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Behavioral diversity assisting obstacle navigation during group transportation in

Shohei Utsunomiya1, Atsuko Takamatsu1*

1 Department of Electrical Engineering and Bioscience, Waseda University, Shinjuku-ku, Tokyo 169-8555, Japan

* atsuko [email protected]

Abstract

Cooperative transportation behavior in has attracted attention from a wide range of researchers, from behavioral biologists to roboticists. Ants can accomplish complex tasks as a group whereas individual ants are not intelligent (in the context of this study’s tasks). In this study, group transportation and obstacle navigation in japonica, an ant species exhibiting ‘uncoordinated transportation’ (primitive group transportation), are observed using two differently conditioned colonies. Analyses focus on the effect of group size on two key quantities: transportation speed and obstacle navigation period. Additionally, this study examines how these relationships differ between colonies. The tendencies in transportation speed differ between colonies whereas the obstacle navigation period is consistently reduced irrespective of the colony. To explain this seemingly inconsistent result in transportation speed, we focus on behavioral diversity in ‘directivity’, defined as the tendency of individual ants to transport a food item toward their own preferential direction. Directivity is not always toward the nest, but rather is distributed around it. The diversity of the first colony is less than that of the second colony. Based on the above results, a mechanical model is constructed. Using the translational and rotational motion equations of a rigid rod, the model mimics a food item being pulled by single or multiple ants. The directions of pulling forces exerted by individual ants are assumed to be distributed around the direction pointing toward the nest. The simulation results suggest that, as diversity in directivity increases, so does the success rate in more complicated obstacle navigation. In contrast, depending on group size, the speed of group transportation increases in the case of lower diversity while it is almost constant in the case of higher diversity. Transportation speed and obstacle navigation success rate are in a trade-off relationship.

Author summary

Cooperative transportation behavior in ants attracts researchers from behavioral biologists to roboticists. Ants can accomplish complex tasks in groups, whereas individuals are not likely intelligent. Recent reports focus on higher-level mechanisms in coordinated transportation, e.g., triggering by an informed ant. In contrast, this study focuses on the simplest form of group transportation where there is no specialist in the group. We analyze group transportation and obstacle navigation combined with mechanical model analysis. Behavioral diversity in ‘directivity’ is found to govern performance effectiveness. Here, directivity is defined as the tendency of an individual

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ant to preferentially transport a food item in its own direction rather than in the nest direction. Such outliers can impede transportation speed but contribute well to the system when a group of ants is hindered by a complex obstacle. It is noteworthy that the two measures of efficiency, transportation speed and success rate of obstacle navigation, are in a trade-off relationship.

Introduction 1

Fully coordinated systems are fragile because, once a part of the system becomes 2

unstable, the whole system can sometimes cease to function. In contrast, a fully 3

decentralized system consisting of simple elements can work well even if some elements 4

are broken. Group transportation in ants is a good example of such a decentralized 5

system. Although individual ants are not intelligent in the context of this study ’s 6

tasks, they can accomplish complex tasks through local interaction, sometimes even 7

without a leader. Consequently, researchers from fields as diverse as behavioral 8

biology, physics, and robotics have studied group transportation in ants [1–7]. In 9

contrast, quantitative analysis of group transportation behavior has been insufficient 10

until recently. 11

Group transportation is a behavior in which two or more worker ants move a large 12

food item that not transportable by a single ant. The behavior was categorized into 13

three kinds or transportation by Czaczkes and Ratnieks: uncoordinated, encircling 14

coordinated, and forward-facing cooperative [1]. In uncoordinated transportation, ants 15

pull a food item in opposite directions [8], resulting in slow-speed transportation 16

and/or frequently reaching a deadlock. In encircling coordinated transportation, ants 17

encircle a food item and the majority of members align at the side of the moving front 18

and pull it, whereas minor members align at the back and pull, lift, or push it. This 19

behavior achieves quick transportation, rarely reaching a deadlock. In forward-facing 20

cooperative transportation, a leader ant lifts the food item as its head faces forward 21

while smaller ants subsequently join behind the leader to stabilize the transportation 22

of the item [9]. In the last case, the role and appearance of each ant are clearly 23

distinct; thus, it is out of scope in this study. Uncoordinated transportation has been 24

categorized by McCreery and Breed [10] as transportation without ‘information 25

transfer,’ and the latter two as transportation with ‘information transfer. ’ 26

Recently, encircling coordinated transportation or transportation with information 27

transfer has been investigated intensively in cases where worker ants appear indistinct 28

but have roles that seem to be different during transportation. The workers of the 29

longhorn crazy ant, Paratrechina longicornis, exhibit fluctuating motion during group 30

transportation. However, they switch to straight movement when closer to the nest 31

once a single informed ant, possessing information on the direction of the nest, attaches 32

itself to the group [3]. Furthermore, this species exhibits excellent performance during 33

obstacle navigation, switching its behavior intrinsically without the attachment of an 34

informed ant [11]. Moreover, under imposition of a constraint such as an obstacle 35

barrier, oscillatory group motion emerges which assists in navigating the obstacle [12]. 36

To achieve the above higher-level strategies for group transportation and obstacle 37

navigation, sophisticated mechanisms are required. In contrast, this study focuses on a 38

lower-level but very simple strategy. By observing the behavior of Japanese wood 39

ants, , the fundamental problem emerging through local interaction 40

was investigated, where individual worker ants follow a simple rule and interact 41

through the food item according to kinematics. 42

The ants, F. japonica are widely distributed in Japan [13,14] and have been 43

investigated particularly with regard to landmark navigation [15], social 44

interaction [16,17], etc. Most ants of this species lives under open ground and 45

