Automated multilevel lineage tracking.

Qualitatively, the behavior of the 1D model is analogous to that of the 2D model. Fig 3A shows a section of the space-time arena for a simulation with default parameters (see Table 1). The left-hand side of the figure (gray scale) presents the population density. The striped pattern clearly reveals the formation of regularly spaced colonies that can persist for thousands of generations. An algorithm was used to detect these colonies automatically and track them in time (see section Materials and methods). The right-hand side of the figure plots the center of mass of the tracked colonies; colors represent the mean level of altruism of the individuals populating the colonies. In the middle part of the figure, density and traces overlap to showcase their consistency. Some traces suddenly end, indicating that the colony went extinct. Such events are detected automatically and indicated with a black square. From the figure, it is apparent that prior to the death of a colony the mean level of altruism always declines, suggesting a tragedy of the commons. In other places, traces suddenly fork, which is also automatically marked, this time with orange circles. Clearly, the colonies in the 1D habitat reproduce by binary fission (like their 2D counterparts); the daughter colonies inherit their color from their parent. Again it appears that more altruistic colonies divide more frequently.

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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 (σ_a = 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.

https://doi.org/10.1371/journal.pcbi.1010612.g003

Colonies emerge due to a linear instability.

The mechanisms behind the emergence of colonies in the 1D habitat can be studied mathematically using linear stability analysis (LSA). We envision a population of individuals with a fixed level of altruism ϕ homogeneously distributed over a large habitat that is populated at carrying capacity. Next we superimpose a tiny periodic density variation with some wavelength λ and derive under which conditions this perturbation is expected to grow exponentially, resulting in “colonies”. This also allows us to make approximate predictions on the wavelength of the emerging pattern, i.e., the distance between colonies. Details are found in section Materials and methods and S2 Fig.

The LSA reveals that colonies are expected to develop only for certain combinations of the scales of altruism, competition, and motility (Fig 3C; remember that σ_a = 1 by definition). As the LSA elegantly demonstrates, the appearance of colonies is determined by a tug of war between these three forces. Altruism by itself tends to amplify differences in local density: areas with a high density contain more altruists, which positively affects the reproduction rate and hence further increases the density. This ultimately drives the emergence of colonies. However, this force is weak for density variations with a wavelength shorter than ∼ σ_a, which average out within the scale of altruism. Resource competition tends to quench density differences because it suppresses reproduction in densely population areas. This force, however, is weak for variations with wavelengths shorter than ∼ σ_rc, which are averaged out within the scale of competition. Lastly, random motility also tends to homogenize the density, but this force is ineffective against variations with wavelengths larger than ∼ σ_m because random motion is famously slow at large scales. Together, this means that colonies form only if σ_m is small compared to the other scales and σ_a is clearly smaller than σ_rc, so that wavelengths exist that are too long to be quenched by motility, long enough to be amplified by altruism, but too short to be suppressed by resource competition (see S2(A) Fig).

To test these predictions, Fig 3C presents results from a large number of simulations using a range of values of σ_m and σ_rc (19 × 21 = 399 simulations in total) in which all individuals are given an immutable value of ϕ = 0.05. Each simulation quantified to what extent colonies developed by simply measuring the variance in the local population density. The results are as expected: when crossing over from the linearly stable (blue) to the linearly unstable (yellow) region of Fig 3B the variance in the local density increases precipitously. The wavelengths of the emerging patterns—typically close to 2σ_rc—also broadly match predictions (S2(C) and S2(F) Fig). We therefore conclude that the LSA accurately describes and explains the emergence of the colonies.

Colony formation by altruists enables evolutionary bootstrapping.

From the observations of both the 1D and the 2D model it appeared that the emergence of colonies is important for the evolution of altruism. If so, appreciable levels of altruism should evolve only in the parameter regime where colonies can emerge given reasonable levels of altruism (the linearly unstable, yellow region of Fig 3B and 3C). This is confirmed by a series of simulations for various scales of motility and competition (Fig 3D). Because the colony formation depends on altruism, but the selection of altruism in turn depends on the formation of colonies, the process must pull itself up by the bootstraps. Random mutations plus local reproduction spontaneously result in unstable colonies with modest levels of altruism and high internal levels of drift. This occasionally produces a colony that is altruistic enough to reproduce, which starts to spread rapidly.

