We model the movement behaviors of each individual in a population of size $N$ using two experimentally-motivated (Berdahl et al., 2013; Katz et al., 2011) behavioral rules: a social response rule and an environmental response rule. The social response rule is motivated by experimental studies of pairwise interactions among golden shiners (Notemigonus crysoleucas) (Katz et al., 2011). Individual fish avoid others with whom they are in very close proximity. As the distance between individuals increases, however, interactions gradually change from repulsive to attractive, with maximum attraction occurring at a distance of two-four body lengths. For longer distances, individuals still attract one another but the strength of attraction decays in magnitude (Appendix section 1; Katz et al., 2011). As found in experimental studies of golden shiners (Katz et al., 2011) and mosquitofish (Gambusia holbrooki) (Herbert-Read et al., 2011) there need not be an explicit alignment tendency; rather alignment can be an emergent property of motion combined with the tendencies for repulsion and attraction described above.

To capture these observed social interactions (or ‘social forces’), we model the acceleration of individuals using a force-based method (Katz et al., 2011). The $i$th individual responds to its neighbors using the following rule:

where $F_{s\mathrm{,}i}$ is the social force on the $i$th individual, $x_i$ is the position of the $i$th individual, $\nabla$ is the two-dimensional gradient operator, the term in brackets is a social potential, $C_a$, $C_r$, $l_a$, and $l_r$ are constants that dictate the relative strengths and length scales of social attraction and repulsion, and the set $N_i$ is a set of the $k$ nearest neighbors of the $i$th individual, where a neighbor is an individual within a distance of $l_{max}$ of the focal individual. Equation 1 does not include explicit alignment with neighbors. A similar model is discussed in D’Orsogna et al. (2006). In Equation 1, $l_{max}$ determines the length scale over which individuals are influenced by social interactions. If $l_{max}$ is greater than $l_r$ but less than $l_a$, individuals repel one another at short distances but do not attract one another. We refer to such individuals as asocial (Appendix section 1). If $l_{max}$ is greater than both $l_r$ and $l_a$, individuals repel one another at short distances and are attracted to one another at intermediate distances as observed by Katz et al. (2011). Finite $k$ ensures that individuals can only respond to a limited number of their neighbors in crowded regions of space and provides a simplified model of sensory-based social interactions (e.g., Rosenthal et al. (2015); Strandburg-Peshkin et al. (2013)). Finite $k$ also ensures that individuals are limited to finite local density (Appendix section 3).

To model the response of individuals to the environment, we develop an environmental response rule based on experimentally-observed environmental responses of golden shiners (Berdahl et al., 2013). In particular, in a dynamic, heterogeneous environment, individual golden shiners respond strongly to local sensory cues by slowing down in favorable regions of the environment, and speeding up in unfavorable regions. In contrast, fish respond only weakly to spatial gradients in environmental quality and instead adjust their headings primarily based on the positions of their near neighbors. Accordingly, we model the $i$th individual’s environmental response as a function of the level of an environmental cue (in this case, the level of a resource) at its current position:

where $F_{a\mathrm{,}i}$ is the autonomous force the $i$th individual generates by accelerating or decelerating in response to the environment, $\Psi (\cdot )$ is a monotonically decreasing function of the value of an environmental cue, $S\left(x_i\right)$ is the cue value at the $i$th individual’s position, $\eta$ is a damping term that limits individuals to a finite speed, and $v_i$ is the $i$th individual’s velocity. In the absence of social interactions, individuals travel at preferred speed $v_i^{*}=\sqrt{{\Psi }_i/\eta }$ (for ${\Psi }_i>0$). Changes in speed are crucial in the schooling behavior of fish (Tunstrøm et al., 2013; Berdahl et al., 2013), and as we show below, are also responsible for generating effective collective response in our model. Following the experimental results in Berdahl et al. (2013) we assume that individuals do not change their headings in response to the cue. In what follows, we refer to 'cue' and 'resource' interchangeably as we model the case where the cue is the resource itself (see e.g., Torney et al. (2009); Hein and McKinley (2012) for cases where the cue is not a resource).

Combining social and environmental response rules yields two equations that govern each individual’s movement (in two dimensions):

and

where $m$ is mass. D’Orsogna et al. (2006) explores the behavior of a similar model with ${\Psi }_i=\Psi$ constant over the full parameter space. Here we focus on a parameter regime that yields behavioral rules that match the experimental observations of Katz et al. (2011) and Berdahl et al. (2013).

We simulate a discretized version of the system described by Equations 3 and 4. In particular, we choose a time step, $\tau$, within which the acceleration due to social influences (Equation 1) and resource value $S\left(x_i\right)$ are assumed to be constant. Positions, speeds, and accelerations of all individuals at time $t+\tau$ are then given by the solutions to Equations 3 and 4 at time $t+\tau$, with the values of $S\left(x_i\right)$ and $|x_i-x_j|$ determined at time $t$. A navigational noise vector of small magnitude $\gamma$ and uniform heading 0 to 2$\pi$ is added to the velocity of each agent at each time step. Taking the limit as $\tau$ goes to zero means that individuals are constantly acquiring information and instantaneously altering their actions in response. In Appendix section 3−6, we analyze a continuum approximation of this limiting model and below we discuss results of this analysis alongside simulation results.

The social interaction rule allows us to build an interaction network for the entire population. Two individuals are socially connected if at least one of them influences the other through Equation 1. We define a 'group' as a set of individuals that belong to the same connected component in this network.

The natural environments in which organisms live are often heterogeneous and dynamic (Stephens et al., 2007). Consequently, we simulate populations of individuals in dynamic landscapes, where individuals make decisions in response to local sensory cues (local measurements of a resource) and these decisions have fitness consequences for the individuals within the population (Guttal and Couzin, 2010; Torney et al., 2011). In keeping with experimental observations (Berdahl et al., 2013), we assume individuals follow a simple environmental response function: ${\Psi }_i={\psi }_0-{\psi }_{1}S\left(x_i\right)$, where ${\psi }_0$ dictates the $i$th individual’s preferred speed when the level of the environmental cue is zero and ${\psi }_1$ determines how sensitive the $i$th individual is to the cue value (Berdahl et al., 2013). Rather than prescribing values of ${\psi }_0$ and ${\psi }_1$, we use an evolutionary framework similar to that developed by Guttal and Couzin (2010) to allow these two behavioral traits to evolve along with the maximum interaction length $l_{max}$, which determines whether individuals are social ($l_{max}>$ length scale of social attraction) or asocial ($l_{max}<$ length scale of social attraction, Appendix section 1).