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transport large food items in groups without dissection [18]. During group 46

transportation, the ants pull the food item in their own preferential directions (Fig 1); 47

Some pull the item toward their nest (denoted by a black arrowhead) and others pull 48

it in the opposite direction of the nest (denoted by a white arrowhead). This results in 49

tug-of-war movement (S1 Video). Thus, their manner of transportation could be 50

categorized as ‘uncoordinated’ [1] and this species has been believed to exhibit 51

low-efficiency transportation [1,8,10]. This is, however, a good model for 52

investigating how the lowest-level strategy can solve tasks using simple rules. We show 53

that the key mechanism is behavioral diversity in ‘directivity, ’which is defined as the 54

tendency of an individual ant to transport a food item in its own preferential 55

direction. How diversity in directivity supports obstacle navigation is discussed. 56

Fig 1. Tug of war observed in ants, Formica japonica, transporting a food item. (A–C) are arranged chronologically in 1-s intervals. The arrow indicates the direction of the nest. The ant denoted by the black arrowheads attempts to drag the food item toward the nest while the ant denoted by the white arrowheads acts against it, resulting in moving the food item in the opposite direction from the nest. See also S1 Video.

Results 57

The behavior of F. japonica ants during group transportation of a food item toward 58

their nest and their obstacle navigation were observed on a stage placed in an indoor 59

artificial environment termed colony (I), or in the outdoor natural environment termed 60

colony (II) (Fig 2; See the Materials and Method section for details). Then the 61

transportation speed, obstacle-passing period, and direction of transportation were 62

analyzed. 63

In individual transportation, ants moved the food items, which were 5–10 times 64

longer and 20 times heavier than a worker ant, by pulling them. During long periods of 65

group transportation with two or more members, the composition of the members was 66

dynamically substituted as reported in the above literature [3,11,19]. This behavior 67

can cause randomness, fluctuating behavior, and/or ‘information transfer’ into the 68

group by newly attached ants. In contrast, in this study only cases of fixed member 69

composition were analyzed because we focused on the basic level of group behavior. 70

Fig 2. Experimental setup. (A) Setup for observation. An observation stage (W 40 × D 30 cm) equipped with a ‘rat-proof ’structure was placed 150–200 cm away from the nest entrance. (B) An obstacle made of polystyrene (W 6.8 × D 1.5 × H 3.7 cm) was placed in front of the ants transporting a food item. Open and closed circles represent positions where an ant group touched the obstacle and passed through it, respectively. The part shaded with oblique lines on the obstacle represents the target area that one of the ants touch it first, whose length was 3.4 cm. (C) Illustration for data analysis.

Transportation in a group and obstacle navigation 71

The transportation speed of a food item by multiple ants in a group depended on the 72

group size in colony (I), as shown in Fig 3A, while it did not in colony (II), as shown 73

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in Fig 3B. When the ant group encountered an obstacle placed on the way to the nest, 74

they initially struggled but finally succeeded in passing through the obstacle 75

(S2 Video). The obstacle-passing period showed a decreasing tendency depending on 76

the group size in both colonies (Fig 3C and D), although the tendency is unclear in 77

the results of colony (II). It is noteworthy that 25% (colony (I); denoted as ”Abn.” in 78

Fig 3E) and 67% (colony (II)) of transportation trials by single ants failed; that is, the 79

ants finally abandoned the task and released their food items (S3 Video), whereas the 80

groups consisting of more than two ants rarely abandoned their food items (Fig 3E). 81

The distribution of the obstacle-passing period by single ants was broad (Fig 3C). 82

This could be caused by the variation of pulling power among individual ants. The 83

passing period in transport by an individual ant decreased depending on speed and 84

ranged from 5 to 45 s, whereas in multiple-ant group transportation, it reduced to less 85

than 25 s, and there was no significant dependency on transportation speed (Fig 3F). 86

Taken together, it can be concluded that transportation by more than two ants 87

reduces the period passing the obstacle and increases the probability of success. 88

Fig 3. Transportation in a group and obstacle navigation. (A and B) Dependence of transportation speed on group size. (C and D) Dependence of obstacle-passing period on group size. (E) Behavioral patterns when a group of ants faces an obstacle. ”Pas.” denotes the case where the group succeeded in the task with its initial members and arrangement. ”Sub.” denotes the case where at least one of the members left and/or was substituted for another ant but where the group eventually succeeded in the task. ”Abn.” denotes the case where the members abandoned the transportation task and released the food item. Only the ”Pas.” cases are included in C and D. Trials of A, C, and E were performed in the indoor artificial environment (colony (I)). Trials of B and D were performed in the outdoor natural environment (colony (II)). (F) Relation between transportation speed and obstacle-passing period. Closed and open marks represent data obtained from colonies (I) and (II), respectively. Black circles, red squares, and blue triangles denote group sizes of 1–3, respectively. In box-and-whisker diagrams, the bottom and top of the boxes denote the first and third quartiles, respectively; the thick band inside the box shows the median; the end and the top of the whiskers represent the lowest and highest data points within 1.5 times the interquartile range (IQR). Scatter plots show the original data. All data sets in A and B passed normality and equal variance tests. Part of the data sets in C and D did not pass them. Thus, both one-way ANOVA and Kruskal-Wallis (KW) tests were applied. A: ANOVA, p = 0.04838 < 0.05; KW, p = 0.03399 < 0.05. B: ANOVA, p = 0.957 > 0.05; KW, p = 0.9102 > 0.05. C: ANOVA, p = 0.004097 < 0.05; KW, p = 0.05354 < 0.1. D: Statistical analysis was not applied because the size of the data for a group size of 1 is too small (n = 2).

