The evolution of distributed sensing and collective computation in animal populations.

Many animal groups exhibit rapid, coordinated collective motion. Yet, the evolutionary forces that cause such collective responses to evolve are poorly understood. Here, we develop analytical methods and evolutionary simulations based on experimental data from schooling fish. We use these methods to investigate how populations evolve within unpredictable, time-varying resource environments. We show that populations evolve toward a distinctive regime in behavioral phenotype space, where small responses of individuals to local environmental cues cause spontaneous changes in the collective state of groups. These changes resemble phase transitions in physical systems. Through these transitions, individuals evolve the emergent capacity to sense and respond to resource gradients (i.e. individuals perceive gradients via social interactions, rather than sensing gradients directly), and to allocate themselves among distinct, distant resource patches. Our results yield new insight into how natural selection, acting on selfish individuals, results in the highly effective collective responses evident in nature.

Introduction

In many highly coordinated animal groups, such as fish schools and bird flocks, the ability of individuals to locate resources and avoid predators depends on the collective behavior of the group. For example, when fish schools are attacked by predators, 'flash expansion' (Pitcher et al., 1993) and other coordinated collective motions, made possible above a certain group size, reduce individual risk (Handegard et al., 2012). Similarly, fish can track dynamic resource patches far more effectively when they are in a group (Berdahl et al., 2013). When an individual responds to a change in the environment (e.g., predator, resource cue), this response propagates swiftly through the group (Rosenthal et al., 2015), altering the group’s collective motion. How are such rapid, coordinated responses possible? These responses may occur, in part, because the nature of social interactions makes animal groups highly sensitive to small changes in the behavior of individual group members; theoretical (Couzin et al., 2002; D’Orsogna et al., 2006; Kolpas et al., 2007) and empirical (Tunstrøm et al., 2013; Buhl et al., 2006) studies of collective motion have revealed that minor changes in individual behavior, such as speed (Tunstrøm et al., 2013), can cause sudden transitions in group state, reminiscent of similarly sudden phase transitions between collective states in physical systems (such as the solid-liquid-gas transitions as a function of increasing temperature). It has been proposed that individuals may trigger such changes in collective state by responding to the environment, thereby initiating a coordinated response at the group level (e.g., Couzin et al. (2002); Kolpas et al. (2007); Couzin and Krause, 2003). This mechanism requires that the behavioral rules of individual animals within a population have evolved in a way that allows groups to transition adaptively among distinct collective states. The evolutionary processes that could lead to this population-level property, however, remain poorly understood.

The feedback between the behavioral phenotypes of individuals, the collective behaviors that these phenotypes produce, and individual-level fitness consequences has made it challenging to study how complex collective behaviors evolve (Torney et al., 2011). Many species, including fish and birds, form groups in which members have low genetic relatedness, which implies that kin selection alone cannot explain the evolution of collective behavior. Moreover, while natural selection acts on the behavioral phenotypes of selfish individuals, collective behaviors are group-level, or perhaps even population-level, properties rather than heritable individual phenotypes. To understand how collective behaviors evolve, then, one must first understand the mapping between individual phenotypes and collective behavior, and between collective behavior and individual fitness.

Here, we take advantage of detailed studies of the social interaction rules and environmental response behaviors of schooling fish (Berdahl et al., 2013; Katz et al., 2011) to develop a biologically-motivated evolutionary model of collective responses to the environment. Using analytical methods and evolutionary simulations, we study how individual behavioral rules produce collective behaviors, and how collective behaviors, in turn, govern the fitness and evolution of selfish individuals. To relate individual and collective behaviors to fitness, we consider a fundamental task faced by fish and other motile organisms: finding and exploiting dynamic resources (Stephens et al., 2007). In our model, individuals respond to the locations of near neighbors and also to local measurements of resource quality. Each individual achieves a fitness determined by the resource level it experiences over its lifetime. We use this framework to explore the evolution of complex collective responses to the environment, and how such responses are related to transitions in collective state.