In each generation, $N$ individuals are located in a two-dimensional environment in which each point in space is associated with a resource value that changes over time (see Materials and methods). Individuals move through the environment using the interaction rules described above, and each individual has its own value of the ${\psi }_0$, ${\psi }_1$, and $l_{max}$ parameters. At the end of each generation, we compute each individual’s fitness as the mean value of the resource it experienced during that generation. Each individual then reproduces with a probability proportional to its relative fitness within the population. $N$ offspring comprise the next generation where each offspring inherits the traits of its parent modified by a small mutation (Appendix section 2). For reference, we compare the evolution of populations in which ${\psi }_0$, ${\psi }_1$, and $l_{max}$ are allowed to evolve, to the evolution of populations of asocial individuals, for which $l_{max}$ is set to a constant (Appendix section 1).

In populations of asocial individuals, the baseline speed parameter and environmental sensitivity increase consistently through evolutionary time (Figure 1A–B). Asocial individuals move through the environment, slowing down in regions where the resource value is high and speeding up when the resource value is low (Video 1). As one would expect from random walk theory (Schnitzer, 1993; Gurarie and Ovaskainen, 2013), individuals more rapidly encounter regions of the environment with high resource value when they travel at high preferred speeds (Equation A65; Gurarie and Ovaskainen, 2013), and the more they reduce speed in regions of the environment with high resource quality, the more time they spend in these regions (Schnitzer, 1993). Because of these two effects, the fittest asocial individuals have high baseline speeds (i.e., high ${\psi }_0$) and accelerate and decelerate rapidly in response to changes in the resource value (i.e., high ${\psi }_1$; Figure 1A–B, Appendix).

Figure 1

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When populations are allowed to evolve sociality, the evolutionary process selects for very different behaviors (Figure 1C–E). Selection quickly favors sociality, and individuals evolve large maximum interaction lengths (Figure 1C). Over evolutionary time, selection removes individuals with high and low values of ${\psi }_0$ and ${\psi }_1$ from the population and an evolutionarily stable state (ESSt; Maynard Smith, 1982) emerges that is characterized by a single mode at the dominant value of each trait (Figure 1D–E; Appendix section 2). The ESSt resulting from selection on ${\psi }_0$, ${\psi }_1$, and $l_{max}$ is robust in that it is resistant to invasion by phenotypes near the ESSt, and by invaders with trait values far from the ESSt (Appendix section 2). Throughout evolution, populations of social individuals achieve mean fitness values that are approximately five times higher than those of asocial populations, and a coefficient of variation in fitness approximately four times lower than that of asocial individuals (Figure 1F).

Notably, a single individual drawn from a population at the ESSt can invade a resident population of asocial individuals and the social strategy quickly sweeps through the population (Appendix section 2). To understand why this invasion occurs, consider a population of asocial individuals that slow down in favorable regions of the environment. If the environment does not change too rapidly, such individuals will accumulate in regions where the resource level is high. This phenomenon has been studied mathematically in the context of position-dependent diffusion (Schnitzer, 1993), and will occur, in general, when individuals lower their speeds in response to the value of an environmental cue. A social mutant that responds to the environment, and to its neighbors, can take advantage of the correlation between density and resource quality by climbing the gradient in the density of its neighbors (Equation 1). In this case, the positions of neighbors contain information about the value of resources and social mutants quickly invade asocial populations leading to a rapid increase in mean fitness (Appendix section 2).

The high fitness of the evolved phenotype is due, in part, to a collective resource tracking ability, similar to that found in golden shiners (Berdahl et al., 2013). Evolved individuals can find and track resource peaks as they move through the environment (Figure 2A, Video 2; Materials and methods), whereas asocial individuals and social individuals with trait values far from the ESSt cannot (Videos 1, 3–4). Tracking occurs via a dynamic process. Individuals near the edge of the peak move rapidly, whereas individuals nearer to the peak center (where the resource value is high) move slowly (Equation 2). As in fish schools (Berdahl et al., 2013), individuals turn toward near neighbors (Equation 1) and travel toward the peak center. This collective tracking behavior is particularly important when the resource field changes rapidly over time. As a resource peak moves, individuals at its trailing edge experience a resource value that becomes weaker through time (Figure 2A). As the resource value becomes weaker, these individuals accelerate (Equation 2), but turn toward neighbors on the peak (Equation 1) and thus travel toward the moving peak (Figure 2A). When the environment contains multiple resource peaks, evolved populations fuse spontaneously to form groups whose sizes correspond to that of the peak they are tracking (Figure 2B), even though no individual is able to assess peak size, or know whether there are multiple peaks in the environment. This behavior is consistent with recent sonar observations of foraging marine fish showing that fish form shoals that match the sizes of dynamic resource patches (Bertrand et al., 2008; Bertrand et al., 2014). Our model demonstrates that collective tracking behavior similar to that observed in real fish schools can evolve through selection on the decision rules of individuals.

