, ants might not need to sample the whole space but instead only visit discrete locations in the neighborhood of the nest. In this case fewer days/simulation steps are necessary to reach the stable state of the exploitation and it is possible that shorter column lengths already minimize the exploitation deviation (ξN). Both hypotheses can explain why column lengths in M. pergandei are optimal regarding the spatial distribution of their nests for individual colonies as well as a population of colonies. We tested the robustness of the random behavior model under different distributions of nests, while maintaining the density at Casa Grande (125 nests/s2 ) (Fig. 7). The general relationship between offsets (lof f ) and measures of foraging success (ξN or l f ), is not affected by changes in nest distribution. Increasing the regularity of spacing minimizes the exploitation deviation and minimizes the number of intersections, thus maximizing the S average length of successful foraging columns (l f and l ).

This effect is reversed when nests are more clumped than at Casa Grande. Next we studied learning behavior by fixing the distribution and density of nests to the values in the Casa Grande data (N ¼125, Clark–Evans ratio ¼ 0.96). An increase in the length of foraging columns results in a roughly linear increase in the rates of intersections with other columns, for both random and learned behavior (Fig. 8). In Phase I (lof f A ½5; 0 m), lower values of G also result in a lower proportion of intersections. This pattern changes in Phase II and Phase III, where frequent intersections (  50%) negate the benefits of avoidance given by the learning behavior model. There is little variation in exploitation deviation (ξ) between the different behavioral models over a range of length offsets (lof f ) (Fig. 9a). The behavioral model does not influence the shape of the curve, which has a minimum (for all behavioral models) at lof f ¼ 10 m. With longer foraging columns (lof f Z 5 m), decreases in relative exploitation deviation (ξN =MEANðξN Þ) gained by learning behavior are lessened, and by lof f Z 10 m there is no difference between learning and random behavior (Fig. 9b). This supports an

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Fig. 8. The effect of changing lof f on the proportion of all columns involved in aggressive encounters. The number of nests is 125 and the distribution was held constant with a Clark and Evan′s ratio of 0.96. Simulations were run with both random behavior (solid black line ○) and learned behavior with a variety of values of G (dashed black lines with ▽ G ¼0.1, ◆ G ¼ 0.07, ▲ G ¼ 0.01and ● G ¼0.001). Mean and upper bar of SE for 30 simulations shown.

lof f ¼ 0 m as a reasonable choice for real ants. For lof f o5 and G Z 0:07 the relative exploitation deviation is smaller than for random behavior or learning behavior with G ¼ 0:001 (Fig. 9b). Thus learning behavior with a short memory favors more spatial exploitation than random behavior and learning behavior with a long memory. For lof f r 10 m, learning behavior results in higher values of l f (length of successful foraging column per simulation step) compared to random behavior for (Fig. 10). For higher offsets, learning behavior does not result in l f values different to random behavior. The length of successful foraging columns per simulation step is at a maximum when lof f  5 m regardless of the behavioral strategy. The lengths of successful foraging columns vary significantly between the two behavioral models: from random behavior simulations usually shorter columns emerge, whereas most simulations with learned behavior revealed longer columns (Fig. 11).

4. Conclusions Contrary to previous studies (Rissing and Wheeler, 1976; Went et al., 1972; Bernstein, 1975, 1971), M. pergandei colonies at Casa Grande do not show a predictable change in compass angle or a regular clockwise/anticlockwise pattern during the formation of foraging columns (Plowes et al., 2013). Foraging is affected, however, by interactions between foraging columns from neighboring colonies (Table 1). The function of columns is to guide workers to foraging grounds, where they harvest 10,000– 22,000 cal of food per day during the summer months (Gordon, 1978). The amount of food that a colony retrieves has direct fitness consequences in that well fed colonies produce more reproductive propagules (Ode and Rissing, 2002), and thus there is an imperative to minimize costly neighbor interactions. Our model demonstrates that natural colonies must employ behavioral mechanisms to decrease the number of interactions with neighbors (Fig. 5), thus increasing colony fitness. We proposed a simple behavioral mechanism by which colonies could maximize their foraging success- learning behavior. While the