Directivity during transportation 89

To investigate the behavioral variation of individual ants, directions of food item 90

transported single ants were analyzed. Fig 4 A shows an example of a trajectory of a 91

food item transported by a single ant. The ant transported it in a certain direction 92

while keeping an almost constant angle relative to the nest direction. The direction 93 angle was defined as the directivity of the ant using θ0. It was estimated using the 94 mean value of the transportation directions θ at each measurement time, represented 95

as a histogram in Fig 4B. In the mean value calculation, the outliers were excluded; 96

these correspond to sudden changes of direction denoted by black arrow heads in 97

Figs 4A and B (see also the Materials and Method section). The cause of these sudden 98

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changes is unknown; however, they could occur when part of the food item is caught 99

by friction with the ground, or by some internal mechanism of the ant. Even after 100 excluding these exceptions, the transportation direction still fluctuated around θ0, 101 which was defined as the magnitude of fluctuation σ equal to the standard deviation of 102

the θ distribution (assumed to be normal distribution and represented with a red line 103

in Fig 4B). The median values of the σ distribution were 12.1 (10.24, 15.6) and 13.5 104

(7.7, 19.9), in colonies (I) and (II), respectively, where the first and second values in 105

parentheses represent the first and third quartiles of the distribution, respectively. 106 The directivity θ0 was quite persistent for individual ants (S4 Video). Thus, θ0 can 107 determine individuality for each ant. Figures 4C and D show the distributions of θ0 in 108 colonies of (I) and (II), respectively, in which θ0 scatters around the direction of the 109 o o nest ( θ0 = 0). The mean values of θ0 in colonies (I) and (II) were −12.48 and 12.27 , 110 respectively. Their variances V , denoting the diversity of θ0, were 0.13 and 0.62 in 111 colonies (I) and (II), respectively. This quantity is defined as V = 1 − R in the domain 112

of 0 ≤ V ≤ 1 using order parameter R (described later in Eq (3)). V = 0 suggests 113

coherence, i.e., ants from the same colony exhibit a high degree of consensus in their 114

moving direction; on the other hand, V = 1 suggests diversity, i.e., individual ants 115

move in dispersed directions. Surprisingly, a considerable number of ants in colony (II) 116

transported the food item in directions quite different from that toward their nest. 117

Fig 4. Directivity θ0 and fluctuation σ in trajectories during transportation by single ants. (A) An example of trajectory of the food item transported by a single ant. The time intervals of the data denoted by black closed circles are δt = 0.1 s. The time duration for estimation of the transportation direction is τ = 1.0 s, whose example is denoted by red open circles and a red line (see also the Materials and Methods section and Fig 2C). (B) Distribution of the transportation directions θ relative to the direction of the nest, estimated using the same data shown in Fig 4A. The distribution was fitted with a normal distribution using all the data (blue dashed line) and using partial data excluding outliers (red solid line; see also the Materials and Methods section). The mean value of the latter data was defined as the directivity θ0 of an ant and the standard deviation as the magnitude of fluctuation σ. The data in Fig 4A and B were obtained from colony (I). (C and D) Directivity θ0. Data shown in Figs 4C and D were obtained from colonies (I) and (II), whose population sizes were n = 25 and n = 50, respectively.

Discussion 118

The results of the experimental observations and analyses can be summarized as 119

follows: (1) the speed of transportation in a group depended on the group size in 120

colony (I) but not in colony (II); (2) the obstacle-passing period tended to decrease 121

with group size and the success rate increased with grouping; (3) the remarkable 122

difference in the properties of colonies (I) and (II) was in their diversity 123

(individuality), measured by the directivity of individual ants. Thus, we hypothesize 124

that diversity in directivity leads to effective obstacle navigation in groups and causes 125

the different tendencies observed in transportation speed between the colonies. 126

To verify this hypothesis, we constructed a simple mechanical model representing 127

ant group transportation of a food item assuming that each member possesses 128

individuality in its directivity. 129

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Model 130

The motion of a food item in traction by single or multiple ants was modeled using 131

translational and rotational motion equations. As already pointed out in the review by 132

McCreery and Breed [10], interaction among ants can be achieved by mechanical force 133

transmitted through the food item, which they referred to as ‘information transfer’ in 134

their paper. The mechanical interaction can be expressed naturally by the mechanical 135

model of the food item, but a mechanism for ‘information transfer’ should also be 136

considered. One of the candidates is adaptive force generation, which is a mechanism 137

by which the performance of individual ants increases according to the load per single 138

ant, as reported in the literature [19]. Additionally, the force fluctuated with direction, 139

as seen in our observations, and the motion of transportation by a single ant 140

sometimes changed suddenly (Figs 4A and B). Note that these behaviors are related 141

closely to ‘persistence,’ defined by McCreery and Breed as “worker’s reluctance to 142

giving up or to change their direction of motion [10].” Therefore, we assumed that 143

traction force exerted by a single ant is generated with fluctuation in directivity, 144

stochasticity, and a simple adaptation mechanism as follows. 145

Basic model equations 146

A food item transported by ants (i = 1, 2, ··· , n) was modeled as a rigid rod with 147 mass M and moment of inertia I (Fig 5A). Each ant grips the rod at a position ri

Fig 5. Model. (A) A rigid rod, mimicking a food item, transported by two ants. The ants exert traction forces at both ends of the rod. (B) A rectangular obstacle placed in front of the food. (C) A U-shaped obstacle. The dashdotted lines in Figs 5B and C represent the goal line used to judge the success of the ants in obstacle navigation.