Factors other than the spatial length scales clearly also affect whether altruism prevails. If the scale of competition becomes too large relative to the scale of motility, the mean level of altruism suffers (Fig 3D, e.g. at σ_rc = 5 and σ_m = 0.1). Also, the stability of colonies against corruption by defectors is affected by the rate with which such defectors are created by mutations. In line with this, the mean level of altruism decreases if the mutation probability is increased (Fig 3E). That said, altruism emerges for a broad range of mutation probabilities.

Definition of the model.

We envision a population or individuals living in a large habitat, which can be one- or two-dimensional. Each individual is fully characterized by its spatial coordinates plus the value of a single quantitative trait ϕ, which indicates its investment in altruistic behavior. The behavior of the individuals is defined by just four stochastic processes: death, motility, reproduction, and heredity with mutation.

Death strikes each individual at a fixed (Poisson) rate d; if an individual dies, it disappears from the population. The average lifespan of an individual is d⁻¹, which we call the generation time.

Motility is modeled as unbiased diffusion with diffusion constant k_D. It follows that, in a generation time, the root-mean-square displacement of an individual in each spatial dimension is , the scale of motility. It can be interpreted as the “typical” distance traveled by an individual during its lifetime. Note that we ignore that individuals take up space: nothing prevents multiple individuals from being at the same position at the same time.

Reproduction is asexual. When an individual reproduces, a new individual is placed at the same position as the parent. The rate of reproduction of each individual is negatively affected by the level of altruism of the individual itself and by competition for resources with other individuals; in contrast, it is positively affected by the altruism of others in its local environment. To implement these effects mathematically, we make use of two quantities that we will now introduce.

First, we define the local population density D(y; σ_rc) at position y as a conventional Kernel Density Estimate (KDE): (1) Here, the summation runs over all individuals i in the population; x_i is the position of individual i; and the kernel function G_rc(x; σ_rc) is the Gaussian (normal) distribution (univariate or bivariate, depending on the dimensionality of the habitat) with standard deviation σ_rc. By this definition, the population density at position y is high if many individuals are found within a distance of order σ_rc from y. The parameter σ_rc is called the scale of competition because, as explained below, it determines the range of competitive interactions.

Second, the altruism experienced by an individual at position y is measured as (2) This is again a KDE, except that each individual i is weighted by its level of altruism ϕ_i. It is convenient to think of A(y; σ_a) as the availability of some public good that organisms secrete locally in proportion to their level of altruism. The summation in Eq 2 runs over all individuals, and the contribution of each individual to the public good at position y decreases with their distance to y according to a Gaussian kernel function G_a. The standard deviation σ_a of the kernel function is referred to as the scale of altruism because it determines the range of altruistic interactions.

In terms of these definitions, the full equation for the reproduction rate g_i of individual i reads (3) Here, g₀ is the basal reproduction rate. In the subsequent factor (labeled “factor 1”), the term −cϕ_i implements a deficit in the reproduction rate owing to the individual’s investment in altruism; the parameter c determines the price of altruism. The last term in factor 1 implements the advantage obtained from the altruism of others. The advantage grows as b₀A(x; σ_a) if A(x; σ_a) is small but saturates at b_max if A(x; σ_a) is large. Factor 2 introduces resource competition: it decreases linearly with the local population density D(x; σ_rc) such that reproduction is locally inhibited when the density approaches K. (In practice, the population density stabilizes somewhat below K, where the average reproductive rate equals the death rate d.) The max[.] function is required because both factors 1 and 2 could in rare cases become negative; in that case, g_i is set to 0.

