1. Ecology
  2. Evolutionary Biology
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The evolution of distributed sensing and collective computation in animal populations

  1. Andrew M Hein  Is a corresponding author
  2. Sara Brin Rosenthal
  3. George I Hagstrom
  4. Andrew Berdahl
  5. Colin J Torney
  6. Iain D Couzin  Is a corresponding author
  1. Princeton University, United States
  2. Max Planck Institute for Ornithology, Germany
  3. Santa Fe Institute, United States
  4. University of Exeter, United Kingdom
  5. University of Konstanz, Germany
Research Article
Cite as: eLife 2015;4:e10955 doi: 10.7554/eLife.10955
6 figures and 4 videos


Evolution of behavioral rules.

(A, B) show evolutionary dynamics of populations of asocial individuals (i.e., maximum length scale of social interactions lmax fixed; see text). (C-E) show evolutionary dynamics of individuals in which the maximum length scale of social interactions lmax is allowed to evolve. Brightness of color indicates the frequency of a phenotype in the population. In asocial populations, baseline speed parameter ψ0 (A) and environmental sensitivity ψ1 (B) increase continually through evolutionary time. When lmax is allowed to evolve (C), individuals quickly become social (lmax approaches maximum allowable value of 30), and baseline speed parameter ψ0 (D) and environmental sensitivity ψ1 (E) 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: C=CrCa=1.1, l=lrla=0.13, N=500, k=25, γ=0.01, τ=1, m=1, ν=1, ρ=0.16, M=2, λ0=10, λ1=20, α=(1,0)β=0.1, and τp=1500.

Collective tracking of dynamic resource and length-scale matching.

(A) 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: Ψ>2.95, blue: 0<Ψ<2.95, green: Ψ<0). Length of tail is proportional to speed. Peak centroid moves according to 2D Brownian motion with drift (see Materials and methods). (B) 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 M=2 and values of ψ0, ψ1, and lmax taken from a population in the ESSt.

Hysteresis plot of the distance to 10 nearest neighbors, averaged over the entire population d10NN (points and error bars) as a function of preferred speed parameter Ψ in a uniform environment.

Figure produced by starting with a population with Ψ=4 in a uniform environment. Population is allowed to equilibrate for 5000 time steps and d10NN is then computed. 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. Regimes where Ψ~0 and Ψ(1.6,2.95) 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 lmax=30.

Evolved populations are positioned near transitions in collective state.

Upper panels show mean distance to 10 nearest neighbors (d10NN, color scale) from simulated populations. A separate populations is simulated in a uniform environment for each value of the social attraction strength (Ca), number of neighbors an individual reacts to (k), and the decay length of social attraction (la) parameters. Red is low density corresponding to dispersed state, and blue is high density corresponding to cohesive state. Points show the mean value of ψ0 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 M=15λ0=10, λ1=1.6, α=(1,0), β=0.1, and τp=1500.

Mean distance to nearest neighbors d10NN (curves) and ESSt value of ψ0 (points) as a function of social parameters.

Points denote mean ESSt value of ψ0. Note abrupt transitions in density as function of Ψ, as shown in Figure 3. In all cases, ESSt value of ψ1 causes populations to cross transition when resource value is high (i.e., ψ0ψ1λ0<0, where λ0 is maximum resource value of each peak). Densities and ESSt values generated as described in Figure 4.

Collective computation and social gradient climbing.

(A) 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. (B) Number of individuals near peak center (within one decay length, λ1, of peak center) as a function of time. Red and blue points and confidence bands represent means ±1 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).



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 α and standard deviation β (see Materials and methods). In Videos 14, view is zoomed in to area surrounding moving resource peak (field of view is 50lr×50lr, where lr is the length scale of repulsion; full environment is projected onto a torus with edge length 346lr). Behavioral parameters as follows: Cr=1.1, Ca=1, lr=1, la=7.5, γ=0.01, τ=1, m=1, η=1, ψ0=3, ψ1=2.54. Environmental parameters in Videos 1are: ρ=0.16, N=300, M=2, λ0=10, λ1=20, α=[0.06 0], β=0.5.

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 k=25, ψ0=3, ψ1=2.45, and lmax=29, where denotes mean over the population. Note rapid accumulation of individuals near peaks and dynamic peak-tracking behavior of groups.

Video 3
Population with mean ψ0 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 ψ0 parameter is lowered so that the population mean ψ0=0.4. Note swarms of individuals form in regions of the environment that are far from resource peaks. Individuals explore poorly and therefore have low fitnesses.

Video 4
Population with mean ψ0 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 ψ0 parameter is increased so that the population mean ψ0=8.8. Note that individuals do not form large groups near resource peaks and fail to track peaks as they move.


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