Corrigendum

When z = 0, helpful mutant natives can invade. Panel (a) presents the level curves of the surface (1 − p2r)/r(1 − p2) for heights 10,20,30,…,90 (left to right). Each level curve shows the pairs of N and p that correspond to the same invasion condition (eqn 5, Wild & Fernandes, 2009). Because we have also assumed b/c<N, the shape of the level curves tells us that helping may, under certain circumstances, be adaptive. For example, consider panel (b). If N=60 (dotted line) and if p belongs to the interval indicated, then (1 − p2r)/r(1 − p2) can be no >52 and so the conditions (1 − p2r)/r(1−p2)<b/c<N can be easily satisfied. Indeed, panel (b) shows us that the scope for invasion by helpful mutants is greatest for intermediate values of p. In the general model, we find that under a range of b/c values allows both complete nonhelping (z = 0) and complete helping (z = 1) to simultaneously be ES native behaviours. In this figure, the endpoints of this range of b/c values are estimated numerically. Panels on the left show how the upper end of the range (estimated to lie between paired circles) depends on the natal dispersal rate for various patch sizes N. Panels on the right do the same for the lower end of the range (estimated to lie between paired crosses). Panels in row (a) assume no cost of dispersal μ = 0, panels in row (b) assume moderate costs of dispersal μ = 0.4, whereas panels in row (c) assume high cost of dispersal μ = 0.8. Interestingly, we find that the estimated upper end of each range of b/c values (bounded by circles), i.e. the value of b/c beyond which complete helping is the only ES native behaviour, corresponds exactly to the critical b/c ratio identified in eqn 5 of Wild & Fernandes (2009) for a special case of the model (‘‘weak fecundity effects’’). The critical b/c ratio is illustrated, here, in all panels as a solid curve. It should be noted that when p=(1 − m), we recover the measure of relatedness used by El Mouden & Gardner (2008) (their RN, eqn 4). This new expression for r, and the associated derivation (not shown), replaces the last two equations of Appendix B in the study by Wild & Fernandes (2009). We sincerely regret any inconvenience our error may have caused.

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