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.