Fig. 9. The effect of lof f on the exploitation deviation, ξN (the standard deviation of number of visits of a cell divided by the average number of visits in the cells of the grid). The number of nests is 125 and the distribution was held constant with a Clark & Evan′s ratio of 0.96. (a) Exploitation deviation varied little over a range of lof f . (b) There is little difference in relative exploitation deviation for lof f 410 m, but for lof f o 5, there are differences in relative exploitation deviation with behavioral model and values of G. Simulations were run with random behavior (solid black line ○) and learned behavior with a variety of values of G (dashed black lines with ▽ G ¼0.1, ◆ G ¼ 0.07, ▲ G ¼ 0.01 and ● G ¼0.001). Mean and upper bar of SE for 30 simulations shown.

mechanisms of choice of column direction in M. pergandei are still unknown, a manifestation of ‘learning behavior’ is biologically reasonable (Johnson et al., 1994). The decision to choose a particular direction could arise through positive or negative feedback. Positive feedback could include a competition between recruitment trails led by successful scouts (oligarchy) e.g., Greene and Gordon(2007), Lopes et al.(2004), Ruano and Tinaut (1999). Negative feedback may include avoidance of areas due to physical interactions with neighboring ants in columns or foraging fans or the detection of colony specific trail markings made by neighbors (as in Pogonomyrmex (Hölldobler et al., 2004)).

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f

Fig. 10. The effect of changing lof f on l , the average daily length of successful foraging columns. The number of nests is 125 and the distribution was held constant with a Clark–Evans ratio of 0.96. Results from simulations with random behavior (solid black line ○) and learned behavior with a variety of values of G (dashed black lines with ▽ G ¼ 0.1, ◆ G ¼ 0.07, ▲ G ¼ 0.01 and ● G ¼0.001). Mean and upper bar of SE for 30 simulations shown.

Fig. 11. There is a significant difference in the mean values of the length of successful foraging columns across behavioral models. Random behavior has significantly shorter foraging columns than all learned behavior except G¼0.09 and 0.03. Significant differences within each grouping are denoted by the letters a4b4c4d. Groupings are based on a one-way ANOVA (F (8261)¼ 8.997, p¼ 6.691-11) followed by a Tukey post-hoc comparisons test (95% CI). The number of nests is 125 and the distribution was held constant with a Clark and Evan′s ratio of 0.96.

Foraging columns in M. pergandei help colonies forage efficiently, by leading foragers quickly to an area where they can search for food individually. The imperative for speed comes from temperature constraints, which limit these ants to short bouts in the early morning or evening during the summer months (Gordon, 1978; Wheeler and Rissing, 1975). The mean length of successful columns in natural colonies is close to the value which in simulations maximizes exploitation, thus column length in natural colonies is optimized. The column length only marginally increases the length of successful columns, but increases the costs incurred by fighting. This is the case in simulations run with random behavior (in which colonies do not