Model development

Behavioral rules

We model the movement behaviors of each individual in a population of size 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 th individual responds to its neighbors using the following rule:

where is the social force on the th individual, is the position of the th individual, is the two-dimensional gradient operator, the term in brackets is a social potential, , , , and are constants that dictate the relative strengths and length scales of social attraction and repulsion, and the set is a set of the nearest neighbors of the th individual, where a neighbor is an individual within a distance of 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, determines the length scale over which individuals are influenced by social interactions. If is greater than but less than , individuals repel one another at short distances but do not attract one another. We refer to such individuals as asocial (Appendix section 1). If is greater than both and , individuals repel one another at short distances and are attracted to one another at intermediate distances as observed by Katz et al. (2011). Finite 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 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 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 is the autonomous force the th individual generates by accelerating or decelerating in response to the environment, is a monotonically decreasing function of the value of an environmental cue, is the cue value at the th individual’s position, is a damping term that limits individuals to a finite speed, and is the th individual’s velocity. In the absence of social interactions, individuals travel at preferred speed (for ). 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 is mass. D’Orsogna et al. (2006) explores the behavior of a similar model with 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, , within which the acceleration due to social influences (Equation 1) and resource value are assumed to be constant. Positions, speeds, and accelerations of all individuals at time are then given by the solutions to Equations 3 and 4 at time , with the values of and determined at time . A navigational noise vector of small magnitude and uniform heading 0 to 2 is added to the velocity of each agent at each time step. Taking the limit as 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.

Evolutionary dynamics

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: , where dictates the th individual’s preferred speed when the level of the environmental cue is zero and determines how sensitive the th individual is to the cue value (Berdahl et al., 2013). Rather than prescribing values of and , 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 , which determines whether individuals are social ( length scale of social attraction) or asocial ( length scale of social attraction, Appendix section 1).

In each generation, 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 , , and 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. 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 , , and are allowed to evolve, to the evolution of populations of asocial individuals, for which is set to a constant (Appendix section 1).

Results

Evolution of behavioral rules

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 ) and accelerate and decelerate rapidly in response to changes in the resource value (i.e., high ; Figure 1A–B, Appendix).

Figure 1.

Evolution of behavioral rules.

(A, B) show evolutionary dynamics of populations of asocial individuals (i.e., maximum length scale of social interactions  fixed; see text). (C-E) show evolutionary dynamics of individuals in which the maximum length scale of social interactions  is allowed to evolve. Brightness of color indicates the frequency of a phenotype in the population. In asocial populations, baseline speed parameter  () and environmental sensitivity (B) increase continually through evolutionary time. When is allowed to evolve (, individuals quickly become social ( approaches maximum allowable value of 30), and baseline speed parameter () and environmental sensitivity () stabilize at intermediate values. Mean fitness of social populations (F, red points) is over five times higher than mean fitness of asocial populations (F, blue points), and the coefficient of variation in fitness is over four times lower in social populations (F inset). Unless otherwise noted, parameter values in all figures are as follows: , , , , , , , , , , , , , , and .

DOI: http://dx.doi.org/10.7554/eLife.10955.003

Video 1.

Asocial population.

Responses of population of asocial individuals (points) and dynamic resource peak (resource value shown in grayscale; dark regions have high resource value, light regions have low resource value). Length of tail proportional to speed. Peak centroid moves according to 2D Brownian motion with drift vector and standard deviation  (see Materials and methods). In Videos 1–4, view is zoomed in to area surrounding moving resource peak (field of view is , where  is the length scale of repulsion; full environment is projected onto a torus with edge length ). Behavioral parameters as follows: , , , , , , , , , . Environmental parameters in Videos 1–4 are: , , , , , , .

DOI: http://dx.doi.org/10.7554/eLife.10955.004

Video 4.

Population with mean  above the ESSt value.

Responses of perturbed ESSt population to dynamic resource peaks. All parameters as in Video 2 except that each individual’s value of  parameter is increased so that the population mean . Note that individuals do not form large groups near resource peaks and fail to track peaks as they move.

DOI: http://dx.doi.org/10.7554/eLife.10955.012

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 and 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 , , and 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).

