File S1 - Altruism Can Proliferate through Population Viscosity despite High Random Gene Flow

Supporting Information. Figure S1, Payoff profiles. Payoffs for types A are represented by black squares, while red circles depict payoffs for types N. From left: Public goods game (PG, Example 1) for n = 20, C = 1 and B = 5. Iterated public goods game (IPG, Example 2) for n = 20, C = 1, B = 5, a = 4 and T = 2. Threshold model (THR, Example 3) for n = 20 C = 1, θ = 4 and A = A′ = 10. Figure S2, Perron-Frobenius eigenvalues ρ as a function of m for δ = 0.1, 0.2 and 0.4. From top to bottom: Public goods game (PG, Example 1) with n = 20, C = 1, B = 5. Iterated public goods (IPG, Example 2) with n = 20, C = 1, B = 5, a = 8 and T = 10. Threshold model (THR, Example 3) with n = 20, C = 1, θ = 4, A = A′ = 10. Critical migration values ms are obtained by solving ρ(ms) = 1. Figure S3, Public goods game (Example 1): Panel A represents critical values ms as a function of the strength of selection δ. Curves correspond to the case C = 1, B = 2 and n = 10 (top, black dotted line), n = 20 (middle, blue dashed line) and n = 50 (bottom, magenta full line). Short horizontal red lines indicate critical values at the weak selection limit obtained from (2) in the paper. The inset shows the same curves within the full range of possible values for ms, illustrating the well known fact that for this model, only under exceptional conditions can the allele A invade. Panel B depicts the same conditions except for B = 5. Figure S4, Public goods game (Example 1): Critical relatedness above which types A proliferate, as a function of the strength of selection δ. (R0(m) = (1– m)2/(n – (n –1)(1– m)2) ≈ 1/(1+2nm) is the relatedness obtained from neutral genetic markers; see Sections S5 and S7). Panels correspond to the same parameter values as in Figure S3: C = 1, B = 2 and n = 10 (bottom, black dotted line), n = 20 (middle, blue dashed line) and n = 50 (top, magenta full line). Panel B depicts the same conditions except for B = 5. Short horizontal red lines indicate critical values at the weak selection limit obtained from Hamilton’s rule , or, equivalently, from (2) in the paper. Note the appreciable effect of the strength of selection. Figure S5, Iterated public goods game (Example 2): Critical values ms as a function of the strength of selection δ. Panel A depicts the case n = 20, C = 1, B = 5, a = 4 with, respectively from bottom to top, T = 1 (dotted black line), T = 10 (dashed blue line), T = 100 (dot-dashed magenta) and T = 500 (green full line). Panel B depicts the same conditions except for a = 8. Short horizontal red lines indicate critical values at the weak selection limit obtained from (2) in the paper. Each curve has T fixed, but to compare different values of T, the product δT is a natural measure of strength of selection, and is used in the horizontal axis. Figure S6, Iterated public goods game (Example 2): Critical relatedness above which types A proliferate, as a function of the strength of selection δ. (R0(m) = (1– m)2/(n – (n –1)(1– m)2) is the relatedness obtained from neutral genetic markers; see Sections S5 and S7). Panels correspond to the same parameter values as in Figure S5: Panel A depicts the case n = 20, C = 1, B = 5, a = 4 with, respectively from top to bottom, T = 1 (dotted black line), T = 10 (dashed blue line), T = 100 (dot-dashed magenta) and T = 500 (green full line). Panel B depicts the same conditions except for a = 8. As in Figure S5, each curve has T fixed, but to compare different values of T, the product δT is a natural measure of strength of selection, and is used in the horizontal axis. Short horizontal red lines indicate critical values at the weak selection limit obtained from (2) in the paper. These values are: Panel A: 0.2000, 0.0865, 0.0402, 0.0243. Panel B: 0.2000, 0.1099, 0.0638, 0.0452. Note the very low values of critical relatedness in Panel A. Figure S7, Threshold model (Example 3): Critical values ms as a function of the strength of selection δ. Panel A depicts the case n = 20, C = 1, θ = 4, A′ = 2A, with, respectively from bottom to top, A = 5 (dotted black line), A = 10 (dashed blue line), A = 50 (dot-dashed magenta) and A = 100 (green full line). Panel B depicts the same conditions except for θ = 8. Short horizontal red lines indicate critical values at the weak selection limit obtained from (2) in the paper. Each curve has A fixed, but to compare different values of A, the product δA is a natural measure