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respond to neighbors), as well as in learning behavior, indicating that learning behavior alone does not allow colonies to escape from the general spatial properties of nest distribution. Our model does not explain how the length of a new foraging column is determined, nor do we know the biological basis for column length determination. The distribution of column lengths observed in the field, appears to be adapted to the nest distribution. In the simulations with learning behavior, aggressive encounters are used to indirectly communicate the positions of neighboring nests. Learning increases the performance of the population by avoiding unnecessary aggressive encounters by modifying the direction of foraging columns, but not column length. If the spatial distribution of colonies remains constant, then forming columns with lengths similar to those observed in the field results in the most consistent harvest for individual colonies and thorough exploitation of space for the whole population, given sufficient time. It is of great interest to know how the lengths of columns are determined in cases where there is no neighbor interaction. Given the density and distribution of nests at Casa Grande, the data from simulations suggest that the observed distribution of lengths of foraging columns in natural colonies result in a balance between aggressive encounters and exploitation of space. Models of army ant behavior (Franks et al., 1991) suggest that emergent behavior can result from external forces, such as the distribution of food, rather than solely on differences in individual behavioral rules. We expect that the directions of foraging columns in natural colonies are driven primarily by interactions with conspecifics e.g., Ryti and Case (1998), but that resource abundance affects other components of column dynamics such as frequency of directional change, column length, or increasing the angle between subsequent foraging bouts to more rapidly survey resources [reviewed in 23]. Other Messor species, such as M. capitatus, appear to be more sensitive to perturbations by neighboring colonies than food resources (Acosta Salmerón et al., 1985). The incorporation of learning in our simulations increases the foraging success by allowing colonies to avoid the direction of the most recent aggressive encounter. Simultaneously, for a range of forgetting rates, exploitation is increased. The optimal forgetting rate is relatively high indicating that fast forgetting works better than slow forgetting. Learning does not increase the length of the successful foraging columns, which means that it does not favor longer or shorter trails. Thus learning seems to provide a benefit for all column lengths. Column length is not a parameter that can be used to improve food income. Changing the direction of columns in response to neighbors, however, decreases aggressive encounters which results in increased exploitation at the level of the population. There are a large number of models for territorial behavior in social animals, which do not specifically apply to our system, but are informative regarding potential costs and benefits to incorporate in spatio-temporal models e.g., Adams (2001). For example, in Solenopsis invicta, the variation in the size and shape of absolute territories is largely determined by the relative sizes of colonies and the distances between nests (Adams, 1998). Some models are based on the economics of defense, where the boundaries of a territory balance the benefits of foraging in an area with the increased costs of transport and defense (Hölldobler and Lumsden, 1980). While Hölldobler and Lumsden (1980) model clearly illustrates the circumstances under which a colony would be expected to demonstrate absolute territoriality, trunk route defense, or spatio-temporal territoriality, their model does not explicitly account for the presence or actions of neighbors. Both cellular automaton models e.g., Edelstein-Keshet et al. (1995) describing the formation of trails by a single colony, and other modeling techniques show that changes in food distribution can affect group foraging patterns (Franks et al., 1991; Deneubourg

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et al., 1989; Adler and Gordon, 2003) but these models do not incorporate the effects of neighbors. Adler and Gordon(2003) model space use by colonies as depending, in part, on the behavior of neighbors. Their models were based on behavioral observations in the harvester ant Pogonomyrmex barbatus, which defend trunk trail territories (Hölldobler, 1974, 1976). They found that their individual and colony level models which did not incorporate conflict costs between neighbors predicted foraging area overlaps. M. pergandei has subtle differences when compared with P. barbatus. In P. barbatus, long lasting trunk trails function to partition space and reduce internidal hostility, resulting in higher nest densities than e.g., P. maricopa which does not form trunk trails (Hölldobler, 1976). M. pergandei′s foraging columns tend to be much more ephemeral than trunk trails, sometimes heading in completely different directions on subsequent days. Thus the M. pergandei system is more dynamic, with colonies responding more frequently to both the environment and neighboring colonies. Our model could be extended to other behavioral systems by incorporating different behavioral strategies and decision-making processes. For example, there are different ways in which colonies may react to enemies (Hölldobler, 1974; Brown and Gordon, 2000; Adams, 1994; Hölldobler, 1983; Schilder et al., 2004). In some species, individuals are attracted by conflict. Colonies of M. andrei are more likely to forage in areas where there had been aggressive encounters on previous days (Brown and Gordon, 2000). Encounters far away from the nest may be evaded, and encounters near the nest might be actively supported. A similar pattern is also seen in Pogonomyrmex (Hölldobler, 1976). Encounters with neighbors provide information about other colonies and the combination of encounters and the strategies used to react on them form a type of communication between colonies (Hölldobler, 1981; Gordon, 1989; Sanada-Morimura et al., 2003; Gill et al., 2012; Dimarco et al., 2010). The effect of these species-specific responses can be tested in simulations through the implementation of different decision rules. Messor pergandei colonies face a general problem which is present in many biological systems: how to compete for resources in a dynamic environment, where both aggression and coexistence are possible options. Avoiding a fight or anticipating a fight can be advantageous, because resources are saved. Avoiding dangerous regions (or directions) maximizes the exploitation of food sources not only on the colony level. The strategies ants use to solve the resource competition problem can inspire the design of algorithms in a variety of applications. For example, swarms of cleaning robots can use these strategies to control the communication between individual robots. The parallels to our model include food/resources corresponding to ‘dirt’ on the floor, and nests are represented by the charging stations which each robot must return to periodically (Lee et al., 2011). The path, the robot walks or drives to clean the area does not necessarily have to be straight as in M. pergandei, there are more sophisticated solutions to the field of patrolling. The problem an ant inspired approach might solve, is to dynamically assign the space to be cleaned to the robots without global knowledge or control such that the exploitation, i.e., a homogeneously clean floor, is maximized and the number of the events of two robots meeting, which means unnecessary cleaning, is low. Avoiding territories where other robots were met should give a benefit in this goal. A similar system which simulates mobile robots searching an area for items and returning them to a central location, while avoiding collisions between robots, found that when the density of robots was high, the algorithm performing best let robots turn in the opposite direction when facing another robot (Rosenfeld et al., 2008). Our model suggests that extending this setup by assigning

