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We recently discovered an error in the study by Wild & Fernandes (2009). The summation in eqn (6) in Appendix B of that paper should have included the index, j = 0. The corrected version of the equation is

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where symbols are defined in the study by Wild & Fernandes (2009). The error changes quantitative features of the model analysis, but qualitative results and the basic conclusions are not affected. The latter point is supported by comparing revised Figs 1 and 2 here to Figs 1 and 2, respectively, in the study by Wild & Fernandes (2009).

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Figure 1.  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.

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Figure 2.  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.

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The main change we must report occurs when z = 0. In that case, eqn (1) implies that, at equilibrium, we have

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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).

For the reader's convenience, we will use an inclusive-fitness argument by Taylor (1996) to show that – other than the new expression for r– the result in eqn 5 of Wild & Fernandes (2009) is unchanged. First, it should be noted that an actor (i.e. a native) in class j is encountered with probability πjj/Np. The deviant investment behaviour displayed by that actor changes its fitness at rate −c, while changing the fitness of its average patchmate (the primary recipient, sensu West & Gardner, 2010), by an amount b. As the primary recipient is, itself, a native with probability j/N, the actor's inclusive fitness changes at rate b(j/N)Rj. The actor's deviant behaviour also changes the extent of local competition on its patch. The change in competition, in turn, changes the fitness of the average parent of those offspring that compete on the actor's patch (the secondary recipient, sensu West & Gardner, 2010) at rate, −(b − c)p. With probability pj/N, the secondary recipient is a native neighbour of the actor, and so the actor's inclusive fitness changes, yet again, at rate, −(b − c)p2(j/N)Rj. Because these rates of fitness change are additive, we find the net change in class-j actor's inclusive fitness to be

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To be clear, ΔWj is equal up to multiplication by a positive constant to

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(in Wild & Fernandes (2009), we note that the reproductive value of each individuals is identical when z = 0).Summing ΔWj with weights πjj/Np gives the net inclusive-fitness effect, ΔW = −c + br − (b − c)p2r presented in the study by Wild & Fernandes (2009). The condition ΔW > 0 rearranges to give eqn 5 in the study by Wild & Fernandes (2009).

We sincerely regret any inconvenience our error may have caused.

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