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).