Figure 2

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That individuals in evolutionarily stable populations have intermediate baseline speeds and intermediate environmental sensitivities (Figure 1D–E) raises a question: what determines the evolutionarily stable values of these traits? It is tempting to conclude that these trait values are determined by the nature of the environment alone. However, the fact that the evolutionary trajectories of social and asocial populations are so different (Figure 1), suggests that the collective behaviors discussed above strongly influence the outcome of evolution. Analysis of Equations 1–4 reveals that the preferred speed parameter divides the dynamical behavior of populations into distinct collective states (Figure 3; analysis in Appendix section 5). For $\Psi <0$, individuals have a preferred speed of zero and the inter-individual distances are governed by initial conditions. In this state, individuals resist acceleration due to social interactions. For small $\Psi >0$, individuals form relatively dense groups that move through the environment as collectives, either milling, swarming, or translating (D’Orsogna et al., 2006), the collective motions exhibited by real schooling fish (Tunstrøm et al., 2013). Individual speeds are relatively low and inter-individual distances are short. For large $\Psi$, inter-individual distances are large, and individuals move through the environment quickly. Dynamic changes among theses states are evident in Video 2. These collective states are also clearly distinguishable in Figure 3 ($0<\Psi <1.6$ and $\Psi >2.9$) and Appendix Figure 9 ($\Psi <0$), and are separated by abrupt changes in the distances between near neighbors (the inverse of local density, Figure 3) or potential energy (Appendix Figure 9). The location of transitions between states depends on the parameters of the social response rule (e.g., number of neighbors an individual pays attention to $k$; Figure 4). The transitional regimes between these states are reminiscent of the first-order phase transitions that occur in some physical systems, for example at the transition between liquid water and water vapor. As in the liquid-vapor phase transition, transitions in collective state are characterized by strong hysteresis (Figure 3). If the population begins with large $\Psi$, mean distance to neighbors remains stable for decreasing $\Psi$ and then decreases abruptly (Figure 3, Appendix Figure 9 upper curve). If $\Psi$ is then increased, mean distance to neighbors increases but follows a different functional relationship with $\Psi$ (Figure 3, lower curve). We refer to the collective states as station-keeping ($\Psi <0$; see Appendix Figure 9), cohesive (small $\Psi$), and dispersed (large $\Psi$). The analogy between transitions in collective state in our system and first order phase transitions in physical systems can be made more precise by analyzing the formation rate of groups when $\Psi$ is in the hysteresis region. In the hysteresis region, the rate at which groups of individuals form spontaneously (and therefore nucleate a transition from the dispersed to cohesive state) depends strongly on $\Psi$; when $\Psi$ is near the upper bound of the hysteresis region, the time required for a group to form spontaneously is very long (see Appendix section 5.4). From a thermodynamic perspective, this makes the spontaneous formation of groups extremely unlikely, which explains why populations that begin in the dispersed state follow the upper branch of the hysteresis curve shown in Figure 3.

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Figure 4

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For a wide variety environmental conditions (Appendix section 2) and social parameters (Figure 4), the evolutionarily stable trait values have a notable feature: the evolved values of the baseline speed parameter, ${\psi }_0$, place individuals in the population slightly above the transition between cohesive and dispersed states when $S=0$ (Figure 4, upper panels, Figure 5; points in both figures show mean ${\psi }_0$ values of population in the ESSt), and the evolved environmental sensitivity, ${\psi }_1$, is large enough that locally, groups of individuals cross from the dispersed state through the cohesive and station-keeping states in regions of the environment where the resource value is high (Figure 2A, colors indicate instantaneous value of $\Psi$ for each individual). In other words, the evolved values of ${\psi }_0$ and ${\psi }_1$ allow local subpopulations to undergo sudden changes from one collective state to another in the proximity of favorable regions of the environment. Importantly, the approximate location of the transition between cohesive and dispersed states can be predicted by directly analyzing Equations 1–4 without considering details of the environment, or the mapping between behavior and fitness (Figure 4 compare upper panels [simulation] to lower panels [analytical prediction]). While the precise evolutionarily stable values of ${\psi }_1$ depend on the parameters of the environment (Appendix section 2), the evolutionarily stable values of ${\psi }_0$ place the population near the cohesive-dispersed transition in many different kinds of environments (Appendix Figure 5). As we show below, being near this transition allows groups to respond quickly to changes in the environment. Our results demonstrate, that such locations in behavioral state-space are, in fact, evolutionary attractors.

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The evolutionary results presented in Figure 1 assume that individuals do not appreciably deplete the resource. We can explore an alternative scenario in which resource peaks are depleted through consumption (Appendix section 2.8). In that case, the $i$th individual consumes resources at a rate $uS\left(x_i\right)$ per time step. We repeated evolutionary simulations assuming either a high or low rate of resource consumption $u$. For high consumption rate (100 individuals can deplete a peak in roughly five time steps), $l_{max}$ still increases so that individuals are attracted to one another through social interactions, but selection for large $l_{max}$ is much weaker than the case shown in Figure 1C (see Appendix Figure 7). Moreover, ${\psi }_0$ and ${\psi }_1$ increase continually through evolutionary time. This result is intuitive because when resources are depleted rapidly, the locations of neighbors convey little information about the future location of resources and transitioning from the dispersed to cohesive state may actually be maladaptive. By contrast, when individuals consume the resource at a more moderate rate (Appendix Figure 7), evolutionary trajectories parallel the trajectory shown in Figure 1C–E; there is strong selection for high $l_{max}$, ${\psi }_0$ reaches a stable value that is situated directly above the hysteresis region shown in Figure 3, and ${\psi }_1$ evolves to a stable value that is large enough to allow individuals to cross from dispersed to cohesive, and station-keeping states in regions of the environment where the resource value is high.

Why do populations of selfish individuals evolve behavioral rules that place them near the transition between collective states? Dispersed, cohesive, and station-keeping states are each associated with a characteristic density (low, intermediate, and high, respectively; Figure 3, Appendix Figure 9). If individuals enter the cohesive and station-keeping states where the resource level is high, the density of individuals becomes strongly correlated with the resource distribution (Figure 6A). The similarity between the distribution of individuals and the distribution of the resource can be quantified by the Kullback-Leibler divergence (KL divergence), an information-theoretic concept that measures the distance between two distributions (Figure 6A inset). Though individuals cannot sense resource gradients, they can detect gradients in the density of their neighbors (Equation 1), and can therefore move up the resource gradient.