148 relative to the rod ’s mass center and drags it with a maximum traction force fi. We 149 assumed adaptative force generation wherein the ants control the traction force 150

depending on the speed of the food item. This is expressed by damping terms 151

proportional to the velocity of the mass center of the rod v using a damping coefficient 152 γi. Then the motion of the rod is written using the following translational and 153 rotational motion equations: 154

dv ∑n M = (f − γ v) − Γ e, (1) dt i i t i=1 dω ∑n I = (r × f ) − Γ ω, (2) dt i i r i=1

v where e = |v| is the unit vector of the travel direction and ω is the angular velocity 155 around the mass center. In both motions, kinetic friction in force and torque were 156 considered, where Γt and Γr are the coefficients of kinetic friction for translational and 157 2 rotational movements, respectively. The latter constant was calculated as Γr = r0Γt 158 (see Eq (5) in the Materials and Methods section). 159

Directivity with fluctuating traction force 160

As observed in the experimental results, ants possess individual directivity when 161 0 0 pulling a food item which is defined as θi . The values of θi in each colony were 162 assumed to follow a von Mises distribution, which is a circular-statistics version of the 163

normal distribution [20] defined by two parameters: the mean and the concentration 164

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parameter κ. In the numerical calculations, the parameters were set with the mean 165

equal to 0 (nest direction), and κ = 4.4 (for a colony with small diversity) or κ = 0.9 166

(for a colony with large diversity). The parameter values were estimated roughly from 167

the experimental results. 168

In addition to the individual directivity, the pulling direction fluctuated in time 169 0 around θi with magnitude σ. Therefore, the pulling direction of ant i at time t is 170 defined as follows: 171

0 θi(t)= θi + ξi(t),

where ξi(t) is the fluctuation of the pulling direction. The second term on the 172 right-hand side of the equation was randomly changed in time as a Poisson process so 173

that the mean interval was equal to λ = 0.1 s and the magnitude was randomly 174

generated according to a Gaussian distribution with a mean value of 0 and a variance 175 2 o 2 of σ = (15 ) . The variance was approximately estimated from the experimental 176

result, and was set uniformly among the ants. Therefore, the maximum traction force 177

is defined as follows: 178

T fi(t)= fizi = fi(cos θi(t), sin θi(t)) ,

where zi is the pulling direction vector. 179

Ant alignment and transportation type 180

As reported by Czaczkes et al. [21], ants preferentially bite the corners or the 181

protrusions of a food item. In our observations, the first and second ants tended to 182

bite both ends of a rod-shaped food item. The others bit the remaining exposed areas. 183

The body direction (equal to the direction of traction) of an ant’s first bite seemed 184

sometimes to be unmatched with its preferential direction (S5 Fig). The ant pulled the 185

food in the body-backward direction first, and later corrected the pulling direction to 186

its preferential direction. Buffin et al. also observed in Novomessor cockerelli that ants 187

pulling the food item in the nest direction persisted to bite, but that ants biting the 188

opposite side leave their positions more easily [19]. Based on the above observation, 189

simulations of group transportation were performed according to the following rule. 190

The first and second ants bite the food item at both ends; then the other ants 191

follow at equally spaced positions along the food item. The initial direction θ(0) was 192

randomly set according to a uniform distribution, which is sometimes inconsistent 193

with an ant ’s preferential direction. During the simulation, ants biting at the equally 194

spaced positions were forbidden from crossing to the opposite side of the food item. 195

Consequently, if an ant is positioned at the side contradictory to its preferential 196 directivity θ0, the ant is not able to contribute effectively to transportation based on 197 the above rule. In practice, such an ant would eventually leave the transportation 198

task; then the resulting open space would become available for another ant. Thus, this 199

case was categorized as ‘Sub.’ (for the definition, see the legend of Fig 3). 200

Transportation speeds were estimated using all groups except for those categorized as 201

‘Sub.’ using the same method. 202

Stochastic force generation 203

As seen in the Results section and in Figs 4A and B, the motion direction of 204

transportation by a single ant suddenly changes on occasion. This could be caused by 205

friction with the ground, resulting in stochastic force generation. The stochastic effect 206 was introduced by setting the magnitude of the traction force fi to 0 for each time step 207 in the simulation for probability p = 0.25 only in the U-shaped obstacle simulation. 208

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Comparison with mechanical models in the literature 209

There have been several reports on mechanical models through physical interaction for 210

group transportation in the literature [2,22,23]. In most of them, the adaptation 211

process in individual ants was described as optimization to an assigned fitness value 212

using an evolutionary algorithm or reinforcement learning. These approaches do not 213

truly mimic real ant behavior but rather prioritize the effectiveness of the system as a 214

machine. 215

Model result: Speed in group transportation 216

In the literature, there is less consistency in the relation between speed and group size 217

during transportation of a constant-weighted load. The speed is simply proportional 218

to group size in P. longicornis [11,12]. It is proportional at small group size and then 219

becomes saturated at large group size in the neotropical ants Pheidole oxyops [21] and 220

Aphaenogaster cockerelli [2]. It seems to be almost constant irrespective of group size 221

in Eciton burchelli [9] and Formica rufa [24]. Some researches implied that larger 222

group size causes slower speed in Pheidole pallidula [25] and Novomessor cockerelli [19] 223

(note that the weights of the loads were not regulated in these experiments). 224

In contrast, two types of tendencies, proportional and constant, were found in our 225

observations using the same species of F. japonica and food items of almost equal 226

weight. This difference in tendencies could have been caused by difference in the 227

diversity of directivity. This conjecture is verified using our model as follows. 228

To estimate roughly how the speed in group transportation depends on group size, 229