of strength of selection, and is used in the horizontal axis. Figure S8, Threshold model (Example 3): Critical relatedness above which types A proliferate, as a function of the strength of selection δ. (R0(m) = (1– m)2/(n – (n –1)(1– m)2) is the relatedness obtained from neutral genetic markers; see Sections S5 and S7). Panels correspond to the same parameter values as in Figure S7: Panel A depicts the case n = 20, C = 1, θ = 4, A′ = 2A, with, respectively from top to bottom, A = 5 (dotted black line), A = 10 (dashed blue line), A = 50 (dot-dashed magenta) and A = 100 (green full line). Panel B depicts the same conditions except for θ = 8. As in Figure S7, each curve has A fixed, but to compare different values of A, the product δA is a natural measure
of strength of selection, and is used in the horizontal axis. Short horizontal red lines indicate critical values at the weak selection limit obtained from (2) in the paper. These values are: Panel A: 0.012, 0.017, 0.044, 0.071. Panel B: 0.061, 0.075, 0.137, 0.194. Note the extremely low values of critical relatedness in Panel A. The large values of A can result from contingent cooperation, based on feedback, as for the IPG. For instance, suppose that a certain activity repeats itself T times over a life-cycle. Suppose also that in each repetition the payoff is well described by the threshold model. If types A discontinue the participation when their payoff in the previous round was negative (as in the IPG discussed in Figure 2 in the paper), then the resulting payoff over the T iterations is also given by a threshold model, with the same value of C, but A replaced by (A – C)T+C, and A′ replaced with A′T. This gives plausibility to values of A and A′ as large as those in this figure, since T can be in the hundreds, or thousands (see discussion on the IPG in the paper). Figure S9, Public goods game (Example 1): Perron-Frobenius eigenvectors ν = (ν1, …, νn) represented in each box as a histogram, as a function of the strength of selection δ (rows) and of the migration rate parameter m (columns). Critical migration rates ms are annotated in each row. Perron-Frobenius eigenvalues ρ are also provided for each box. In this picture we have C = 1, B = 2 and n = 20. Figure S10, Self-organization of copies of A. In these pictures we have PG with n = 2, C = 1, B = 3, δ = 0.3, resulting in ms = 0.2889. Pictures show evolution of f(t) = (f1(t), f2(t)), started from several different initial distributions f(0). Circles over the lines mark f(t), with t = 0, 1, …, 500 obtained by iterations of the map f(t +1) = f(t)M(A+B). The direction spanned by the eigenvector ν is represented as a dotted green line. Left side (black): cases with ρ <1, the allele A is eliminated; right side (red): cases with ρ >1, the allele A spreads. In the top row, m is far from ms: (A1) m = 0.3389, ρ = 0.9340, ν = (0.8506, 0.1494); (B1) m = 0.2389, ρ = 1.078, ν = (0.7342, 0.2658). In the bottom row, m is close to ms: (A2) m = 0.2890, ρ = 0.999856, ν = (0.7342, 0.2658); (B2) m = 0.2888, ρ = 1.000014, ν = (0.7999, 0.2001). Note that in all cases f(t) reaches in a few generations a steady state, in which it shrinks (ρ <1), or grows (ρ >1), as a multiple of ν. When m approaches ms, the eigenvalue ρ becomes close to 1, the stationary movement along the direction given by ν slows down and the trajectories towards this direction straighten themselves, but are not slowed down. Figure S11, Self-organization of copies of A. In this picture we have IPG with n = 10, C = 1, B = 3, T = 100, a = 2, δ = 0.01, and m = 0.153, slightly smaller than ms = 0.163. Top part shows evolution of p(t), and bottom part shows corresponding evolution of f(t) = (f1(t), …, f10(t)), displayed as normalized histograms. Two initial conditions are compared: (Red) f(0) = 10–2(1, 0, …, 0), so that p(0) = 10–3. (Black) f(0) = 10–5(0, …, 0, 1), so that p(0) = 10–5. Note that from generation to generation the distribution of copies of A adjusts itself to the same stationary distribution, “losing memory of the initial distribution”. Figure S12, Self-organization of copies of A. This picture corresponds to the same model and situation described in Figure S11, but with a different time-frame, including later times. Note that eventually the two curves of p(t) become parallel straight lines, illustrating the exponential growth of p(t) at rate ρ independently of the initial condition. This picture also illustrates two other important points: 1) The possible non-monotonicity of p(t). 