Figure 1.

Areas of parameter space (defined by the proportion of the queen's eggs that are female f and sister–sister relatedness r) that allow invasion of worker reproduction and policing in a population where worker reproduction is initially absent (ψ ≈ 0) (cf. table 1). The results are shown for (a) Model 1a, where colonies are founded by lone queens and worker-laid eggs randomly compete with male- and gyne-destined queen-laid eggs; (b) Model 1b, where colonies reproduce by swarming and worker-laid eggs randomly compete with male-, gyne- and worker-destined queen-laid eggs (graph is shown for parameter values appropriate for Melipona stingless bees: b = 0.044, q = 0.0847, μ = 0.024; see electronic supplementary material, tables S2–S4); and (c–f) Model 2, where the workers manipulate the sex ratio towards their own optimum via the killing of less related brothers and reinvesting freed-up resources into more related sisters with efficiencies E = 0.3, 0.5, 0.8 and ≈1. The areas in the parameter space are green, no worker reproduction; blue, worker reproduction selected for, but potentially inhibited by worker policing; red, uninhibited worker reproduction. The optimal proportion of female eggs laid by the queen f is indicated with a dashed yellow line. In Models 1a and 1b, two alternative equilibria can be reached depending on initial conditions (initial f greater or smaller than the critical value indicated with the white dotted line, which is here drawn for n = 10; see electronic supplementary material, figure S4).