Evolved populations collectively compute properties of the environment

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.

Collective tracking of dynamic resource and length-scale matching.

() Sequence (left to right, top to bottom) of individuals interacting with moving resource peak (resource value in grayscale, darker = higher resource value). Peak is drifting to the right (grey arrow). Colors indicate the regime into which each agent falls (red: , blue: , green: ). Length of tail is proportional to speed. Peak centroid moves according to 2D Brownian motion with drift (see Materials and methods). () When environments contain multiple resource peaks, evolved populations divide into groups that match peak sizes, e.g., in a two-peak environment, the size of group on each peak is proportional to peak size. Total size of two peaks is constant so that the larger the first peak (Peak 1, x-axis), the smaller the second peak. Peak size computed as the integral of the resource value over the entire peak (see Materials and methods). Group size is mean size of the group nearest each peak (mean taken over the last 2,500 time steps of each simulation). Points (and error bars) represent mean ( 2 standard errors) of 1,000 simulations for each combination of peak sizes. Parameters as in Figure 1 with and values of , , and taken from a population in the ESSt.

DOI: http://dx.doi.org/10.7554/eLife.10955.005

Video 2.

Population at the evolutionarily stable state (ESSt).

Responses of population of individuals evolved for 1500 generations to the ESSt to dynamic resource peaks. Behavioral parameters as in Video 1 with , , , and , where denotes mean over the population. Note rapid accumulation of individuals near peaks and dynamic peak-tracking behavior of groups.

DOI: http://dx.doi.org/10.7554/eLife.10955.006

Video 3.

Population with mean  below the ESSt value.

Responses of perturbed ESSt population to dynamic resource peaks. All parameters as in Video 2 except that each individual’s value of  parameter is lowered so that the population mean . Note swarms of individuals form in regions of the environment that are far from resource peaks. Individuals explore poorly and therefore have low fitnesses.

DOI: http://dx.doi.org/10.7554/eLife.10955.011

Evolved populations are poised near abrupt transitions in collective state

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 , 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 , 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 , 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 ( and ) and Appendix Figure 9 (), 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 ; 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 , mean distance to neighbors remains stable for decreasing and then decreases abruptly (Figure 3, Appendix Figure 9 upper curve). If is then increased, mean distance to neighbors increases but follows a different functional relationship with (Figure 3, lower curve). We refer to the collective states as station-keeping (; see Appendix Figure 9), cohesive (small ), and dispersed (large ). 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 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 ; when 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.

Figure 3.

Hysteresis plot of the distance to 10 nearest neighbors, averaged over the entire population  (points and error bars) as a function of preferred speed parameter in a uniform environment.

Figure produced by starting with a population with in a uniform environment. Population is allowed to equilibrate for 5000 time steps and is then computed. is then lowered. This process is repeated until , at which point the same procedure is used to increase . Upper curve corresponds to decreasing . Lower curve corresponds to increasing . Regimes where and correspond to transitions between collective states. Points and (error bars) correspond to mean ( 2 standard errors) of 50 replicate simulations. Parameters as in Figure 1 with .

DOI: http://dx.doi.org/10.7554/eLife.10955.007

Appendix Figure 9.

Hysteresis plot of potential energy averaged over the entire population (compare with Figure 3 in Main Text).

Figure produced by starting with a population with Ψ = 4 in a uniform environment. Population is allowed to equilibrate for 5,000 time steps and then average potential energy is calculated using Equation 4 in the Main Text. Ψ is then lowered. This process is repeated until Ψ = −1, at which point the same procedure is used to increase Ψ. Upper curve corresponds to decreasing Ψ. Lower curve corresponds to increasing Ψ. Note drop in mean potential energy at Ψ = 0. We refer to the states on either side of this transition as station-keeping (Ψ < 0) and cohesive Ψ > 0 and below upper transitional regime at Ψ ∼ 1.7. Points and (error bars) correspond to mean (and 2 standard errors) of 50 replicate simulations. Parameters as in Figure 1 of the Main Text with lmax = 30.