Eqs (1) are rewritten by neglecting inertia as follows: 230

f 1 ∑n Γ v = · z − t e, γ n i nγ i=1

where ξi(t) ≡ 0, fi ≡ f, and γi ≡ γ were assumed for simplicity. The value 231

1 ∑n R(n) ≡ z , 0 ≤ R(n) ≤ 1, (3) n i i=1

is an order parameter of directivity in the group. R = 0 suggests dispersal; on the 232

other hand, R = 1 suggests coherence. 1 − R represents the degree of a group ’s 233

diversity V , as mentioned in the Results section. Because the vector e is the unit 234

vector of v, the speed can be written as follows: 235 f Γ |v| = R(n) − t . (4) γ nγ

As seen in this equation, the speed consists of two converse terms: total traction force 236

and kinetic friction force. The first term is proportional to the order parameter R(n), 237

which depends on diversity and group size n as follows (see also the Materials and 238

Methods section and S6 Fig). When there is no diversity, the order parameter is 239

always R(n) = 1 irrespective of n ( S6 FigA). Additionally, R(1) = 1 irrespective of 240

diversity. On the contrary, when there is diversity, R(n) statistically decreases 241

depending on n among a number of trials (S6 FigB and C). Larger diversity affects a 242

larger decrease in R(n). The second term on the right-hand side of Eq (4) decreases 243

depending on n irrespective of diversity. Taken together, with the above converse 244

effects, the speed of group transportation |v| increases under small diversity, whereas 245

it decreases or is almost constant under large diversity, as shown in the open 246

box-and-whisker diagrams in Fig 6. 247

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Gray box-and-whisker diagrams in Fig 6 represent the results of numerical 248

simulation using the mechanical model of Eqs (1) and (2). The tendencies in speed 249

dependence on group size are similar to those of the above analytical calculations; 250

however, the absolute values were different. In fact, fluctuation of the pulling force was 251

not considered in the theoretical equation (4), which could have caused the observed 252

difference. 253

Both analyses suggest that the tendency in the speed of group transportation, 254

whether increasing or constant, is simply governed by the diversity in directivity 255

among individual ants belonging to each colony. 256

Fig 6. Theoretical prediction and simulation results of group-size dependence in transportation speed. A: Small diversity in directivity (κ = 4.4). B: Large diversity in directivity (κ = 0.9). Other parameters were set as f = 10 2 2 gmm/s , γ = 0.4 g/s, Γt = 7.5 gmm/s , and r0 = 10 mm. Open box-and-whisker diagrams show the results obtained using the theoretical equation Eq 4. Gray filled box-and-whisker diagrams show the results obtained from the simulation using the mechanical model Eqs (1) and (2).

Simulation results: passing obstacles 257

To investigate the relation between diversity in directivity and performance in obstacle 258

navigation tasks by a group of ants, numerical simulations were performed (see the 259

Materials and Methods section for details). A rod mimicking a food item being pulled 260

by multiple ants was placed in front of a rectangular- or U-shaped obstacle, as shown 261

in Figs 5B and C. 262

Figure 7 shows the result of simulation for a rectangular-shaped obstacle. When 263

there is no diversity in directivity, the passing period was relatively longer and did not 264

depend on group size (S7 Video, Fig 7A). When there was diversity, the passing 265

period was reduced to almost half by grouping two ants (S8 Video, Figs 7B and C). 266

Grouping more than two ants did not contribute further. In addition, the magnitude 267

of diversity did not affect the results. In the simulation without diversity, 90% of 268

single ants were not able to pass the obstacle (denoted as ‘Abn.’ in Fig 7D). The rate 269

of ‘Abn.’ decreased by grouping two or more ants. In addition, for small diversity, the 270

rate of ‘Abn.’ greatly decreased and was minimized for a group size two (Fig 7E). For 271

a large diversity, the rate of ‘Abn.’ was constant irrespective of group size (Fig 7F). 272

These findings suggest that grouping two ants with a small diversity optimize 273

navigation of relatively simple obstacles (rectangular-shaped). 274

Figure 8 shows the result of simulation for a U-shaped obstacle. In contrast to the 275

case of the rectangular-shaped obstacle, the success rate of obstacle navigation was 276

extremely low in the small-diversity case (‘Sub.’ and ‘Pas.’ in Fig 8D) but improved in 277

the large-diversity case (Fig 8E). This suggests that, the larger the diversity, the more 278

able an ant group to pass complicated obstacles. The large diversity in colony (II) in 279

the natural environment could be effective in navigating more complex terrain. 280

Stochastic force generation in addition to diversity in directivity moderately increased 281

the success rate in this task (Fig 8F). In all cases, the obstacle-passing period 282

decreased with group size except in the single-ant case. These results suggest that 283

grouping more than two ants and the presence of large diversity values can assist in 284

navigation of complicated obstacles. 285

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Fig 7. Simulation results for obstacle-passing period through a rectangular obstacle. A–C: Obstacle-passing period depending on group size. D–F: Behavioral patterns after ant groups face an obstacle. A and D: No diversity in directivity. B and E: Small diversity in directivity(κ = 4.4). C and F: Large diversity in directivity (κ = 0.9). Other parameters were set as f = 10, γ = 0.4, and Γt = 7.5. Other notations and classification of behaviors are the same as those in Fig3.

Fig 8. Simulation results for obstacle-passing period with a U-shaped obstacle. A–C: Obstacle-passing period depending on group size. D–F: Behavioral patterns after an ant group faces an obstacle. A and D: Small diversity in directivity(κ = 4.4). B, C, E, and F: Large diversity in directivity (κ = 0.9). Stochastic force generation as p = 0.25 was considered in C and F. Other parameters were set as f = 10, γ = 0.4, and Γt = 7.5. Other notations and classification of behaviors are the same as those in Fig 3.