2) The fact that the asymptotic rate of growth may be smaller than the initial rate of growth. Indeed, computations of Δp only indicate the long term prospects for the allele A, when done under stationary conditions, as in (1). The initial distribution of copies of A in the red line produces neighbor modulated fitness for A below that of allele N, so that Δp(0) <0. In contrast, the initial distribution of copies of A in the black line produces neighbor modulated fitness for A well above that of allele N, so that not only Δp(0) >0, but this growth happens at an unsustainably high rate. The distribution ν, towards which the copies of A self-organize is optimal for their stationary, stable, growth. This is so because (ρ, ν) is the leading eigenpair of the driving matrix M(A+B): ν is the vector ν′ that satisfies the eigenvalue (stationarity) equation ν′M(A+B) = ρ′ν′, with maximum ρ′. Figure S13, Self-organization of copies of A. This picture corresponds to the same model described in Figure S11, but now m = 0.173 is slightly larger than ms = 0.163. Note that again eventually the two curves of p(t) become parallel straight lines, illustrating in this case the exponential decrease of p(t) at a rate independent of the initial condition. Here again one can see that Δp(0) is not indicative of the relevant long term evolution. The self-organized distribution ν is still optimal for the proliferation of the allele A in a stable, sustainable, fashion. But when ρ <1, as in this picture, this optimal stable distribution is still not good enough for A to spread, and instead, its copies are eliminated by natural selection. Figure S14, This diagram illustrates the concept of identity by descent (IBD) in the 2lFW. Two individuals X an Y in a given group in generation t, regardless of their type, are identical by descent (IBD) if their lineages, when followed back in time, coalesce before a migration event (indicated by a dashed arrow in the figure in the right panel). Considering a migration rate of m, migration typically takes place within a random number, of order 1/m of generations back. Figure S15, Perron-Frobenius eigenvectors ν = νδ for selection strengths δ = 0.01 (left column), δ = 0.3 (middle column) and δ = 0.7 (right column). Migration rate
is set to m = 0.1 and group sizes to n = 20. Each line represents a different model. The top row, labeled as PG depicts the Public Goods game (Example 1) with parameters C = 1 and B = 2. The Iterated Public Goods game (Example 2) with parameters C = 1, B = 4, a = 4 and T = 10 is shown in row at the middle, labeled as IPG. The bottom row shows Perron-Frobenius eigenvectors for the Threshold model (THR, Example 3) with C = 1, A = A′ = 5 and θ = 4. The leftmost column emphasizes that the weak selection limit is independent of the model. In contrast, when selection is strong, νδ depends on the model, as illustrated in the other columns. Figure S16, Distribution πk (bars) given by π = πQ and , compared with . Here νδ is the Perron-Frobenius eigenvector of M(A+B), with δ = 0.01, for the Threshold model (THR, Example 3) with parameters n = 20, C = 1, A = A′ = 5 and θ = 4 (red diamonds). The comparison is repeated for migration rates m = 0.01 (top panel) and m = 0.1 (bottom panel). Figure S17, This diagram illustrates why KD evolves as a Markov chain driven by Q. In this picture represents the number of individuals that are IBD to the focal individual , in generation u (red circle). Two scenarios are discernible. MC1 (left panel): the focal individual is a migrant. This happens with probability m and implies that . MC2 (right panel): the focal individual is not a migrant, and she is a child of . Each individual in the focal group in generation u chooses a mother from the group of in the previous generation with uniform probability, as δ = 0. With probability the chosen mother is IBD to (orange circles) and, consequently, her children are also IBD to , provided that they are not migrants. In this case, the number of individuals in generation u that are IBD to the focal is, therefore, 1 (for the focal individual herself) plus a number of individuals given by a binomial random variable with probability of success in n –1 trials. Figure S18, Relatedness R0 as a function of migration rate m, under neutral drift, δ = 0, as given by (S11). From top to bottom, n = 20 (dot-dashed blue line), n = 50 (dashed green line) and n = 100 (full red line). Figure S19, Limit of large n and small m under