DOI: http://dx.doi.org/10.7554/eLife.10955.021

Figure 4.

Evolved populations are positioned near transitions in collective state.

Upper panels show mean distance to 10 nearest neighbors (, color scale) from simulated populations. A separate populations is simulated in a uniform environment for each value of the social attraction strength (), number of neighbors an individual reacts to (), and the decay length of social attraction () parameters. Red is low density corresponding to dispersed state, and blue is high density corresponding to cohesive state. Points show the mean value of of populations in the EESt (populations evolved for 1,000 generations in an environment with dynamic resource peaks). Evolved populations are positioned near transition between cohesive and dispersed states. Lower panels are based on analytical calculations and show the predicted regions in which the dispersed state is stable (white) and unstable (black, Appendix section 5). Parameters as in Figure 1 with , , , , , and .

DOI: http://dx.doi.org/10.7554/eLife.10955.008

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, , place individuals in the population slightly above the transition between cohesive and dispersed states when (Figure 4, upper panels, Figure 5; points in both figures show mean values of population in the ESSt), and the evolved environmental sensitivity, , 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 for each individual). In other words, the evolved values of and 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 depend on the parameters of the environment (Appendix section 2), the evolutionarily stable values of 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.

Figure 5.

Mean distance to nearest neighbors  (curves) and ESSt value of  (points) as a function of social parameters.

Points denote mean ESSt value of . Note abrupt transitions in density as function of , as shown in Figure 3. In all cases, ESSt value of causes populations to cross transition when resource value is high (i.e., , where is maximum resource value of each peak). Densities and ESSt values generated as described in Figure 4.

DOI: http://dx.doi.org/10.7554/eLife.10955.009

Appendix Figure 5.

Trait values after 1500 generations of evolution in randomly generated environments.

Each point represents the mean trait values of a single population that has been allowed to evolve for 1500 generations. Point sizes denote the number of peaks that were present in the environment. Point colors represent the maximum resource value λ0 averaged over all peaks present in the environment. Gray region corresponds to the region of hysteresis shown in Figure 3 of the Main Text. Number of peaks and peak parameters were chosen at random. All other parameters as in Figure 1 of Main Text.

DOI: http://dx.doi.org/10.7554/eLife.10955.017

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 th individual consumes resources at a rate per time step. We repeated evolutionary simulations assuming either a high or low rate of resource consumption . For high consumption rate (100 individuals can deplete a peak in roughly five time steps), still increases so that individuals are attracted to one another through social interactions, but selection for large is much weaker than the case shown in Figure 1C (see Appendix Figure 7). Moreover, and 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 , reaches a stable value that is situated directly above the hysteresis region shown in Figure 3, and 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.

Appendix Figure 7.

Evolution of behavioral traits when individuals consume resource.

Lines show means of independent evolutionary simulations. () High consumption rate corresponding to fast depletion of resource peaks. () Intermediate consumption rate corresponding to slower depletion of the peaks. Note different axis limits in the top panels of and . Grey region corresponds to hysteresis region between collective states shown in Main Text Figure 3. Parameters are as follows: s*=2, high consumption rate = 3.2*10–3 (time step−1), low consumption rate = 8.0*10–5 (time step−1), N = 300. All other parameters as in Figure 1 of the Main Text. These consumption rates correspond to the case where 100 individuals near the peak center can deplete a peak in roughly five time steps (fast depletion, ), and the case where the same task takes 200 time steps (slower depletion, ).

DOI: http://dx.doi.org/10.7554/eLife.10955.019

Changes in collective state allow for rapid collective computation of the resource distribution

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.

Collective computation and social gradient climbing.

() Collective computation of the resource distribution (grayscale represents resource value, normalized to maximum of 1). Curves show local density of individuals at different distances from the resource peak center (maximum value also normalized to 1). Note the rapid accumulation of individuals near the peak center. The distribution of individuals becomes increasingly concentrated in the region where the resource level is highest; inset shows that the Kullback-Leibler divergence between the resource distribution and the local density of individuals decreases through time as the two distributions become more similar. () Number of individuals near peak center (within one decay length, , of peak center) as a function of time. Red and blue points and confidence bands represent means sd. for 100 replicate simulations. Red points and band is ESSt population and blue points and band is an asocial population with the same parameter values. Curves are analytical predictions based on Equations 3 and 4 (Appendix section 6).