Conclusion 286

Two types of colonies, those conditioned in an indoor artificial environment (I) and an 287

outdoor natural environment (II), were analyzed for F. japonica. Their dependences 288

for speed of group transportation on group size differed. Previously, this difference 289

may have been ascribed to differences between species. In contrast, we concluded 290

through mechanical model analysis that this difference in speed dependence can be 291

explained by diversity in directivity for individual ants among colony members. 292

Additionally, results of numerical simulation suggested that broader diversity 293

increased the success rate of obstacle navigation and reduced the obstacle-passing 294

period by grouping, supporting navigation of more complicated terrain. Diversity in 295

directivity was also observed in Novomessor cockerelli [19] and in Solenopsis 296

invicta [26], where the trajectories of transportation by single ants were distributed 297 o o around 45 − 60 , although they were not discussed the effect of diversity in their 298

reports. 299

It is interesting that there is a trade-off relationship in transportation speed and 300

rate of success in obstacle navigation. The speed increased depending on group size in 301

colony (I), which possessed less diversity in directivity. The model analysis suggested 302

that this is unsuitable for obstacle navigation. If the ants’s habitat is in the open and 303

allows for a long line of sight, there are fewer obstacles but the probability of being 304

attacked by predators is higher. Thus, rushing into the nest is a good strategy. In 305

contrast, the speed did not increase depending on group size in colony (II), which 306

possessed broad diversity in directivity; this is suggested as being suitable for obstacle 307

navigation in complicated terrain. In such environments, the probability of being 308

attacked by predators is lower. Thus, the ants can give priority to reliable obstacle 309

navigation rather than transportation speed in such cases. 310

Lastly, the simulation results revealed that stochastic force generation, whether 311

generated coincidentally by friction with the ground or intrinsically by the 312

system [11,12], can also assist in navigating more complicated obstacles. The success 313

rate could be further enhanced by using an informed-ant-attachment mechanism [3]. 314

There are still open questions, such as how colony members learn the configuration 315

around their nest and encode this information into diversity of directivity. 316

Additionally, how ants whose preferential directions are opposite to the nest return to 317

their nest remains unknown (these ants continued food transportation in a direction 318

opposite to the nest over a distance of more than 1 m in the outdoor observations). A 319

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further question is whether distribution directivity depends on distance to the nest? 320

Materials and Methods 321

Ant preparation 322

Two types of colonies of Japanese wood ants, F. japonica, were prepared: ants kept in 323

an indoor artificial environment, named colony (I), and ants that inhabited an outdoor 324

natural environment, named colony (II). 325

Colony (I) was purchased from a vendor (Arinko Spot, Chiba, Japan), which 326

collected a queen ant from sparse grassy ground located in Chiba prefecture, Japan, in 327

July (nuptial flight season), 2015. Then the queen and pupae collected from another 328

nest were kept in an artificial environment until the number of worker ants grew to a 329

few hundred. The ants, including the queen and pupae, were transferred into an 330

artificial nest made using a polystyrene chamber (176 × 80 × 32 mm), whose inner 331

surface was plastered to maintain appropriate humidity and whose outer surface was 332

covered with aluminum foil to maintain a dark environment. The nest was equipped 333

with a single exit 7 mm in diameter. The nest was kept in another larger polystyrene 334 o chamber (300 × 200 × 50 mm) and was maintained under an environment of 26 C and 335

≥50 RH% humidity that was illuminated 12 h per day using LED light (800 lux). The 336

ants were fed with sugar water or dissected mealworms every 3 days. The experiments 337

were performed on the third day before feeding. One hour before the observations, the 338

nest was placed into an arena made of polystyrene boards (1800 × 500 × 100 mm) as 339

shown in Fig 2A. The observations were performed from July to October in 2016 in 340 o the indoor environment 27–28 C in temperature, 40–55RH% in humidity, and under 341

conventional LED neutral white light bulb illumination (Luminous, LDAS40N-GM, 342

Doshisha Corp.) by illuminating the observation stage. 343

Colony (II) was found in a plant pot (about 30 × 30 × 40 cm) located in a shaded 344

place on a roof terrace on the sixth floor of building in Shinjuku-ku, Tokyo. Thus, this 345

place was isolated from other colonies of F. japonica. Although it is known that F. 346

japonica exhibits polydomy (i.e., a single ant colony forms multiple nest sites) [14], 347

this colony exhibited monodomy (a single ant colony forming a single nest site), at 348

least during the observation season. The observations were performed in September 349 o 2013 in the outdoor environment 26–32 C in temperature, 40–55RH% in humidity, 350

and in an unshaded place under fair or slightly cloudy weather. 351

The lengths of workers ranged over 4―6 mm in both colonies. Their living weight 352

is known to be 5–6 mg [13]. 353

Food items 354

Food items consisting of Sakura-ebi (dried small shrimp Sergia lucens, around 40 mm 355

in length and 120  30 mg in weight) and Shirasu (boiled whitebait of sardine, around 356

20 mm in length and 90  20 mg in weight) were respectively prepared for colonies (I) 357

and (II) to observe food item transportation. These were 5–10 times longer and 20 358

times heavier than a worker ant. 359

Observation of group transportation 360

Observation stages were placed at positions 150 cm away from the nest for colony (I) 361

and 150–200 cm for colony (II), as shown in Fig 2A. First, a food item was placed in 362

front of an ant foraging at a site on the observation stage. After the ant found the 363

food item and recruitment was initiated, the food item baited by a single ant or by 364