Heredity and mutation are implemented as follows. Upon reproduction, the offspring usually inherits the value ϕ of the parent, but with probability μ a mutation occurs. In that case, the ϕ-value of the offspring is determined by adding a random change δϕ to the value of the parent. The absolute value of δϕ is drawn from an exponential distribution with mean m, and its sign is positive or negative with equal probability. A concern with this procedure, however, is that the resulting trait value ϕ of the offspring can become negative. In simulations with a 2D habitat this was not permitted and in such events the value was instead set to 0. Although this is a natural choice, it introduces a mutational bias (see S1 Fig), which would complicate some of the analyses performed on the 1D version of the model (in particular, Fig 3D and 3E). In the simulations of the 1D system ϕ was therefore allowed to become negative, but the behavior of the individuals was determined by the “effective” value ϕ_E = max[ϕ, 0] rather than by ϕ itself. In Fig 3D and 3E the mean of ϕ_E is plotted. In Fig 3A the colony mean of ϕ is plotted, but the distinction is immaterial because in this window of the simulation negative values of ϕ are rare.

Units and parameter reduction.

We are free to choose convenient units for length, time, and the trait ϕ; thus, three parameters can be eliminated. First, we choose the unit of length such that the scale of altruism σ_a equals 1 by definition. The two other length scales that exist in the model, σ_rc and σ_m, are therefore expressed relative to σ_a. Second, units of time are chosen such that the generation time d⁻¹ is 1. This implies that the death rate d also equals 1 by definition. Third, the unit of the trait value ϕ is chosen such that the parameter c (see Eq 3) equals 1. This simplifies the interpretation of ϕ: an individuals with trait value ϕ directly sacrifices a fraction ϕ of its basal reproductive rate to the public good. Note, however, that the summation in Eq 2 runs over all individuals, so that each individual also benefits from its own altruism. In the literature, a distinction is sometimes made between soft and hard altruism, depending on whether the direct benefits that altruists reap from their behavior outweigh the direct costs [47]. We always choose parameters such that the direct costs far exceed the direct benefits, modeling hard altruism. The contribution of individual i to the public good A(x_i; σ_a) at its own position x_i is given by (4) From Eq 3 and the fact that σ_a = 1 by definition, it then follows that the reproductive advantage due to one’s own altruism is bounded by (in the 1D habitat) or b₀ϕ_i/(2π) (in the 2D habitat). This reproductive advantage cannot outweigh the deficit of cϕ_i = ϕ_i unless (in the 1D case) or b₀ > 2π (in the 2D case); we steer clear of this regime by choosing b₀ appropriately small.

Simulation scheme.

In the simulations, continuous space is approximated by a linear grid (in the 1D habitat) or a square grid (in the 2D habitat) with grid cells of linear size δx. Periodic boundary conditions are imposed. Time is divided into computational time steps δt.

During each computational time step, the state of the system at time t + δt is constructed based on the state at time t by the following sequence of steps:

1. Step 1. Calculate reproduction rates First, the density D(x; σ_rc) and the availability of public good A(x; σ_a) at each position x are computed, taking into account the periodic boundary conditions. After this, the reproduction rates g_i of all individuals can be calculated.
2. Step 2. Reproduction and mutation Each individual i in the field reproduces with probability g_iδt. The offspring is mutated with probability μ, as described above.
3. Step 3. Death Each individual subsequently dies with probability d δt.
4. Step 4. Motility Each individual is displaced in each spatial dimension by a distance drawn at random from a discrete approximation of a Gaussian distribution with mean 0 and standard deviation .

Initial conditions.

The steady-state population density of a population of defectors (ϕ_i = 0) is approximately (1 − d/g₀)K, which can be derived by solving g_i = d under the assumption of a homogeneous population distribution. Therefore, the initial condition was constructed by placing (1 − d/g₀)KL (in the 1D habitat) or (1 − d/g₀)KL² (in the 2D habitat) defectors at uniformly random positions, where L is the linear size of the habitat.

Default parameters.

