Fig 1.
Role of demographic stochasticity in the evolution of cost-free social traits acting on death rate.
The model uses b ≡ 3(1 − x1 − x2), m ≡ 1 + νx1/(x1 + x2) with Ω = 900; if ν > 0, the trait is spite, whereas if ν < 0, the trait is altruism. Subplot a—stationary distributions corresponding to three different values of ν: altruism (ν = −0.95), neutral (ν = 0), and spite (ν = 0.95), revealing the close match between our analytic results and simulations of the full stochastic process. Mutation rate is μ = 0.006 in all cases. The distribution is skewed towards a higher frequency of the social actor in the case of altruism and towards a lower frequency of the social actor in the case of spite (distribution is symmetric in the neutral case). Underlying contour plot shows the value of the ratio T(p)/[Ωn(p)]. Subplots b and c shows the degree to which the social actor is disfavoured (spite, ν = 0.95; subplot b) or favoured (altruism, ν = −0.95; subplot c) for different mutation rates: as mutation rate decreases, the effect of demographic stochasticity increases. Subplots d-g show how changing mutation rate alters the shape of the stationary distribution. When mutations are low (subplot d), the stationary measure is U-shaped, but skewed in favour of the social actor if the trait is altruism (red curve) or non-social actor if the trait is spite (black curve). As mutation rate increases, the distribution is initially pushed into the interior at the boundary for which the ratio T(p)/n(p) is minimized (p = 1 and red curve on subplot e; p = 0 and black curve on subplot f), before the distribution ultimately becomes unimodal with distribution favouring the type which minimizes T(p)/n(p) (subplot g). In all plots, the curves/bars are analytic predictions, while circles are the average of 3 × 104 simulations of the full stochastic process (see S1 Appendix). For subplot b and c, simulations were terminated after 5 × 105 and 7.5 × 105 time units, respectively.
Fig 2.
Role of demographic stochasticity in the evolution of cost-free social traits acting on birth rate.
In subplots a-b, the social trait is altruism. In subplots c-d, the trait is spite. Subplot a—stationary distributions corresponding to three different strengths of altruism ν, showing how altruism can be disfavoured. Subplot c—stationary distributions corresponding to three different strengths of spite ν, showing how spite can be favoured. Underlying contour plot shows the value of the ratio T(p)/[Ωn(p)]. Subplots b,d show that in some cases an intermediate level of social action is optimal. Subplot b—a type that uses the intermediate level of altruism that minimizes the ratio T(p)/n(p) in a monomorphic population (in this case ν = 0.75) is favoured over all other levels of altruism. Subplot d—a type that uses the intermediate level of spite that minimizes the ratio T(p)/n(p) in a monomorphic population (in this case ν = 0.75) is favoured over all other levels of spite. Curves are analytic predictions and each circle is 6 × 104 simulations of the full stochastic process; simulations were run for 103 and 5 × 104 time units for subplots b and d respectively. Parameters values: {β, d, κ1, κ2, Ω, μ} = {1, 0.5, 0.75, 0.01, 250, 0.01} (subplots a-b) and {β, d, a, κ1, κ2, Ω, μ} = {8, 1, 0.05, 0.05, 0.2, 900, 0.005} (subplots c-d).
Fig 3.
Role of demographic stochasticity in the evolution of cost-free social traits.
Each subplot is the model indicated by the per-capita birth and death rates, b and m, with n = ∑i xi. The black circles are the results of 104 simulations of the system of SDEs (S1 Appendix) and represent the probability of observing the simulation in a given state (left y-axis). The blue curve is the expected population density of a population monomorphic for the trait value (right y-axis). If population size alone was sufficient to predict which trait is favoured, we would expect a close match between the stationary distribution (black circles) and population size (blue curve)—this does not occur because what is important is the ratio T/n. Indeed, as predicted by consideration of (1), the stochastically favoured trait for subplot a is altruism, whereas for subplot b and c it is the trait value at the red dashed line. Parameter values used: subplot a, {β, d} = {3, 1}, subplot b, {β, d, κ1, κ2} = {1, 0.5, 0.75, 0.01}, and subplot c, {β, d, κ1, κ2, a} = {8, 1, 0.05, 0.2, 0.05}. All simulations used Ω = 104 and assumed type i mutates to type j at a per-capita rate μ = 10−6.
Fig 4.
Can stochasticity reverse selection? Here we compare two models of altruism having the same per-capita growth rate, but differing in variance in per-capita growth (Model 1-red, Model 2-black).
As a consequence, this can lead to a stochastic reversal of selection for model 2 but not model 1. Subplots a-c: predicted stationary distribution for both models from Eq 1 (curve) compared to 2 × 104 simulations of the full stochastic process (circles) for decreasing habitat size, Ω (i.e., increasing levels of demographic stochasticity). The distribution is always skewed towards the non-altruist in model 1. For model 2 the distribution changes from being skewed towards the non-altruist to being skewed towards the altruist as demographic stochastic increases (i.e., a selective reversal occurs). Subplot d: magnitude of the selective reversal plotted against the cost of altruism. Parameters used were {β, d, κ, ν} = {3, 2.4, 1.2, 1}, with ϵ = 0.003 for subplots a-c.