DOI: http://dx.doi.org/10.7554/eLife.10955.010

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 , where 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, , increases exponentially with time: , where and 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 and are, by default, less responsive to their neighbors. Perturbing the values of and of individuals in populations at the ESSt show that, in populations with high mean , 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 , individuals form groups (Appendix section 2.7), but fail to exploit regions with the highest resource quality. Individuals with low values of or form groups but do not effectively track dynamic resources (Appendix section 2.7).

Appendix Figure 6.

Performance of populations near the evolutionarily stable state.

() The number of individuals on each peak (the starting peak, blue; the second peak, red) as a function of the mean baseline speed parameter, of a population perturbed from the ESSt. Below a  of roughly 2.2, individuals do not form a large group on the second peak. () Mean resource value of individuals on the starting peak (blue) and second peak (red). () Resource value averaged over all individuals in the population (individuals in groups nearest each peak and all other individuals in the environment). Note maximum value occurs in the regime where individuals aggregate on both the starting and second peaks (3.6). Orange point indicates values corresponding to ESSt. (-) Group size (), mean resource value of individuals on peaks (), and mean resource value of all individuals () as a function of the mean environmental sensitivity parameter of a population perturbed from ESSt. Orange point in () indicates values corresponding to ESSt. Note rapid decrease in mean fitness for perturbations in both directions. Semitransparent points are results of 2000 individual simulation runs. To compute means and standard errors, simulation runs were divided into 50 evenly spaced bins. Bolded points and error bars show mean of each bin  2 standard errors.

DOI: http://dx.doi.org/10.7554/eLife.10955.018

Discussion

Our model demonstrates that selection on the behavioral phenotypes of selfish individuals can lead to the rapid evolution of distributed sensing and collective computation. The mechanism that promotes this evolution involves the use of public information: when individuals respond to the environment by slowing down in regions of high resource quality – a behavior that is adaptive even in the absence of social interactions (Appendix Figure 2) – their positions become correlated with the locations of resources. Social individuals can exploit this public information by climbing gradients in the density of their neighbors. As in simple, game-theoretic models of social foraging (e.g., Clark and Mangel, 1984), social individuals gain a fitness advantage by using information about the environment gleaned by observing neighbors. Because of this, asocial populations are readily invaded by social mutants and collective behaviors evolve (Appendix section 2).

Appendix Figure 2.

Fitness of asocial population through evolutionary time.

Blue points indicate mean fitness of population in each generation. Horizontal red line indicates mean fitness of population over first ten generations. Corresponding and values for each generation are shown in Figure 1A,B of the Main Text.

DOI: http://dx.doi.org/10.7554/eLife.10955.014

Evolutionarily stable populations occupy a distinctive location in behavioral state space: one in which small changes in individual behavior cause large changes in collective state (Figures 4, 5). When individuals respond to local environmental cues by accelerating or decelerating, local populations transition between the collective states shown in Figure 3 (e.g. Figure 2A). This creates the strong spatial gradient in population density (Figure 6A) and allows groups to track dynamic features in the environment rapidly. Perturbations of this evolutionarily stable state cause individuals either to weigh social information too heavily (i.e., small and/or ), in which case groups fail to explore effectively (Video 3, Appendix Figure 7), or to weigh personal information too heavily (i.e., large and/or ), in which case individuals fail to exploit the social information that enables dynamic resource tracking (Video 4, Appendix Figure 7). Because of this, mutants with phenotypes far from the evolutionarily stable state are removed from the population by natural selection. The transitions we observe in collective state bear a resemblance to phase transitions in physical systems, and our results lend credence to the hypothesis that natural selection can result in the evolution of biological systems that are poised near such bifurcation points in parameter space. Importantly, we show that these high-fitness regions of parameter space can be predicted a priori from the structure of individual decision rules, even without knowledge of the environment.