Figure 6

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The abrupt transitions in the density of individuals between dispersed and cohesive states (Figure 3) mean that there is a strong density gradient in regions of the environment where individuals in the dispersed state border individuals in the cohesive state (e.g., Figure 2A, 6A, Video 2). This suggests that the behavior of an individual in this region can be approximated by considering only its interactions with individuals that are on the resource peak (i.e., where density is high). Using this assumption, we derive analytically the rate at which new individuals join (or rejoin) a group on the resource peak (Appendix section 6.5). Asocial individuals arrive at a resource peak at a rate ${\kappa }_a$, where ${\kappa }_a$ is a constant (Figure 6B, blue curves and points; Equation A65). However, social individuals initially arrive at a rate that increases as more individuals reach the peak, such that the number of individuals on the peak, $N_s$, increases exponentially with time: $N_s\approx {\kappa }_{s\mathrm{,1}}+\text{exp}\left({\kappa }_{s\mathrm{,2}}t\right)$, where ${\kappa }_{s\mathrm{,1}}$ and ${\kappa }_{s\mathrm{,2}}$ are positive constants (Figure 6B, red curves and points; Equation A68–A70). Analytical calculations (Figure 6B, solid lines) agree well with results of numerical simulations (Figure 6B, points and confidence bands). The rapid accumulation shown in Figure 6 is especially important when the environment changes quickly with time; it allows groups to respond swiftly to changes in the resource field and enables the emergent resource tracking behavior described above.

The form of Equations (3–4) implies that an individual’s behavioral response combines personal information about the environment (Equation 2) with social cues (Equation 1). In fact, under a time rescaling, our model is equivalent to one in which the relative strength of social forces varies across the environment (Appendix section 4). The tradeoff between using social information and personal information is inherent in social decision-making (Couzin et al., 2005; Couzin, et al., 2011). This tradeoff means that individuals with large ${\psi }_0$ and ${\psi }_1$ are, by default, less responsive to their neighbors. Perturbing the values of ${\psi }_0$ and ${\psi }_1$ of individuals in populations at the ESSt show that, in populations with high mean ${\psi }_0$, individuals fail to form large groups and are poor at tracking resource peaks (Appendix section 2.6, Appendix Figure 6). In populations with high mean values of ${\psi }_1$, individuals form groups (Appendix section 2.7), but fail to exploit regions with the highest resource quality. Individuals with low values of ${\psi }_0$ or ${\psi }_1$ form groups but do not effectively track dynamic resources (Appendix section 2.7).

Our model of the resource environment incorporates three salient features of the resource environments that schooling fish and other social foragers encounter in nature. These features are: 1) spatial variation in resource quality, 2) temporal variation in resource quality, and 3) characteristic length scales of resource patches (Stephens et al., 2007; Bertrand et al., 2008; Bertrand et al., 2014). Accordingly, we model a two-dimensional environment in which the resource is distributed as a set of $M$ resource peaks. We assume the boundary of the environment is periodic such that individuals, inter-individual potentials, and resource peaks are all projected onto a torus. Each of the $M$ peaks decays like a Gaussian with increasing distance to the peak center. The value of the resource in a single peak at a location, $x_i$, is given by

where ${\lambda }_0$ is a constant that determines the resource value at the peak center and ${\lambda }_1$ is a decay length parameter, and $x_s$ is the location of the centroid of the peak of interest. The total resource value the $i$th individual experiences $S\left(x_i\right)$ is the sum over all peaks in the environment. Each peak moves according to Brownian motion with drift vector $\alpha$ and standard deviation $\beta$. At each time step, each peak has a probability $1/{\tau }_p$ of disappearing and reappearing at a new location, chosen at random from all locations in the environment.

Agents obeying the equations described in the Main text exhibit several distinct collective states. One such state, which we call the cohesive state, is characterized by dense groups of agents occupying a fixed fraction of the environment. One of the salient properties of these groups is that they eventually reach a fixed density that becomes independent of group size. Using a small number of simple assumptions about the behavior of agents within a cohesive group, we are able to predict the density of agents directly from the model parameters. The motivation of our calculation comes from the structure of the equations, which include social potential terms and velocity-dependent self-propulsion terms. The social force on an agent is given by Equation A1. We will rewrite the social potential in Equation A1 (i.e., the term in brackets) as

The effect of the potential term in the equations is to exert a force on the entire system towards configurations where the potential energy is lower. The propulsive forces are non-conservative, causing phase-space volumes to contract, allowing the system to approach a potential energy minimum. We model the cohesive state as $N$ agents occupying a circular region of radius $l$. Further, we assume that the probability distribution of agents within this circular region is uniform, so that agent density is given by:

The density $\rho$ lets us define an interaction radius $l_I$ which is expected to contain $k$ individuals:

This expression is valid when $N>\mathrm{}k$, which is the case we are interested in. When $N<\mathrm{}k$ the interaction radius is simply the group radius, $l=\mathrm{}{l}_I$, and each agent interacts with every other agent. We calculate the expected potential by integrating over a circle of radius $l_I$:

This integral evaluates to the following expression:

It is illustrative to write this expression after the substitution $l=\mathrm{}\sqrt{\frac{N}{\pi \rho }}$:

The density that minimizes ${\Phi }_i$ will be the density of agents in the cohesive state. When written this way, it is clear that $N$ will not influence the location of the minimum of ${\Phi }_i$, as $N$ only appears in the expression as a constant multiplier. Thus, when $N>\mathrm{}k$ we expect cohesive groups to have a constant density, so that the radius of a group grows like $\sqrt{N}$. These predictions match the results of our simulation quite closely, which one can see from comparisons between Appendix Figure 8 to the lower branch of the hysteresis plot in Appendix Figure 9.

Appendix Figure 8

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Appendix Figure 9

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That density is necessarily constant with increasing $N$ is a hallmark of topological interaction laws which are repulsive at short range. When there is no restriction on the number of interaction neighbors, an interaction law of the type that we use can give rise to catastrophic behavior, where the group density increases without bound with increasing $N$ (D’Orsogna et al., 2006). One feature of the topological interaction is that it allows for biological realism for agent parameter values that would otherwise lead to catastrophic behavior.

In order to better understand how changing parameters affect our model, we ignore stochasticity and consider the equations for the acceleration and velocity in a homogeneous environment without resources (so that the background velocity is constant). First we define:

The equations become:

Let $l_0$ be a characteristic length scale in this problem, let $v_0$ be a characteristic velocity scale, and let $t_0=\mathrm{}\frac{l_0}{v_0}$ be a characteristic time scale. Then, let $C_a$, the attraction coefficient, be the scale of the potential. We non-dimensionalize our equations by rewriting them in terms of the dimensionless variables:

The resulting dimensionless equations are:

The non-dimensional number $\frac{l_0^2{C}_a}{v_0^2}=\mathrm{}\frac{\eta l_0^2{C}_a}{{\psi }_0}$ measures the relative strength of the social potential. When ${\psi }_0$ becomes large, social forces become negligible. The reason for this effect is that agents begin moving too quickly for the social forces to have any appreciable effect on their trajectories. Therefore, for constant $l_0$ and $C_a$, an alternative interpretation of ${\psi }_0$ is as a term that dictates the relative strengths of autonomous versus social forces.

