Version of Record online: 19 JAN 2012
© 2012 The Authors. Journal of Evolutionary Biology © 2012 European Society For Evolutionary Biology
Journal of Evolutionary Biology
Volume 25, Issue 3, pages 599–600, March 2012
How to Cite
WILD, G. and FERNANDES, A. D. (2012), Corrigendum. Journal of Evolutionary Biology, 25: 599–600. doi: 10.1111/j.1420-9101.2011.02448.x
- Issue online: 13 FEB 2012
- Version of Record online: 19 JAN 2012
Vol. 22, Issue 5, 927–941, Version of Record online: 27 FEB 2009
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
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).
The main change we must report occurs when z = 0. In that case, eqn (1) implies that, at equilibrium, we have
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
To be clear, ΔWj is equal up to multiplication by a positive constant to
(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.
- 2008. Nice natives and mean migrants: the evolution of dispersal dependent social behaviour in viscous populations. J. Evol. Biol. 21: 1480–1491. &
- 1996. Inclusive fitness arguments in genetic models of behaviour. J. Math. Biol. 34: 654–674.
- 2010. Altruism, spite, and greenbeards. Science 327: 1341–1344. &
- 2009. Investment in the public good through conditional phenotypes of large effect. J. Evol. Biol. 22:927–941. &