Collective computation is a notion that has strongly motivated research on animal groups (Berdahl et al., 2013; Couzin, 2007; Cvikel, et al., 2015). In our model, populations perform a collective computation through their social and environmental response rules. When individuals are exposed to a heterogeneous resource environment, their responses to the environment cause a modification of the local population density; individuals aggregate in regions where the resource cue is strong. The population performs a physical computation in the formal sense (Schnitzer, 2002): physical variables – the positions and relative densities of neighbors – represent mathematical ones – spatially resolved estimates of the quality of resources in the environment. The environments considered in our study bear a strong resemblance to those encountered in dynamic coverage problems in distributed control theory (Bachmayer and Leonard, 2002), dynamic optimization problems (Passino, 2002), and Monte Carlo parameter estimation (McKay, 2003). Combining an evolutionary approach to algorithm design with collective interactions may therefore be a useful starting point for optimization schemes or control algorithms for autonomous vehicles, particularly if the structure of social interactions leads to bifurcation points in behavioral parameter space as in the model studied here.

Understanding the feedback loop between individual behavior, collective behavior of populations, and selection on individual fitness is a major challenge in evolutionary theory (Guttal and Couzin, 2010; Torney et al., 2011; Pruitt and Goodnight, 2014). Our framework closes this loop and demonstrates how distributed sensing and collective computation can evolve through natural selection on the decision rules of selfish individuals.

Materials and methods

Resource environment

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 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 peaks decays like a Gaussian with increasing distance to the peak center. The value of the resource in a single peak at a location, , is given by

where is a constant that determines the resource value at the peak center and is a decay length parameter, and is the location of the centroid of the peak of interest. The total resource value the th individual experiences is the sum over all peaks in the environment. Each peak moves according to Brownian motion with drift vector and standard deviation . At each time step, each peak has a probability of disappearing and reappearing at a new location, chosen at random from all locations in the environment.

eLife posts the editorial decision letter and author response on a selection of the published articles (subject to the approval of the authors). An edited version of the letter sent to the authors after peer review is shown, indicating the substantive concerns or comments; minor concerns are not usually shown. Reviewers have the opportunity to discuss the decision before the letter is sent (see review process). Similarly, the author response typically shows only responses to the major concerns raised by the reviewers.

Thank you for submitting your work entitled "The evolution of distributed sensing and collective computation in animal populations" for peer review at eLife. Your submission has been favorably evaluated by Ian Baldwin (Senior editor), a Reviewing editor, and two reviewers.

The reviewers are enthusiastic about your paper, and we would like to invite you to revise your manuscript according to the suggestions made by the reviewers.

Summary:

This paper considers a model for collective motion in the presence of a time-varying resource environment, in which the characteristics of future generations of the agents evolve according to how well the current generation obtained resources. The model is based on two behavioral rules: a social response rule and an environmental response rule. It is a force-based model, in which the interactions with other agents and the environment are captured in terms of potentials that encode these rules. Parameters that are allowed to evolve include an individual's preferred speed when there is no environmental cue, and the sensitivity of an individual to an environmental cue.

It is found that the parameters evolve to robust evolutionarily stable states, and these are typically places in parameter space near transitions to other collective states. This allows groups to respond quickly to changes in the environment. It is argued that such changes in the collective state resemble a phase transition in physical systems.

The manuscript presents an important and substantial advance in understanding the evolution of collective motion of animals.

Reviewer #1:

This paper considers a model for collective motion in the presence of a time-varying resource environment, in which the characteristics of future generations of the agents evolve according to how well the current generation obtained resources. The model is based on two behavioral rules: a social response rule and an environmental response rule. It is a force-based model, in which the interactions with other agents and the environment are captured in terms of potentials that encode these rules. Parameters that are allowed to evolve include an individual's preferred speed when there is no environmental cue, and the sensitivity of an individual to an environmental cue.