The default values of model parameters are listed in Table 1. Here, we also provide the computational parameters used to simulate the model.

For simulations of the 2D model, a square habitat of linear size L = 102.4 was used; using δx = 0.1 this amounted to ≈ 10⁶ grid cells. Using the default parameter values K = 40 and g₀ = 5, the total population size was approximately n = (1 − d/g₀)KL² ≈ 3.4 × 10⁵. The simulations were run for T = 8 000 generations, with time steps δt = 0.08.

For simulations of the 1D model, a habitat of size L = 819.2 was used with δx = 1/80, resulting in 65 536 grid cells. Using the default parameter values K = 100 and g₀ = 5, the total population size was approximately (1 − d/g₀)KL ≈ 6.6 × 10⁴. These simulations were run for T = 160 000 generations, again with time steps δt = 0.08.

Additional settings were used to calibrate the automated recognition of colonies; see below.

Calculating cumulative effects of selection, drift, and mutational bias.

The mathematical framework used to quantify selection, drift and mutational bias is described in S1 Text Section 3. We applied this calculation to each time step of the simulations, so that the cumulative effect of each of the evolutionary forces could be tracked (Fig 2A and S1 Fig).

For the analysis we need to obtain, for each individual i present right before the computational time step, the expectation value of the number of offspring W_i it will have after the time step (also counting the individual itself if it survives). To do so, first the growth rate g_i was calculated and subsequently the reproduction and death probabilities P_r = g_iδt and P_d = d δt over this time step. From the simulation scheme (see above) the expectation value can then be derived: (5) This expression is used in the calculations.

We note that this expectation value is conditioned on the current state of the simulation, in particular the population density D(x_i) and availability of public good A(x_i) at the position of the individual. In other words, only the effects of the inherent randomness of reproduction and death given the state of the local neighborhood are accounted as random drift; the fact that the state of the local neighborhood itself is also affected by random events in the past, such as the stochastic motility and demographics of others, is not. (See also S1 Text Section 3.)

Calculating the local population density.

To efficiently calculate the local population density (Kernel Density Estimate or KDE) D(x; σ_rc) (Eq 1), first a matrix was constructed that specifies, for each position in the habitat, the number of individuals at that position. The KDE at each position, taking into account the periodic boundary conditions, is the circular convolution of this occupancy matrix with the periodic summation of the (discretized approximation of the) Gaussian kernel G_rc(x;σ_rc). To perform this convolution, we use the Circular Convolution Theorem, which states that the circular convolution of two matrices can be obtained by first calculating their Discrete Fourier Transform (DFT) and then calculating the inverse DFT of their element-wise product.

Calculating the availability of public good.

The public good available at each position, A(x; σ_a), is calculated in a similar way. First a matrix is constructed that contains, for each position in the habitat, the sum of the trait values of all individuals present at that position. The value of A(x; σ_a) for each position is now obtained as the convolution of this matrix with the (discretized approximation of the) Gaussian kernel G_a(x; σ_a), again using the Circular Convolution Theorem.

Calculating the radial distribution function.

The radial distribution function or pair-correlation function g(r) is defined as the observed number of pairs of individuals separated by a distance r, relative to the expected number under the null model assuming that each individual is placed at a random position.

In the 1D case, the distance r can only take on values kδx, where k is a non-negative integer. Call the population size n and the size of the habitat in grid cells X. To calculate the expected number of pairs at distance r = kδx, written as E(r), we note that the number of individuals o_x at position x is binomially distributed under the null model. Its expectation value is and hence . (Here, we used that the occupancies of different sites are to good approximation independent.) The observed number of pairs of individuals found at a distance r = kδx, called O(r), is precisely given by the auto-correlation of the occupancy matrix, which is again efficiently calculated using the Circular Convolution Theorem. For each value r = kδx, the radial distribution function is then obtained as g(r) = O(r)/E(r).

