Fig 1.
(A) The model considers a population of individuals, here represented as circles, in an explicitly spatial habitat. Individuals reproduce, die, and move stochastically, and are characterized by their level of altruism, indicated in color. Altruism is costly to the actor, but beneficial to recipients: it increases their reproduction rate. (B) For concreteness, imagine that each altruist produces a public good and secretes it locally. The contribution of a particular altruist to the concentration of public good falls with the distance (green curve) and increases with the level of altruism of the actor (vertical arrow). At the same time, individuals compete for a limiting resource: the reproduction rate of each individual is inhibited by each individual in its neighborhood (red curve). The scale of altruism σa and the scale of resource competition σrc are indicated. (C & D) The competition experienced at any coordinate (panel C, red contour plot) and the availability of public good at any position (panel D, green contour plot) are obtained by summing up the contributions of all individuals.
Table 1.
Model parameters and their default values for simulations with the 1D and 2D habitat.
Fig 2.
Altruism and colonies emerge in the two-dimensional habitat.
Results are shown from a representative simulation run (see S1 Fig for replicates) with default parameters (see Table 1). (A) Mean level of altruism versus time (thick colored line) as well as the cumulative contribution of natural selection (black), which is consistently positive (see S1 Text, section 3). (B) Snapshots of the simulation habitat; also see the S1–S3 Videos. In time, the population self-organizes into a hexagonal pattern of discrete colonies. A section of a hexagonal grid is superimposed in the right-most panel. (C) The hexagonal pattern is also apparent from the radial distribution function at t = 8000, the distribution of distances between pairs of individuals normalized by the random expectation. Black arrows indicate the distances occurring in an exact hexagonal grid with grid constant a = 8.4 and their relative frequency. (D) Enlargements of a small domain of the habitat, showing that the colonies behave like Darwinian entities: they disappear as a result of a within-colony tragedy of the commons [38] (red circle and cross), and reproduce by binary fission (green arrows).
Fig 3.
The origins of altruism and colony formation in the one-dimensional habitat.
(A) Dynamics of a representative simulation run (see S4 Fig) with default parameters (Table 1). A small domain of space-time is visualized. The left-hand part of the figure shows the local population density; the striped pattern indicates that regularly spaced colonies develop. The right-hand side plots the center of mass of each colony; color indicates mean level of altruism. The two representations overlap in the middle of the figure to demonstrate their consistency. Black squares mark the deaths of colonies; orange circles indicate reproduction of colonies by binary fission. (B) Prediction from linear stability analysis. (See also section Materials and methods and S2 Fig.) Colonies are expected to emerge in the yellow part of the phase diagram where the scale of competition is clearly larger than the scale of altruism (σa = 1 by definition) and the scale of motility is small. The red cross marks the default parameters used in panel A. (C) Simulation results testing the prediction of panel B. As predicted, colonies emerge only in the region to the right of the red line, which is copied from panel B: the variance of the local population density increases precipitously when the line is crossed. (D) Mean level of altruism at the end of evolutionary simulations. Altruism evolves only in the regime where colonies can form. Each data point plotted represents the mean of three independent replicate simulations. (E) Same as panel D, but as a function of mutation probability μ. Two independent replicates are plotted in gray; colored circles represent their mean value.
Fig 4.
Quantification of multilevel selection: two approaches.
Two approaches to quantify multilevel selection are applied to the simulation of Fig 3A. S4 Fig shows the results of two more replicate simulations. (A) Population mean level of altruism versus time. (B) The first approach (MLS 1) is to mathematically split the natural selection measured acting on the organisms into two parts: selection within and selection among colonies [42]. (See S1 Text Section 4.) For the simulation shown in Fig 3A, this calculation was done for each subsequent interval of 80 generations (see section Materials and methods). Plotted is the change in the mean level of altruism (black), and the within-colony (red) and among-colony (blue) components of selection. The rotated histograms on the right-hand side show distributions based on second half of the simulation, indicated with the gray background. In this part of the simulation, the population mean level of altruism no longer changed systematically. However, within-colony selection is nearly always negative, compensated by positive among-colony selection. (C) The second approach (MLS 2) describes evolution at the level of the colonies. (See S1 Text Section 5.) Evolution taking place within colonies then appears as a transmission bias: a bias in the change between ancestor and offspring colonies in the colony mean level of altruism. This transmission bias (red) tends to be negative, but is compensated by positive selection at the colony level.