It is found that the parameters evolve to robust evolutionarily stable states, and these are typically places in parameter space near transitions to other collective states. This allows groups to respond quickly to changes in the environment. It is argued that such changes in the collective state resemble a phase transition in physical systems.

The paper has a nice combination of results from computations and theory, much of the details of which are described in the Appendix. Overall it is very well written and describes important properties of collective motion systems. I feel that it is a very strong contribution to the literature.

My only "major" comment is the following:

It is common for phase transitions to be characterized in terms of critical exponents that describe how quantities of interest scale near the phase transition. Is it possible to do this for the system in the paper? This would help to strengthen the somewhat vague claim that the changes in the collective state "resemble phase transitions".

Reviewer #2:

In my opinion, the manuscript "The evolution of distributed sensing and collective computation in animal populations" presents an important and substantial advance in understanding the evolution of collective motion of animals and could be published in eLife. The authors build up on the relatively well-studied dynamics of non-evolving swarms of interacting particles, introduce important empirically-derived features, and create a comprehensive evolving model which reproduces several common and intuitively-appealing behavioural patterns of such animal groups. In addition to a thorough modeling effort, I especially appreciate the extensive analytic and simplified computational estimates of each important facet of the observed dynamics. Such highly-focused "sub-models" help a reader to verify that the contributions of basic mathematical and biological mechanisms to the rather complex observed phenomenology are understood correctly.

A major issue that naturally comes to one's mind while reading the manuscript is how the results would be affected by adding a realistic feature of a gradual depletion of well-localized resources by swarming consumers. How a rate of such depletion would affect the evolutionary patterns? In the present form, the manuscript is already loaded with new insights, hence, I think, that a qualitative and perhaps even speculative discussion of this topic would suffice.

Reviewer #2 (Minor Comments):

The analogy to the 1-order phase transition based on the density contrast and hysteresis curve could be developed a step further: In traditional first-order phase transitions the hysteresis is caused by the instability of small-size nuclei of an emerging phase. Could something similar be said about the transition between dispersed and cohesive states?

The word "criticality" seems to become too fashionable and abused in many not directly relevant contexts. In its true meaning, a critical state and critical point terminate the line of first-order phase transition rather than sits "close to it". In addition, it is characterized by many unique properties (such as non-analytic behaviour of potentials, scale-free correlations, etc.) which are not mentioned and apparently are irrelevant here. So I would recommend to simply refer to such evolutionary stable state as to one being close to the localized-delocalized transition line.

Same with the word "computation", which, in my understanding, describes some mathematical actions. I think a more appropriate word to describe the collective behaviour would be "collective sensing".

Unless dictated by the journal format requirements, it would be easier on a reader to have the description of the resource dynamics in the Model Development section and to get rid of the Materials and methods one.

The captions to some figures, especially Figure 4 are very difficult to understand. Unless desperately pressed by the size limit, it would definitely help to expand the caption and even split the figure into several ones.

Would it be possible to embed the movies into the main text somewhere near the description of three typical dynamical regimes?

Reviewer #1:

[…] It is common for phase transitions to be characterized in terms of critical exponents that describe how quantities of interest scale near the phase transition. Is it possible to do this for the system in the paper? This would help to strengthen the somewhat vague claim that the changes in the collective state "resemble phase transitions".

We appreciate the reviewer’s point that the analogy between sharp transitions in density/potential energy we observe in collectives, and phase transitions in physical systems could be clarified further. We observe that agents undergo what appears to be a discontinuous change in local density as a function of , so the observed transition more closely resembles a first order phase transition than a second order phase transition. Because of this, we do not expect the kind of power law scaling near the transition point that is a characteristic of second order transitions, and the search for a critical exponent would not be meaningful. On the other hand, hysteresis is a hallmark first order phase transitions so that Figures 3 and Appendix Figure 9 provide evidence that the analogy to phase transitions is sound. Additionally, we have expanded discussion of the hysteresis behavior present in our model and added additional numerical results (Appendix section 5.4) to further clarify the connection between the behavior of our system and first order phase transitions in physical systems (subheading “Evolved populations are poised near abrupt transitions in collective state”, see also the response to reviewer #2’s comment).