In the 2D case, the rectangular grid imposes that r can only take on values such that r² = (a² + b²)δx², where a and b are integers. In addition, in calculating the expectation under the null model, the frequency F(r) with which each distance occurs in the grid has to be taken into account. (E.g., the distance 5δx occurs three times more often than the distance 6δx.) Under the same assumptions as made for the 1D case, the expectation is E(r) = F(r)n²/X². To calculate the observed number of pairs O(r), the auto-correlation matrix of the occupancy matrix is used. Then g(r) = O(r)/E(r) is calculated for each admissible value of r. To obtain the plot in Fig 2C, the distances were subsequently binned.

Calculating the terms of MLS 1 and MLS 2.

To obtain Fig 4 and S4 Fig, we divided the simulation into 2 000 time intervals of Δt = 80 generations and applied the analyses of MLS 1 and 2 to each time interval. The mathematical expressions for MLS 1 and 2 are briefly summarized in S1 Text Section 4 and Section 5. Here follows a description of the computational methods used.

For concreteness, let us focus on a particular interval (t₁, t₂]. The first step is to calculate the selection differential S, defined as the covariance of ϕ and relative fitness w (Eq 2 in S1 Text Section 2). For the purpose of this analysis, the relative fitness w_i of an individual i living at time t₁ is the number of offspring it has at time t₂ (the absolute fitness W_i), divided by the population mean . To find these offspring numbers, each individual at time t₁ was assigned a unique ID that was subsequently inherited by all offspring. At time t₂, a frequency table of ID values was constructed, which directly provided the fitness of each each individual at time t₁. With this information, S can be calculated directly.

The analysis of MLS 1 splits S into two parts (see Eq 8 in S1 Text Section 4). It is sufficient to calculate the second term, S_among, after which the first follows as S_within = S − S_among. To calculate S_among first the geographical borders of all colonies at time t₁ were identified; the algorithm used for this is described in the next section. Next, each individual at t₁ was assigned to a colony. Then for each colony j we calculated its population size n_j, its mean relative fitness {w; j}_w and its mean trait value {ϕ; j}_w. At this point, S_among could be calculated from its definition.

The analysis of MLS 2 describes the dynamics from the perspective of the colonies (see Eq 9 in S1 Text). To determine the fitness of the colonies present at time t₁, we had to count how many offspring colonies they have at time t₂. This requires that we defined the borders between the colonies at time t₂, but also that we traced the ancestor colony at t₁ for each offspring colony at t₂; the algorithm used is described in the next section. The other ingredient of Eq 9 in S1 Text is the trait value Φ of the colonies. Because Φ_j = {ϕ; j}_w (see S1 Text Section 5), these quantities were already calculated for MLS 1 and both terms of Eq 9 in S1 Text can be evaluated directly.

Automated recognition and tracking of colonies.

To perform the multilevel selection analysis, the simulation had to automatically recognize colonies and track their ancestry. Because existing clustering algorithms are inefficient for 1D systems and/or difficult to adjust to our needs, we used our own heuristics.

Where to draw the border between neighboring colonies, and when to conclude that one colony has divided into two, is to some degree arbitrary. The results of the analyses, however, do not depend sensitively on such details as long as we use reasonable definitions and apply them consistently.

The basic idea is to identify the borders between colonies with local minima of the population density. However, local minima can also occur temporarily within colonies due to random fluctuations, and such minima should not be confused with true borders between colonies. To solve this, one might exclude local minima if the density at their position exceeds a set threshold, so that only “deep” minima are considered. Such a simple threshold rule can identify most colonies correctly, but issues arise during the binary fission of colonies. During this process the depth of the local minimum that separates the two daughter colonies fluctuates, and hence it is likely to cross the threshold multiple times. Consequently, the threshold rule tends to record multiple events of fission and fusion during a single process of colony division. Similarly, when a dwindling colony is about to disappear, the threshold rule tends to infer series of deaths and resurrections of the same colony.