Figure 3 in the Main Text illustrates that populations exhibit distinct regimes, which we refer to as collective states, as a function of the preferred speed parameter. Two states are evident in Figure 3 in the Main Text: a state with short inter-individual distances for small $\Psi$ and a state with large inter-individual distances for large $\Psi$. A third state is evident if the mean potential energy of all individuals in the population is plotted as a function of $\Psi$ in a uniform environment, where potential energy is calculated from Equations 3 and 4 in the Main Text. For $\Psi <0$ there is a distinct drop in potential energy for decreasing $\Psi$ (Appendix Figure 9). We refer to the state that occurs for $\Psi <0$ as station-keeping in the Main Text.

In order to better understand the behavior of agents in the context of our model, we have developed a continuum equation for the time evolution of agent density. In the context of a homogeneous environment this description can be used to predict the points in parameter space at which the uniform, purely solitary state becomes unstable, and to demonstrate that heterogeneous states cannot be stable at high enough background velocity. Although the continuum description is only an approximation, it is able to qualitatively predict many of the features of our multi-agent simulations, which makes the mechanisms responsible for this behavior mathematically more explicit. In order to derive continuum equations, we begin with a Liouville equation for the probability density of all the $N$ particles within the full phase space, and derive a hierarchy of equations by taking moments (Flierl et al., 1999; Born and Green, 1946). This hierarchy can be closed by assuming that stochastic forces are sufficiently strong to ensure independence of the individual agents. For the analysis presented here, we assume that the agents travel at a constant velocity $v_0=\mathrm{}\sqrt{\Psi }$ (using the angular variable $\theta$ to describe the direction of the velocity), and that there is noise in the angular velocity driven by a Wiener process with variance $\sqrt{2\epsilon }$ per unit time. The assumption of a constant velocity implies that we have taken a limit where $\eta$, ${\psi }_0$, and ${\psi }_1$ go to infinity, though their ratios remain constant. We will let ${\psi }_0$ and ${\psi }_1$ stand for the limiting ratios of the original model. We will denote the space of positions by $V$, which will be a 2-torus with length $L_D$. The assumption of a constant velocity and of angular noise lead only to small quantitative changes in agent behavior, and they make it possible to analyze the resulting equations.

Therefore we begin with the following set of stochastic differential equations:

Here $F_i$ is the social force on agent $i$, and ${\Gamma }_i$ is the angular direction of the social force on agent $i$. We assume that this force is produced through a topological interaction of the following form:

Here $N_i$ is the set of $k$ closest neighbors to agent $x_i$, and $w$ is an interaction kernel.

From these equations, we are able to write a Liouville equation by introducing a probability density on phase space, $P\left(\right\{\mathrm{}x\}, \{\theta \left\}\right)\mathrm{}$, where $\left\{\mathrm{}x\right\}\mathrm{}$ is the set of $N$ agent positions and $\left\{\mathrm{}\theta \right\}\mathrm{}$ is the set of $N$ agent directions. The value of $P$ at a given set of positions and directions is the probability that the each of the agents have the specified positions and directions. The Liouville equation is:

In order to simplify this equation, we assume that the probability density function $P$ can be factorized into the product of $N$ identical single particle probability density functions:

This assumption is equivalent to assuming statistical independence of the positions and directions of the agents, a condition which could be reached either through large stochasticity $\epsilon$ or ergodic single particle trajectories. The assumption allows us to derive a closed equation for the single particle probability density function $p$, in a similar fashion to a closure of the usual BBGKY hierarchy in kinetic theory.

Then, we are able to write the following equation for the single agent probability density function $p\left(\mathrm{}x\mathrm{,}\theta \mathrm{,}t\right)\mathrm{}$ (where we have replaced a binomial distribution with a Poisson distribution):

Here, the expression $\lambda (\mathrm{}x\mathrm{,}x\text{'})\mathrm{}$ is the expected number of agents within a distance $|\mathrm{}x\text{'}|\mathrm{}$ from the point $x$, the function $w\left(\right|\mathrm{}x\left|\right)\mathrm{}$ is the social potential between two particles, and the expression $\Gamma (\mathrm{}x\mathrm{,}x\text{'})\mathrm{}$ is the angle of force from an agent located at $x\text{'}\mathrm{}$ to an agent located at $x$:

The presence of the terms involving $\lambda$ are a consequence of the topological interaction law between the agents. This equation is most accurate in the limit where ${\rho }_c{l}_c^2\gg 1$, where ${\rho }_c$ is a characteristic density and $l_c$ is the characteristic length scale of the interaction. In the examples which we considered in our study, this ratio is typically only slightly larger than 1 (see Peshkov et al., 2012; Chou et al., 2012) for derivations of continuum descriptions of topological interactions in a more collisional regime). Despite that, we find both quantitative and qualitative agreement between the continuum description and the agent based model. This kinetic description can be converted into a hierarchy of fluid equations by taking moments with respect to the angular direction variable $\theta$.

We introduce the phase space particle density $f$:

The Fourier series for $f$ gives the important macroscopic variables, for instance:

Here $u$ is the mean velocity of agents at each point in space. We will take moments of the kinetic equation through Fourier series:

The time evolution equation for the $m$th Fourier coefficient is:

Here, we have simplified this expression by introducing two new functionals of the density, $F\left(\mathrm{}x\mathrm{,}\rho \right)\mathrm{}$ and $\Theta \left(\mathrm{}x\mathrm{,}\rho \right)\mathrm{}$, which represent the social force exerted at the point $x$ due to the density $\rho$ and the direction of that social force. Explicitly these are given by:

The evolution of the $m$th moment depends on the value of the $m+1$st moment, so that we have an infinite hierarchy of equations. Moments with high values of $\left|\mathrm{}m\right|\mathrm{}$ experience strong damping, and we can use this to justify discarding all moments with $\left|\mathrm{}m\right|\mathrm{}$ above a given threshold. In the following treatment we will set to zero all Fourier coefficients with $\left|\mathrm{}m\right|>1$, which is the simplest truncation of the hierarchy that leads to non-trivial equations.

