Table 2.

Simulation results. In all simulation, there were g = 100 000 groups each with carrying capacity n0 that exchanged migrants with all other groups at a rate m. The global carrying capacity was n0g. As described in the text, the local share of population regulation was a, so a = 1 means that regulation only depended on group size. The initial frequency of A in all simulations was set to 0.001 and then the simulation was run for tsim generations. Fitnesses in a step fitness function were given by Embedded Image Embedded Image and Embedded Image, and Embedded Image Averages of Fst and group size n were taken over the last tsim − 50 generations, to allow for quasi-equilibrium. This average value of Fst was used to compute Embedded Image that was used in two ways. (i) To calculate the predicted critical values of T using (5.7), labelled Tsf (for ‘simple formula’). (ii) Using (3.3), to numerically calculate the value of T for which F(0.001) = 0. This value is labelled Tβ. Finally, the observed average fitness difference between A and N types over the last tsim − 50 generations were fit to a straight line using linear regression and the intersection with the x-axis is reported as the observed critical value of T, labelled T0. Negative values of T indicate that A cannot invade for any positive value of T because the population is in the crowded regime.

tsimθmr0an0nFstTsfTβT0
10000.20.050.1180760.1169.07.47.7
1000.20.10.50.180800.06325.023.929.6
10000.20.050.5180790.114−37.8−41.1−32.4
1000.20.10.1180770.06627.326.735.3
1000.20.10.10.580780.06423.422.626.9
10000.20.10.10.580780.06423.422.426.3
10000.20.10.5080800.06219.818.922.9
1000.20.10.05180770.06423.420.824.8
1000.10.10.102002000.02731.128.242.7
1000.150.10.111201160.04439.939.248.2