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multiple ants was transferred onto the observation stage. The observation stages were 365

made of a polystyrene board (for colony (I)) or cardboard (colony (II)) and were 366

equipped with a ‘rat-proof ’structure to prevent additional ants from joining in the 367

transportation of the target. Their behaviors were recorded using digital cameras 368

(EXLIM EX-100, Casio Computer Co., Ltd. for colony (I); D-LUX4, Leica Camera 369

AG for colony (II)) and saved as MPEG-4 videos or JPEG linear PCM photos. 370

Obstacle 371

An obstacle made of polystyrene (W 6.8 × D 1.5 × H 3.7 cm) was prepared. Its lateral 372

walls were coated with fluoropolymer resin (Fluon®PTFE, Asahi Glass Co., Ltd.) for 373

colony (I), and with fragrance-free baby powder made of zinc oxide (baby powder, 374

Pigeon corp., Japan) for colony (II) in order to prevent ants from climbing the walls. 375

Data analysis 376

Images 377

Movies were converted into sequential TIFF format images at 10 fps in 8-bit gray-scale 378

using image processing software (ImageJ [27]), and were then abstracted at the center 379

position of the food item or the ants in each frame. The trajectories during 380

transportation were obtained as demonstrated in Fig 4A. 381

Estimation of transportation speed 382

Speed of transportation was estimated as d/T using the period T that the food item 383

moved a fixed distance d before the ants touched obstacles or the number of ants in 384

the group changed (Figs 2B and C). 385

Estimation of obstacle-passing period 386

Periods for passing the obstacle were estimated as the difference between the time 387

when one of the group ants touched a target area of the obstacle (an open circle 388

denotes the touching point and the part shaded with oblique lines on the obstacle 389

denotes the target area in Fig 2B) and the time when all group members transited 390

across the line of the obstacle surface (denoted by a closed circle in Fig 2B). 391

Classification of obstacle navigation 392

Behavioral patterns when ant groups faced the obstacle were manually categorized 393

into the following three types by watching the videos: the case where the ants 394

succeeded in the task with the initial ant members and the initial arrangement 395

(denoted by ”Pas.” in Fig 3E), the case where at least one of the members left the 396

food item and/or was substituted by another ant and the group succeeded in the task 397

(denoted by ”Sub.” in Fig 3E), and the case where all members abandoned the 398

transportation task, releasing the food item (denoted by ”Abn.” in Fig 3E). 399

Estimation of transportation direction 400

The angle of the transportation direction θ(t) at time t was defined using the direction 401

of the position at time t + τ from the position at time t relative to the nest, as shown 402

in Fig 2C. By moving the window of τ with the interval δt = 0.1 (equal to the image 403

interval), a data set of θ values for a single trajectory was obtained. Then the 404 directivity θ0 and the magnitude of fluctuations σ were estimated as explained in the 405

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following subsection. The value τ = 1.0 s was set so that θ0 could be captured more 406 accurately and σ could be captured with moderate accuracy as follows. Various values 407

of τ ranging from 0.1–2.0 s were tested as shown in S9 Fig. The mean value of the 408

distribution of θ was saturated at around τ = 1.0 (S9 FigA), suggesting that larger 409 values than this (denoted as (a) in S9 FigA ) would be appropriate for capturing θ0 410 more accurately. If τ were too small, measurement errors would be detected (S9 FigB 411 and C), leading simultaneously to worse estimation of θ0. On the contrary, if τ were 412 too large, fluctuations in transportation directions would not be captured (S9 FigB 413

and E). Thus, the range between them would be appropriate (denoted as (b) in 414

S9 FigB). Therefore, τ = 1.0, which satisfied both conditions, was used. 415

Note that the magnitude of fluctuations would be underestimated in this scheme. It 416

is, however, difficult to evaluate this magnitude because its dependence on τ exhibits a 417

continuous curve as shown in S9 FigB. Therefore, we tested the effect of fluctuations 418

with low resolution in terms of presence or absence. Without fluctuation, most trials 419

failed in obstacle navigation. Thus, all simulations were performed with fluctuations. 420

Definition of directivity and fluctuation 421

From the data set θ(t) obtained by the above method, the distribution was analyzed 422

as follows. The distribution was fitted with a normal distribution function using all 423

the data first, e.g., as denoted by the blue dashed line in Fig 4B. Next, a normality 424

test (Shapiro-Wilk test) was applied. If the p value was smaller than a level of 425

significance (α=0.05 was used), rejection of normality was suggested and the data 426

value farthest away from the mean value was removed (denoted as an outlier). Then 427

the application of the normality test and outlier removal were applied until the 428

distribution could fit the normal distribution (p > α, shown by the red solid line of 429

Fig 4B). The data set of outliers corresponded with irregular movements such as 430

sudden changes of the moving direction denoted by arrow heads and open blue circles 431 in Fig 4A. Consequently, the final mean value was defined as the directivity θ0, and 432 the standard deviations defined as the magnitude of the fluctuation σ were obtained. 433

Analysis of directivity 434

The directivity θ0 can be a measure of individuality of each ant, and was distributed 435 around the nest direction for each colony as shown in Fig 4C and D. To characterize 436 the distribution, mean values of θ0 and of the variance (V ) were estimated using 437 directional statistics [20] using the “circular” package in R [28]. The variance was 438

defined in the domain of 0 ≤ V ≤ 1. The value V = 0 represented all data pointing in 439

the same direction, suggesting the smallest level of diversity. The value V = 1 440

represented a uniform distribution in all directions, suggesting the largest level of 441

diversity. 442

For application to the mathematical model in the discussion section, the 443 distribution of θ0 was fitted with a von Mises distribution, a circular version of the 444 normal distribution using the “circular” package in R [28]. The mean value, equal to 445

that of the above circular statistics, and a concentration of 0 ≤ κ, which is a measure 446

of diversity, were analyzed. 447

Numerical simulation 448

Derivation of rotation friction torque 449

In the Eqs (1) and (2), the speed v(s) of a position at a distance s from the center on 450

the rotating rod with rotation speed ω is represented as v(s) = sω. Therefore, the 451