To prevent this, the algorithm that was used in the simulations in fact uses two density thresholds: a low and a high one, T_low and T_high. When a new local minimum appears, a new border (and hence the birth of a new colony) is inferred only when the local density at the minimum drops below T_low. In contrast, when an existing border is about to disappear, this is acknowledged only when the density at the associated local minimum rises above T_high. The result is a hysteresis of sorts: when the density of a new minimum drops below T_low for the first time, a colony is born; if afterwards the density at the border temporarily exceeds T_low, the border is maintained unless it also exceeds T_high.

To be precise, when performing the MLS analysis on the interval (t₁, t₂], the borders of the colonies at time t₂ were constructed as follows:

1. Calculate a smoothed density. The smoothed density at each position was defined as a KDE with bandwidth σ_a/2, taking into account the periodic boundary conditions.
2. Identify local minima. If the smoothed density at grid point x is written as ρ_x, each x such that ρ_x < ρ_x+1 and ρ_x < ρ_x−1 marks a local minimum. (Because of the periodic boundary conditions, all indices should be read modulo X, the size of the grid.)
3. Determine tentative borders between colonies. First, local minima were selected with a density ρ_x < T_high; the other minima were discarded. Each of the selected minimal was then associated with a tentative border which would later be further scrutinized. To ensure that no individuals can sit exactly at a border (causing ambiguity as to which colony it belongs to), borders were positioned between grid points. First, the derivative of the density at the position of each minimum was approximated as ρ′(x) = (ρ_x+1 − ρ_x−1)/(2δx). If a minimum was located at grid point x, then a tentative border was placed between x and x + 1 if the derivative was negative, and between x − 1 and x if the derivative was positive.
4. Assign an ancestor to each tentative colony. The tentative borders implicitly also defined tentative colonies. For each tentative colony at time t₂ an ancestor colony at time t₁ was determined. To do so, we exploited that we have already traced back the ancestry of the individuals in the colonies. We then used the expectation that the ancestor colony P of a colony Q contains most, if not all, ancestors of the individuals that belong to Q. Based on this, we identified P as the ancestor colony that contains the largest fraction of the ancestors of the individuals belonging to Q.
5. Reject tentative borders that reflect fluctuations or incomplete divisions. If the colonies on either side of a tentative border had the same ancestral colony, this suggested that a colony division might have taken place. In this case, we compared the density at the corresponding minimum to the low threshold T_low; if the density was above that threshold, the border was rejected. All other tentative borders were now accepted, so that the identification of colonies at t₂ and their ancestor at time t₁ also became final.
6. Count the number of offspring of each ancestral colony. Now that the ancestors of colonies at time t₂ had been identified, the number of offspring—the absolute fitness—of each ancestral colony could be tabulated. If an ancestral colony had fitness 0, it must have died between t₁ and t₂. If an ancestral colony had absolute fitness > 1 it must have divided. (In practice, a fitness above 2 did not occur because Δt is too short to support multiple consecutive divisions.) If a colony had fitness 1, the ancestral colony most likely survived the time interval without reproducing.

The thresholds T_high and T_low are parameters; we found that T_high = 0.7K and T_low = 0.2K worked well.

Inclusive fitness theory and Hamilton’s rule.

In S6 Fig we demonstrate that the simulations can also be analyzed using inclusive fitness theory. To do so, we essentially use a formulation originally due to Queller [68]; we briefly describe it in S1 Text Section 2. We applied these calculations separately to every computational time step of the simulation.

The method involves a regression variable , which is usually defined as the sum or the mean of the trait values of all social interaction partners of individual i. In our simulation, each individual interact with all other individuals, albeit with an interaction strength that decays with their separation (see Eq 2). We therefore define as a weighted sum: (6) With this definition, the theory of S1 Text Section 7 can be applied directly.

Estimating the lattice constant of the hexagonal lattice by counting colonies.

The lattice constant of the hexagonal pattern that emerges in the 2D model can be estimated by counting the number of colonies in the habitat. A hexagonal lattice is composed of equilateral triangles with side a and area . The number of triangles is twice the number of nodes ν. In a large enough habitat of area L², the number of nodes can then be estimated as . Conversely, after counting the number of nodes, a can be estimated as . At the end of the simulations of Fig 2 we find approximately 179 colonies, which, given L = 102.4, corresponds to a ≈ 8.2. This is consistent with the estimate based on the radial distribution function (Fig 2C).