each robot a unique deposition location will produce similar results. The parallels between collective behaviors in ants and algorithmic problems in robotics control mean that further analysis in either domain will also increase knowledge in the other domain. The simple example of cleaning robots can also be generalized to an abstract system where agents have to perform tasks which involve coverage of a (2-dimensional) space, e.g., multi-robot patrolling (Portugal and Rocha, 2011). Ant inspired algorithms have been successful in creating robotic swarms with efficient foraging and low energy use, but can be constrained by interference between robots (Krieger et al., 2000). Our model provides a way for individual robots (column analogs) to minimize interactions further. To conclude, our simple empirically based model of column foraging provides a theoretical framework for how ants solve a complex spatial partitioning problem at both the colony and population level. In addition, the results from the learning behavior model are robust- generating similar exploitation patterns across different densities and distributions of nests. This system has strong potential application in the coordination of swarms in multi-robot systems e.g., Krieger et al. (2000) in addition to its utility to test hypotheses of the effects of behavioral mechanisms in social insects. Acknowledgments We would like to thank the Social Insect Group at ASU for hosting the Biomimicry conference, which led to our collaboration, and the Ohio State University Mathematical Biosciences Institute for hosting a workshop on “Insect Self-organization and Swarming”. We also thank Robert Johnson for helpful suggestions concerning the ecology of Messor pergandei, and two anonymous reviewers for their contribution to the manuscript. Supported by funding from Arizona State University, the Arizona State University Foundation (B.H.) and the Human Frontier Science Program Grant RGP51/2007 (M.M/K.R.). Appendix A A 1. Generation of nest distributions The algorithm developed by Wu et al. (1987) can generate nest distributions controlled by parameter η. To match particular Clark– Evans ratios for different η we let η A f1=eg [ f10i=10 ji ¼ 4; 3; …; 15g and created 100 nest distributions each. Then we measured the Clark–Evans ratios and used the resulting map as empirical correspondence between η and the Clark–Evans ratio. A 2. Simulation All simulations providing results for this paper executed the program simulation_step() with the following pseudo code for each iteration: NAME simulation_step() INPUT nests: list of nests with locations in the 2D plane BEGIN columns ¼{} FOR EACH nest IN nests DO IF active(nest) THEN column ¼createColumn(nest) add(columns, column) END IF DONE