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rotation friction torque was calculated by the following integration: 452 ∫ ∫ r0 r0 2 Γrω = 2 Γtv(s)ds = 2 Γtsωds = Γtr0ω, (5) 0 0

where 2r0 is the length of the rod. 453

Theoretical analysis of transportation speed using Eq4 454

Eq (4) and the order parameter R(n) defined by Eq (3) were statistically analyzed. 455

The directivities for individual ants among a group were randomly generated 456

according to a von Mises distribution with a mean parameter of 0 and a concentration 457

parameter of κ =0, 4.4, and 0.9 for calculations corresponding to no diversity, low 458

diversity, and high diversity, respectively, as shown in S6 Fig. Then 1,000 trials for 459

each group size were performed and distributions of R(n) and |v| were analyzed using 460

their medians and box-and-whisker diagrams. 461

Simulation of transportation 462

Computer simulations using Eqs (1) and (2) were performed. For simplicity, inertia 463

was neglected; namely, left-hand terms in Eqs (1) and (2) were approximated as 0. 464

The Euler method was applied with a time step size of ∆t = 0.1 s. Initial positions of 465 food items were set 3r0 apart from the center of the obstacle. 466

Simulation of obstacle navigation 467

A rectangular (Fig 5B) or U-shaped obstacle (Fig 5C) was placed in front of the 468

transported food item so that the center of the food item faced near the center of the 469 obstacle within the range of 2r0. In our observations, even when ants reached the wall 470 of the obstacle, they persisted in moving in the same direction but failed to climb the 471

wall owing to slipping; thus, the position did not move. Consequently, we assumed 472

that the ants exerted a force from the obstacle equal to the normal vector according to 473

Newton’s third law. 474

We judged the time when whole body of the food item passed through a line 475

denoted by the dash-dotted line in Figs 5B and C, as the time when an ant group 476

succeeded in navigation of an obstacle. The obstacle-passing period was then 477

estimated using the same method as in the experiments. When the period exceeded 60 478

s, we judged that the group had abandoned the transportation task (‘Abn.’) in Figs 7 479

and 8. The critical period was estimated according to the experimental results, in 480

which the period required for groups to succeed in the task was always less than 50 s. 481

The category of ‘Sub.’ denotes the case where the initial direction of an ant’s first bite 482

was unmatched with its preferential direction but where the group succeeded in 483

passing the obstacle nevertheless. The category of ‘Pas.’ denotes the case where there 484

was no directional mismatch and the group succeeded in the task. 485

Supporting Information 486

S1 Video. Tug-of-war in group transport. Intervals between time stamps are 487

1/30 s. Time stamp 968, 998, and 1028 correspond to the photos of A, B, and C in 488

Fig 1, respectively. 489

S2 Video. Success in passing the obstacle. Three ants in a group transported a 490

food item and eventually passed the obstacle. 491

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S3 Video. Abandonment of obstacle navigation. A single ant tried to 492

transport a food item across the obstacle but abandoned the task, releasing the food 493

item. 494

S4 Video. Persistence of directivity. A single ant was transporting a food item 495

in a certain direction. Even when interrupted and moved to another location, the ant 496

continued to transport the food item in almost the same direction. 497

S5 Fig. Mismatch between the first bite direction and the ant ’s 498

preferential direction. The ant pulled the food item in the body-backward 499

direction first (red arrow) and then corrected the pulling direction to its preferential 500

direction (blue arrow). 501

S6 Fig. Order parameter R(n) depending on group size n. (A) No diversity. 502

(B) Low diversity. κ = 4.4. (C) High diversity. κ = 0.9. The values R(n) were 503 calculated using Eq (3). The directivity values θ0 of individual ants were randomly 504 generated from the von Mises distribution with the mean equal to 0 and with the 505

concentration parameter κ. Thus, the values of R(n) among trials were distributed, 506

and were obtained from 1,000 trials and represented by box-and-whisker diagrams. 507

S7 Video. Simulation results for obstacle navigation with no diversity. 508

The preferential directions of two ants were set to the direction of the nest (upper area 509

in the movie). It took longer to finish obstacle navigation in this case. The directions 510

of the traction force of individual ants (represented with magenta lines) fluctuated 511

around their preferential directions (represented with black lines). 512

S8 Video. Simulation results for obstacle navigation with high diversity 513

(κ = 0.9). The preferential directions of two ants were set randomly relative to the 514

direction to the nest. It was easy for the group ants with high diversity to succeed in 515

the obstacle navigation task. 516

S9 Fig. Estimation of transportation direction. (A) Mean value of the 517

distribution of the transportation directions θ depending on the time step τ used in 518

calculations. (B) Standard deviation of the distribution of θ depending on τ. (C–E) 519

Examples of transportation trajectories. The values τ =5, 10, and 20 were set for (C), 520 (D), and (E), respectively. (a) An appropriate range for estimating θ0. (b) An 521 appropriate rage for estimating σ. 522

Acknowledgments 523

The authors wish to thank Professor H. Nishimori, Hiroshima University, and Mr. S. Hisamoto, Waseda University, for useful information on keeping ants and on the experimental setup.

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