weak selection. This figure compares tail probabilities for the distribution π (stairs) and for Beta distributions with parameters α = 1 and β = 2mn. Panel A shows the case n = 20 for, from top to bottom, m = 0.01 (red dotted line), m = 0.1 (blue dashed line) and m = 0.5 (black dot dashed line). Panel B depicts the same scenarios for the case n = 100. Figure S20, Limit of large n under weak selection for the threshold model (THR, Example 3). Panels represent critical migration rates (A and C) and critical relatedness (B and D) for the THR with C = 1, A = A′ = 10 as a function of . Top panels A and B depict the case n = 20. Bottom panels C and D depict the case n = 100. In each panel critical values obtained by the viability condition under weak selection derived from (A2) (black full lines) are compared with the approximation for large n given by (S25) (, , approx.1, dashed blue lines) and with the approximation (S26) (approx.2, dotted red lines). Figure S21, Limit of large n under weak selection for the Iterated public goods (IPG) game (Example 2). Panels represent critical migration rates (A and C) and critical relatedness (B and D) for the IPG with C = 1, B = 5 and T = 100 as a function of . Top panels A and B depict the case n = 20. Bottom panels C and D depict the case n = 100. In each panel critical values obtained by the viability condition under weak selection derived from (2) (black full lines) are compared with the approximation for large n given by solving (S27) in R (approx., dashed blue lines). In panel B we have when , and in panel D we have when . Types A are altruistic in the strong sense of (S31) when . Figure S22, Limit of large n under weak selection for the Iterated public goods (IPG) game (Example 2): behavior of solutions for (S27) - Part 1. Top panel: H(R) corresponds to the l.h.s. of (S27) while G(R) depicts the r.h.s. of (S27). H(R) is strictly decreasing and it is positive for R<C/B. Derivatives of G(R) converge to 0 as R → 0. H(R) and G(R) are equal to each other at exactly one point that is a decreasing function of C/B. Curves depicted correspond to the cases C/B = 0.5 (full black line), C/B = 0.2 (dashed red line) and C/B = 0.1 (dot-dashed blue line) with and T = 100. Bottom panel: as a function of for C/B = 0.5
(top, full black line), C/B = 0.2 (middle, dashed red line) and C/B = 0.1 (bottom, dot-dashed blue line) and T = 100. is continuous in the interval , takes the value C/B on both end-points of this domain and has a minimum at . Figure S23, Limit of large n under weak selection for the Iterated public goods (IPG) game (Example 2): behavior of solutions for (S27) - Part 2. Top panel: G(R) and H(R) for C/B = 0.5, and T = 10 (leftmost, full black line), T = 103 (dashed red line) and T = 105 (dot-dashed blue line). is a decreasing function of T. Bottom panel: in the limit T → ∞, if then (full magenta line). If then very slowly. Figure S24, Limit of large n under weak selection for the Iterated public goods (IPG) game (Example 2): behavior of solutions for (S27) - Part 3. H(R) (strictly decreasing straight line) and G(R) for C/B = 0.5 and T = 10 for , 0.1, 0.3, 0.4, 0.5 from right to left in Panel A and for , 0.6, 0.7, 0.9, 0.999 from left to right in Panel B. The graph of G(R) moves upwards for and downwards for . G(R) → (T –1)(BR – C) as (dashed magenta line in Panel A). In Panel B it can be seen that G(R) → 0 as . Figure S25, Limit of large n under weak selection for the Iterated public goods (IPG) game (Example 2): behavior of solutions for (S27) - Part 4. In all panels C/B = 0.5. Panel A depicts as a function of 1/log(T) for (full black line) and for (dashed red line). For (this value is approximately 0.286 for the case shown). If then converges to 0 very slowly as T increases, more specifically (dotted magenta line). Bottom panels show the behavior of G(R) as T increases. Panel B: case for, from right to left, T = 2, 10, 100, 500. Panel A: case for T = 2, 10, 100, 500, from right to left. G(R) stays at zero for and goes monotonically to infinity for . Figure S26, The solid lines provide the solution of C/B – R = (T –1)R(1– (C/B))1/R, as a function of C/B, for (top to bottom) T = 1 (black), 10 (blue), 100 (magenta), 1000 (green) and 10000 (cian). The corresponding dashed lines with same colors (no black one) provide the approximation (A5), R = –ln(1– (C/B))/ln T. This figure is an expanded version of Panel C of Figure 2 in the paper.
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