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events ¼{} FOR EACH column1 IN columns DO FOR EACH column2 IN columns DO IF intersects(column1, column2) (aggressor, defender) ¼ determineAggressorDefender(column1,column2); event ¼createEvent(aggressor, defender) add(events, event) END IF END FOR END FOR sort(events) FOR EACH event IN events DO aggressor ¼getAgressor(event) reduceAggressorColumn(aggressor) removeEvents(events, aggressor) END FOR FOR EACH event IN events DO evaluate(event) END FOR END First the list of foraging columns is created. In function active() each nest is checked whether or not that nest establishes a foraging column in that iteration according to the foraging experience on the previous step and the transition rules in Table 1. In the case where the nest forages in a column, a column is created either in a random direction (random behavior) or using the dynamics in Eqs. (1) and (2) (learned behavior). After that the algorithm outlined in Section 3.1.1 is applied. The columns are tested pairwise for intersections in function intersects(), and events are created from intersecting columns. For this determineAggressorDefender() labels the nest with the shorter distance to the intersection point as defender and the respective other nest as attacker, resolving (improbable) equalities by the order of their columns in the list columns. After that all events are sorted by the distance of the aggressor to the intersection point in function sort(), resolving equalities in the same way. Then the foraging column of the aggressor is cut at the intersection point in reduceAggressorColumn() and all remaining events containing the aggressor are removed from the list of events in removeEvents(). At this time, events contain all pairs of aggressors and defenders which participate in an intersection in the simulation step. Finally evaluate() updates statistics which are used to calculate the measures presented in this paper. A 3. Unoccupied space in field data To measure the nest density in the field site that was to be simulated in the model, we moved the border of the sampling region to exclude the area which was unoccupied by M. pergandei nests. In detail: we sampled parallels of the western edge in the high density region of the field site and calculated the mean distance, d4 , of the four closest nests to each of these parallels. Then we oriented the new western border so that the mean distance to its four closest nests was matching d4 (Fig. 2). A 4. Eliminating border effects When evaluating empirical field data (or simulated data) of an area that is selected as a part of a larger area, border effects occur. In order to reduce the border effects in our simulations the simulated square area A (with side length s), was embedded into the center of a larger surrounding square area A'. The side length of A' is larger than s by factor K ¼ max fK 1 ; K 2 g where K 1 is a

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parameter of the simulation and K 2 ¼ ðs þ 3 U ðE 〈lcg 〉 þ lof f ÞÞ=s. Parameter K 1 can be used to easily define the relative side lengths of both areas. In our simulations we set K 1 :¼ 1:25. Parameter K 2 guarantees that the distance between the borders of A and A' is at least three times the expected length of a foraging column, i.e., 3 U E 〈l〉. The size of the border was chosen to be 3 times the mean column length, because in the field we do not find interactions between colonies which are that far apart. The latter is necessary because with longer foraging columns the size of the simulated area A which is strongly affected by nests outside the simulated area A' becomes larger. To measure the length of successful foraging columns only the nests positioned in the simulated area A are considered.

A 5. Convergence time of simulations Our simulations needed to run until the exploitation deviation converged to a stable state. In order to determine when simulations should be stopped, we looked at when the normalized exploitation of cells reached a steady state (converged). In regular intervals (100 time steps/days) called evaluation intervals we measured the normalized exploitation of each cell. The convergence rate at the end of an evaluation interval is the absolute value of the difference of the normalized exploitation of the two consecutive evaluation intervals, averaged over all cells in the grid. The convergence rate is still prone to the noise induced by the irregular pattern of foraging columns. However, as soon as the fluctuations from this noise exceed the level of the convergence rate, it is safe to assume, that the exploitation distribution will not change except by the inherent noise. In our pilot simulations we found that after approximately 12,000 time steps random bursts in the convergence rate (maximum) exceeded the mean þ3  standard deviation in a sequence of evaluation intervals for relevant parameter ranges (N A ½30; 130; number of successive evaluation intervals A ½20; 40, Clark–Evans ratio A ½0:82; 2:0, number of simulation steps¼30,000).

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