The right hand side of the momentum equation leads to rapid equilibration, and we can eliminate the time derivative in this equation. This allows us to find an expression for $\rho u$ in terms of $\rho$ only:

We can use this to write a single closed equation for $\rho$. In order to facilitate the analysis of this equation, we make one further approximation, replacing the sum over Poisson factors with a Heaviside function that is equal to $1$ if the expected number of agents between inside a ball of size $x\text{'}$ is less than $k$, and $0$ otherwise. This captures the dominant qualitative feature of the topological interaction in a simple way: the effective interaction radius is a function of the local density. This approximation is quantitatively consistent with the assumptions of the previous section and the results of our simulations. The resulting equation is:

The advection-diffusion equation will be used to understand the behavior of our multi-agent simulations. From its form one can see the effects of $v_0$: large $v_0$ enhances the diffusivity and reduces the effects of the potential.

5.1 Formation of the cohesive state and stability of the dispersed state

Any constant density function ${\rho }_0$ is an equilibrium solution of Equation A36. A crucial property of our multi-agent model is that depending on the background velocity, the agents have the ability to spontaneously form a dense state which we call the cohesive state. In order for this to be possible, the uniform state must be unstable. One advantage of the continuum description is that it allows us to investigate such questions within a much simpler framework than in the original agent based model. In order to do so, we select a uniform state with value ${\rho }_0$ and linearize around it, neglecting terms second order or higher in the deviation away from ${\rho }_0$:

(A38) $\frac{\partial {\rho }_1}{\partial t}=\mathrm{}\nabla \cdot \left(\frac{v_0^2}{2\epsilon }\nabla {\rho }_1-\frac{{\rho }_0}{2\epsilon }{\int }_{|\mathrm{}x-x\text{'}|<\mathrm{}{L}_0}\nabla w\left(\right|\mathrm{}x-x\text{'}\left|\right)\mathrm{}{\rho }_1(\mathrm{}x\text{'})\mathrm{}{d}^2{x}^{\text{'}}\right)$

(A39) $L_0=\mathrm{}\sqrt{\frac{k}{\pi {\rho }_0}}$

This is translationally invariant, and we have periodic boundary conditions, so we consider the Fourier coefficients of ${\rho }_1$:

(A40) $\frac{\partial {\tilde{\rho }}_{\mathrm{1,}j}}{\partial t}=\mathrm{}-{\tilde{\rho }}_{\mathrm{1,}j}\left(\left|\mathrm{}{j}^2\right|\mathrm{}\frac{2{\pi }^2{v}_0^2}{\epsilon L_D^2}-G\left(\right|\mathrm{}j\left|\right)\mathrm{}\right)$

The term $G\left(\right|\mathrm{}j\left|\right)\mathrm{}$ can be calculated by application of the convolution theorem and integration by parts, using the fact that the integrals are radially symmetric (which is true as long as $L_0<\mathrm{}{L}_D$):

(A41) $G\left(\right|\mathrm{}j\left|\right)=\mathrm{}\frac{-{\rho }_0\pi ij}{\epsilon L_D}\cdot {\int }_V{e}^{\frac{-2\pi ij\cdot x}{L_D}}\nabla w\left(\right|\mathrm{}x\left|\right)\mathrm{}H(\mathrm{}{L}_0-|\mathrm{}x\left|\right)\mathrm{}{d}^{2}x$

(A42) $=\mathrm{}-{\int }_0^{L_0}\frac{2{\pi }^2{\rho }_0\left|\mathrm{}j\right|\mathrm{}}{\epsilon L_D}drr\frac{\partial w}{\partial r}{J}_1\left(\frac{2\pi \left|\mathrm{}j\right|\mathrm{}r}{L_D}\right)$

Here $J_1$ stand for the corresponding Bessel functions of the first kind. Linear stability is determined by the sign of the coefficient on ${\tilde{\rho }}_{\mathrm{1,}j}$ on the right hand side of the following expression:

(A43) $\frac{\partial {\tilde{\rho }}_{\mathrm{1,}j}}{\partial t}=\mathrm{}-\frac{2{\pi }^2{v}_0^2|\mathrm{}j{|}^2}{\epsilon L_D^2}{\tilde{\rho }}_{\mathrm{1,}j}\left(1-\frac{L_D{\rho }_0}{\left|\mathrm{}j\right|\mathrm{}{v}_0^2}{\int }_0^{L_0}rdr\frac{dw\left(\mathrm{}r\right)\mathrm{}}{dr}{J}_1\left(\frac{2\pi \left|\mathrm{}j\right|\mathrm{}r}{L_D}\right)\right)$

We can use our formula to determine the stability or instability of an arbitrary homogeneous equilibrium solution. We have plotted an example of this in Appendix Figure 10. A number of general features emerge from these diagrams. Increases in the background velocity $v_0$ always promotes stability of the dispersed state. The agents make use of this feature to enable themselves to transition from the dispersed state to the cohesive state in regions where $\Psi$ crosses below the stability threshold. Increases in the number of interaction neighbors $k$, the decay length of the attractive interaction $l_a$, and the strength of the attractive interaction $C_a$ all promote instability of the dispersed state, though at large $k$ further increases in $k$ have little effect. The background density of agents has a more complicated effect on the stability of the dispersed state, when ${\rho }_0$ is low the social forces are very weak because the distance between agents is large, so that a very small $v_0$ is required for formation of the dispersed state. When ${\rho }_0$ gets too large, the repulsive part of the interaction becomes more important and stability of the dispersed state is promoted.