Estimating error bars.

In the Results section and S1 Table we provide 95% confidence intervals for the means of all quantities plotted in Fig 4 and S4 Fig. Because data points in these time series are auto-correlated and the distributions of some quantities are skewed, the standard methods for calculating confidence intervals could not be used. Therefore, we applied the method described in Ref. [69].

Briefly, the idea is to divide the data series into blocks if length l and use the means of these blocks (rather than the original data points) to estimate the standard error of the mean (SEM). Starting with l = 2, if l is increased, the correlations between block means eventually become negligible and the estimates stabilize around a sensible value, which we determined by manual inspection and then rounded off conservatively. Moreover, because of the central limit theorem, the distribution of block means converges to a normal distribution, which justifies the use of t-statistics to estimate confidence intervals. Although the correct number of degrees of freedom to be used is poorly constrained (as it depends on the minimal value of l that is deemed large enough to remove correlations), it is in all cases large enough to ensure that the critical t-value for t_0.05(2) is near 2. We therefore estimated the 95% confidence interval as the sample mean ± twice the estimated SEM.

Software.

Simulations were performed with custom software written in Fortran; the code is made available on the following GitHub repository: https://github.com/rutgerhermsen/altruism.git. Statistics were performed in R version 3.6.1. Visualization was done in R, using ggplot2, and in Wolfram Mathematica 12.

Mathematical analysis.

Consider a population of individuals that each have the same level of altruism ϕ. If the carrying capacity is large, the dynamics if the density ρ(x, t) can be approximated by the following mean-field equation: (7) Here, G_a and G_rc are the kernel functions used in Eqs 1 and 2 to define the availability of public good and the local density, respectively. The notation f * h stands for the convolution of functions f and h. Eq 7 has a homogeneous equilibrium solution ρ(x, t) = ρ₀ > 0; we ask under what conditions this solution is (linearly) unstable to periodic perturbations so that colonies can form spontaneously.

To find out, we first identify ρ₀ by equating Eq 7 to zero and solving for ρ(x, t) = ρ₀. Ignoring the trivial solution ρ₀ = 0, the equation is quadratic and can be solved straightforwardly. Out of the two solutions, one is negative and hence irrelevant. The remaining solution depends on all parameters except for the diffusion constant k_D.

We then consider a periodic perturbation (8) with a very small (infinitesimal) amplitude ϵ(0) and ask whether ϵ(t) will grow or decay. To obtain a dynamic equation for ϵ(t) we insert Eq 8 into Eq 7. In doing so, we have to work out the convolutions of the Gaussian kernel functions G_a and G_rc with the sine wave of Eq 8. From the Convolution Theorem it follows that, for any real-valued, normalized, symmetric kernel function f(x) that has a Fourier transform , the convolution with a sine wave is again a sine wave, but with a reduced amplitude: (9) In the specific case where f(x) is Gaussian with standard deviation σ, we get (10) The convolutions with G_a and G_rc follow directly from Eqs 9 and 10.

We then expand the resulting equation to first order in ϵ(t). Because ρ₀ is the homogeneous solution, the zeroth-order term vanishes. The result is a linear equation of the form: (11) where the factor E(λ) can be written as: (12) The solution of Eq 11 is exponential, with growth rate or eigenvalue E(λ). Hence, if Eq 12 is positive for some wavelength λ, perturbations with this wavelength are predicted to grow exponentially. Because demographic noise produces perturbations of any wavelength, this is expected to eventually give rise to periodic density fluctuations with a similar wavelength.

Eq 12 provides considerable insight. It consists of three terms, expressing the effects of altruism, motility, and resource competition, as indicated. The three length scales in the system —the scale of altruism σ_a, the scale of motility σ_m, and the scale of competition σ_rc— each appear in their appropriate term.