Appendix Figure 3.

Evolution of traits under invasion by mutants far from the ESSt.

Example evolutionary progression for , and . Note that invaders (blue points introduced across phenotype space) do not establish and the dominant trait values in the population do not change over evolutionary time. Color indicates frequency of phenotype in population (white = 0, blue = low frequency, orange = high frequency).

DOI: http://dx.doi.org/10.7554/eLife.10955.015

Reviewer #2:

[…] A major issue that naturally comes to one's mind while reading the manuscript is how the results would be affected by adding a realistic feature of a gradual depletion of well-localized resources by swarming consumers. How a rate of such depletion would affect the evolutionary patterns? In the present form, the manuscript is already loaded with new insights, hence, I think, that a qualitative and perhaps even speculative discussion of this topic would suffice.

To address the reviewer’s comment, we performed two additional sets of evolutionary simulations under alternative assumptions about resource depletion. In the first alternative scenario, we assumed that individuals deplete the resource very quickly. As one might expect, social cues provide little useful information about the locations of resources when resources are quickly depleted, and as a consequence, the evolutionary trajectories under fast resource depletion resemble those of asocial individuals. Under more moderate levels of resource depletion, however, the evolutionary trajectories shown in Figure 1 of the main text are largely unchanged; social individuals evolve to an evolutionarily stable state in which 0 values are near the transition point in collective state when the resource is at a low value, and individuals cross the transition shown in Figure 3 in regions where the resource value is high. We have added a discussion of these new results to the manuscript (paragraph three, subheading “Evolved populations are poised near abrupt transitions in collective state”).

Reviewer #2 (Minor Comments):

The analogy to the 1-order phase transition based on the density contrast and hysteresis curve could be developed a step further: In traditional first-order phase transitions the hysteresis is caused by the instability of small-size nuclei of an emerging phase. Could something similar be said about the transition between dispersed and cohesive states?

As the reviewer mentioned, when is increased from a low value to a higher value, in the transitional regime, small nuclei of tightly clustered individuals coexist with individuals in the dispersed state. These nuclei eventually become completely unstable at large enough values of . We have added additional numerical results describing how the rate of nucleus formation depends on in the hysteresis region and described these in subheading “Evolved populations are poised near abrupt transitions in collective state” of the revised manuscript.

The word "criticality" seems to become too fashionable and abused in many not directly relevant contexts. In its true meaning, a critical state and critical point terminate the line of first-order phase transition rather than sits "close to it". In addition, it is characterized by many unique properties (such as non-analytic behaviour of potentials, scale-free correlations, etc.) which are not mentioned and apparently are irrelevant here. So I would recommend to simply refer to such evolutionary stable state as to one being close to the localized-delocalized transition line.

We have removed the term “criticality” throughout the manuscript to avoid confusion about terminology.

Same with the word "computation", which, in my understanding, describes some mathematical actions. I think a more appropriate word to describe the collective behaviour would be "collective sensing".

While we agree with the reviewer that the term “computation” is often misused or used casually, in this manuscript we have strived to define precisely what we mean when we use this term (subheading “Changes in collective state allow for rapid collective computation of the resource distribution” and paragraph three, Discussion). One of our motivations for doing this is to help develop a more rigorous definition of “collective computation” as it is a term that is frequently used in the literature on collective behavior.

Unless dictated by the journal format requirements, it would be easier on a reader to have the description of the resource dynamics in the Model Development section and to get rid of the Materials and methods one.

As far as we can tell, the Material and methods section is required by eLife. If it is allowed, we would be willing to move the details of the resource environment to the Model Development section and omit Materials and methods altogether.

The captions to some figures, especially

We have divided Figure 4 into two figures and worked to clarify the other figure captions.

Would it be possible to embed the movies into the main text somewhere near the description of three typical dynamical regimes?

We have referenced Video 2 in the section of the text that describes dynamical regimes (subheading “Evolved populations are poised near abrupt transitions in collective state”).