Appendix figure 10

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5.4 Some additional properties of the transition between collective states: nucleation rates and hysteresis

In the theory of first order phase transitions, hysteresis often arises because there is a free energy penalty for small droplets of the stable phase. This leads to extremely low probabilities of critical droplet formation near the transition temperature. In order to test whether this effect is responsible for the hysteresis in our model equations, we performed long time numerical simulations using values of $\Psi$ within the hysteresis region, allowing us to estimate the nucleation rate of the cohesive phase. We performed replicate simulations with ${10}^3$ agents, restarting the simulations each time the agents were able to form the cohesive state. The results of this simulation are shown in Appendix Figures 11 and 12, illustrating the super-exponential growth in the mean nucleation time as $\Psi$ increases. This growth in nucleation times corresponds to an increase in the minimum radius of a stable group. When $\Psi$ increases above 1.7, the expected time for nucleating the cohesive state becomes extremely large, leading to strong hysteresis. We also computed $\frac{1}{\sqrt{log\overline{T}}}$ to illustrate the approximate scaling of the nucleation time (Appendix Figure 12). Because we use a topological interaction, we do not necessarily expect this scaling to hold for much larger values of $\Psi$, as groups with $N<25$ will have an increasing, rather than constant, potential energy per particle.

Appendix Figure 11

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Appendix Figure 12

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In this section we derive a model of collective exploration and exploitation that allows us to understand how the ability of agents to find the resource peak changes when model parameters like ${\psi }_0$ or sociality are varied. We model agents as being either in a cohesive state near a resource peak or in a dispersed state. Using the model parameters and some simple assumptions about the dynamics, we calculate the fraction of particles approaching the resource peak that are able to enter the cohesive state. Using this model, we quantitatively estimate the rate at which agents are able to find the peak and the advantage of social agents versus asocial agents. We begin by stating a number of assumptions, each of which arises from some feature of our multi-agent model, that help make our theoretical analysis tractable.

6.2 Critical angle for capture by the peak

Consider a particle reaching $r_M$, the radius where it begins to feel the influence of the agents on the environmental resource peak, as is depicted in Appendix Figure 13. If the angle of the agent’s trajectory is sufficiently directed towards the resource peak, the agent will reach the peak, and if the angle is directed sufficiently away, the agent will not reach the peak. There is a critical angle ${\Delta }_i$ at the boundary between these two scenarios. The size of ${\Delta }_i$ will determine the fraction of agents captured by the resource peak after crossing $r=\mathrm{}{r}_M$, and it will also determine the flux of agents onto the resource peak.

Appendix Figure 13

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To derive an expression for ${\Delta }_i$, we write equations for an agent traveling with a velocity of magnitude $v\left(\mathrm{}r\right)=\mathrm{}\sqrt{{\psi }_0-{\psi }_{1}A{e}^{-r^2/\mathrm{}{\sigma }^2}}$, in direction $\omega$, experiencing the potential force $-\nabla {\Phi }_{\text{p}\mathrm{eak}}\left(r\right)\mathrm{}$:

Our question is the following: given initial radius $r=\mathrm{}{r}_M$, initial angle ${\theta }_0$, and initial direction ${\omega }_0$, does the agent reach the zero velocity radius? The equations of motion are:

(A53) $\frac{dr}{dt}=\mathrm{}v\left(\mathrm{}r\right)\mathrm{}\cos (\mathrm{}\theta -\omega )\mathrm{}$

(A54) $\frac{d\theta }{dt}=\mathrm{}\frac{v\left(\mathrm{}r\right)\mathrm{}}{r}\sin (\mathrm{}\omega -\theta )\mathrm{}$

(A55) $\frac{d\omega }{dt}=\mathrm{}\frac{{\Phi }_{\mathrm{peak}}^{\text{'}}\left(\mathrm{}r\right)\mathrm{}}{v\left(\mathrm{}r\right)\mathrm{}}\sin (\mathrm{}\omega -\theta )\mathrm{}$

We define the angle $\Delta =\mathrm{}-\pi -\omega +\theta$, which is the angle between the velocity vector and the vector directed from the position of the agent to the origin. Then we can rewrite our equations in terms of the variables $r$ and $\Delta$ alone, leading to the following planar system:

(A56) $\frac{dr}{dt}=\mathrm{}-v\left(\mathrm{}r\right)\mathrm{}\cos \left(\mathrm{}\Delta \right)\mathrm{}$

(A57) $\frac{d\Delta }{dt}=\mathrm{}-\left(\frac{{\Phi }_{\mathrm{peak}}^{\text{'}}\left(\mathrm{}r\right)\mathrm{}}{v\left(\mathrm{}r\right)\mathrm{}}-\frac{v\left(\mathrm{}r\right)\mathrm{}}{r}\right)\sin \left(\mathrm{}\Delta \right)\mathrm{}$

The system of equations described here has the following properties:

1. If $\Delta =0$, then $\frac{d\Delta }{dt}=0$, so that $\Delta =0$ for all further times. Similarly, $\Delta =0$ implies $\frac{dr}{dt}<0$, so that any agent with $\Delta =0$ will reach the zero velocity radius.

2. If $-\frac{\pi }{2}<\Delta <\mathrm{}\frac{\pi }{2}$, then $\frac{dr}{dt}<0$ and the agent will move closer to the peak.

3. If both $\left(\frac{{\Phi }_{peak}^{\text{'}}\left(\mathrm{}r\right)\mathrm{}}{v\left(\mathrm{}r\right)\mathrm{}}-\frac{v\left(\mathrm{}r\right)\mathrm{}}{r}\right)>0$ and $-\frac{\pi }{2}<\Delta <\mathrm{}\frac{\pi }{2}$, then $\frac{d\left|\Delta \right|\mathrm{}}{dt}<0$, and $\Delta$ becomes closer to $0$.

We make one additional assumption, which has been true in most practical cases, that allows us to make progress with the analysis.