The first term, describing the effect of altruism, is the only positive one, and it scales with ϕ. This shows that altruism is required to obtain a positive eigenvalue for any wavelength λ. Indeed, altruism tends to amplify density differences: because the benefits of altruism grow with the number of altruists in the local neighborhood, its effect is to increase the reproduction rate in regions of high density, which tends to further increase that density. However, the equation shows that this positive contribution is exponentially suppressed if the wavelength λ is small relative to the scale of altruism σ_a; this is because the kernel function averages out such short waves within the social neighborhoods of individuals.

The second term reflects the effect of motility. Random motion (diffusion) is a homogenizing force and therefore quenches density fluctuations, as reflected in the negative sign of this term. However, because diffusion is famously slow on large length scales, only short wavelengths are strongly affected: if the wavelength λ exceeds the scale of motility (the typical distance traveled by an individual in a generation time) the contribution becomes small.

The third term describes the effect of resource competition. Resource competition reduces the reproduction rate in areas with a larger density and thus suppresses density differences, which explains that its contribution is negative. However, if the wavelength is small relative to the scale of competition σ_rc, the density wave averages out within the competitive neighborhood of individuals and the homogenizing effect becomes weak.

Together, this clearly indicates in which regime we ought to expect colonies. Density fluctuations are suppressed by diffusion if their wavelength λ is smaller than σ_m, and by resource competition if λ is increased far beyond σ_rc. Instabilities are therefore expected only if there is a gap between these two regimes. At the same time, the positive contribution of altruism becomes weak if λ is smaller than σ_a. For altruism to be effective in the “gap”, σ_a therefore should be chosen smaller than σ_rc. In summary, instability requires that the diffusion constant is small enough, the scale of competition is large enough, and the scale of altruism is smaller than the scale of competition. Apart from these rules of thumb, Eq 12 can of course be evaluated numerically to make precise predictions; see S2 Fig.

Validation of predictions.

In Fig 3B and 3C and S2 Fig we test predictions based on the linear stability analysis using simulations. The key predictions are (i) the region of parameter space where colonies can form, and (ii) the wavelength of the resulting pattern. The following methods were used.

As illustrated with the red dot and arrow in S2(A) Fig, both predictions are found by maximizing E(λ). S2(B) Fig shows a contour plot of the maximal value of E(λ) under variation of the scales; S2(C) Fig the corresponding wavelengths. Both values were obtained by differentiating Eq 12 and numerical root finding.

To test the predictions we performed a large number of simulations using different values for σ_rc and σ_m. (We used σ_rc ∈ {1, 1.2, 1.4, …, 5} and σ_m ∈ {0.0671, 0.100, 0.134, …, 0.671} in all 21 × 19 = 399 combinations.) As a simple proxy for the presence of colonies, we measured the variance of the (smoothed) population density (KDE) over space. To identify the dominant wavelength in the density pattern, we calculated the Fourier transform of the KDE and selected the mode with the largest amplitude.

The simulations were performed as usual and using default parameters except for the following adjustments:

1. All individuals were initialized with a trait value ϕ = 0.05.
2. In these simulations, we were interested in the ecological patterns of a population with fixed ϕ; therefore the mutation rate was set to μ = 0 to disable evolution.
3. The simulation was run for T = 2 400 generations. (The colonies establish very rapidly.)
4. Starting at t = 400 generations, after each time interval of 80 generations the following analysis was performed:
   1. a. Calculate a KDE using a Gaussian kernel with standard deviation/bandwidth σ_a/2.
   2. b. Calculate the variance of this KDE.
   3. c. Calculate the Fourier transform of the KDE and identify the wave number with the largest amplitude.

After the simulation, the mean value of the variance was reported. It is this variance that is plotted in Fig 3C and S2(E) Fig. Also, the mean value of the wave number with the largest amplitude was reported; this wave number was transformed to a wave length, which was plotted in S2(F) Fig.