Assumption: The function ${\Phi }_{\mathrm{peak}}^{\text{'}}\left(r\right) - \frac{v(r{)}^2}{r}$ has exactly one sign change on the interval $(r_0,\infty )$. The location of the sign change occurs when ${\Phi }_{\mathrm{peak}}^{\text{'}}\left(r\right) = \frac{v(r{)}^2}{r}$, a point which we denote by r*. At r = r*, there is a balance between centrifugal and potential forces. We have the following inequalities:

(A58) $\frac{{\Phi }_{\mathrm{peak}}^{\text{'}}\left(r\right)}{\nu \left(r\right)}<\frac{\nu \left(r\right)}{r} \mathrm{for} r<r*, \frac{{\Phi }_{\mathrm{peak}}^{\text{'}}\left(r\right)}{\nu \left(r\right)}>\frac{\nu \left(r\right)}{r} \mathrm{for} r>r*$

This assumption allows us to divide the $(r, ∆)$ plane into four regions:

1. $-\frac{\pi }{2}<∆<\frac{\pi }{2}\text{and}\text{r < r*}$
2. $∆<-\frac{\pi }{2}\text{or}\frac{\pi }{2}\text{< ∆}\text{and}\text{r > r*}$
3. $-\frac{\pi }{2}<∆<\frac{\pi }{2}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\text{and}\text{r > r*}$
4. $∆<-\frac{\pi }{2}\text{or}\frac{\pi }{2}\text{<∆}\text{and}\text{r < r*}$

In region 1, the agents move towards the peak and the potential force is stronger than the centrifugal force. In region 2, the agents move away from the peak, and the potential force is weaker than the centrifugal force. In region 3, agents move towards the peak but the potential force is weaker than the centrifugal force. In region 4, the agents move away from the peak but the potential is stronger than the centrifugal force. We conclude that:

1. Any trajectory that enters region 1 will reach the zero velocity radius.

2. Any trajectory that enters region 2 will escape to $\infty$.

Consider again the hypothetical agent in region 3 and at radius r = r_M. The agent will eventually either reach region 1, region 2, or the boundary points between the two regions (which are unstable equilibria).

1. The points $\left(r*,\pm \frac{\pi }{2}\right)$ are unstable equilibrium points, each corresponding to a periodic orbit around the resource peak.

2. There are two values of Δ such that the solution of the initial value problem with initial condition (r_M, Δ) reach these equilibria. We call these angles $\pm {∆}_i$, where we define ${∆}_i>0$. Any trajectory with initial value $(r_M, ∆)$, with $|∆|<{∆}_i$ will enter region I and be captured by the resource peak, and any trajectory with initial value satisfying $|∆| >{∆}_i$ will enter region II and escape the resource peak.

The angle ${\Delta }_i$ is the critical angle that we seek.

6.3 Solving the reduced system

Instead of considering the time dependent differential equation, we search for an equation that describes the shape of a trajectory, that is, we assume $\Delta$ is a single valued function of $r$, and use the original equation in combination with the chain rule to write a differential equation for $\Delta \left(\mathrm{}r\right)\mathrm{}$. This method is valid in regions where $\Delta$ is actually a single-valued function of $r$, and for this to be the case integration must be restricted to regions where ${\Phi }_{\mathrm{peak}}^{\text{'}}\left(\mathrm{}r\right)\mathrm{}-\frac{v^2\left(\mathrm{}r\right)\mathrm{}}{r}$ has only one sign. The resulting equation will be valid only while a trajectory is in region $3$, and the coefficients of this equation will blow up at the border of region $3$.

The quotient of Equation A56 and A57 is the desired equation:

(A59) $\frac{dr}{d\Delta }=\mathrm{}\frac{\cot \left(\mathrm{}\Delta \right)\mathrm{}}{\frac{{\Phi }_{\mathrm{peak}}^{\text{'}}\left(\mathrm{}r\right)\mathrm{}}{v{\left(\mathrm{}r\right)\mathrm{}}^2}-\frac{1}{r}}$

Equation A59 can be solved by integrating along a trajectory beginning at $\left(\mathrm{}{r}_M\mathrm{,}{\Delta }_i\right)\mathrm{}$, and ending at $\left(r^{*}\mathrm{,}\frac{\pi }{2}\right)\mathrm{}$, leading to:

(A60) ${\int }_{r_M}^{r^{*}}\left(\frac{{\Phi }_{peak}^{\text{'}}\left(r\right)\mathrm{}}{v{\left(r\right)\mathrm{}}^2}-\frac{1}{r}\right)dr={\int }_{{\Delta }_i}^{\frac{\pi }{2}}\mathrm{}\cot \left(\Delta \right)\mathrm{}d\Delta$

To simplify the resulting expressions, let $F\left(\mathrm{}r\right)\mathrm{}$ be the anti-derivative of $\frac{{\Phi }_{\mathrm{peak}}^{\text{'}}\left(\mathrm{}r\right)\mathrm{}}{v{\left(\mathrm{}r\right)\mathrm{}}^2}$.

(A61) $F\left(\mathrm{}r\right)={\int }^r\mathrm{}\frac{{\Phi }_{\mathrm{peak}}^{\text{'}}\left(\mathrm{}r\right)\mathrm{}dr}{v{\left(\mathrm{}r\right)\mathrm{}}^2}$

This leads to an integral equation:

(A62) $F\left(r*\right)-F\left(r_M\right)-\log \left(\frac{r*}{r_M}\right)=-\log \left(\sin \left({∆}_i\right)\right)$

This equation can be solved for the critical angle ${\Delta }_i$:

(A63) ${∆}_i={\sin }^{-1} \left(\frac{r*}{r_M}{e}^{F\left(r_M\right)-F\left(r*\right)}\right)$

The critical angle ${\Delta }_i$ is a function of the parameters defining the agent behavior, such as ${\psi }_0$ and ${\psi }_1$, and the parameters defining the clump, such as the peak occupancy $N$. When ${\psi }_0$ is very small, trajectories spend much more time under the influence of the potential, and consequently it is much more likely that they are captured by the peak. Thus, for small ${\psi }_0$, ${\Delta }_i=\mathrm{}\pi /2$. As $N$ increases, the potential becomes stronger, and the values of ${\Delta }_i$ are increased for all ${\psi }_0$. When ${\psi }_0$ is too large ${\Delta }_i$ goes to $0$ and agents cannot find the peak. Appendix Figure 15 contains a plot demonstrating the aforementioned properties of ${\Delta }_i$.

Appendix Figure 14

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Appendix Figure 15

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Appendix Figure 16 contains a plot of the direction field Equation A59 and plots of trajectories that reach $\left(\mathrm{}{r}^{*}\mathrm{,}\pm \frac{\pi }{2}\right)\mathrm{}$, demonstrating the trapping of trajectories that enter region $I$, and providing numerical confirmation of our formula for the critical angle ${\Delta }_i$.

Appendix Figure 16

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