Supplementary Material – ‘Models of social evolution: Can we do better to predict “who
helps whom to achieve what”?’
António M. M. Rodrigues1,2,* and Hanna Kokko3,†
1
Department of Zoology, University of Cambridge, Downing Street, Cambridge CB2 3EJ,
United Kingdom.
2
Wolfson College, Barton Road, Cambridge CB3 9BB, United Kingdom.
3
Institute of Evolutionary Biology and Environmental Studies, University of Zurich,
Winterthurerstrasse 190, CH-8057 Zurich, Switzerland
Authors for correspondence, emails:
*
ammr3@cam.ac.uk; † hanna.kokko@ieu.uzh.ch
Contents:
Appendix A. Methodology - Selection gradients
Appendix B. Figures S1, S2, S3.
References
Appendix A. Selection gradients
Assuming the life-cycle outlined in the main text, our aim is to understand how selection acts on
the social behaviour of group living breeders, and to contrast the selection pressures acting on
their fecundity with the selection pressures acting on their survival. We take the neighbourmodulated approach to kin selection to derive the selection gradients acting on social behaviour
[1-3]. For this purpose we need to describe the class-specific reproductive success of each
breeder. Thus, the contribution of a solitary breeder to the class of solitary breeders is given by
w11 = f1(1–g)+s1(1–a). The contribution of a solitary breeder to the class of group living
breeders is given by w12 = f1g+s1a. The contribution of a group living breeder to the class of
solitary breeders is given by w21 = f2(1–g)+s2(1–s2). Finally, the contribution of a group living
breeder to the class of group living breeders is given by w22 = f2g+s2s2. The expected fitness
matrix is then given by
𝑤1→1
𝐰 = (𝑤
1→2
𝑤2→1
𝑤2→2 ).
(A1)
The elements of the right eigenvector of the fitness matrix give the frequency of solitary and
group living breeders in the population, denoted by u1 and u2, respectively. The elements of the
left eigenvector of the fitness matrix give the reproductive value of solitary and group living
breeders, denoted by v1 and v2, respectively. The fitness of a focal group living breeder is then
given by
1
𝑊i = 𝑣 ∑2j=1 𝑤i→j 𝑣j .
(A2)
j
We start by considering a general social behaviour. The marginal fitness effects of such
behaviour is then given by
1
d𝑊i
d𝐺
1
= 𝑣 ∑2j=1
j
d𝑤i→j
d𝐺
𝑣j ,
(A3)
where G is the breeding value of the social behaviour. Expanding the right-hand-side of equation
(A3), we find that the conditions for natural selection to favour the social behaviour, which are
given by conditions (4.1-4.5) in the main text.
Appendix B. Figures
In figures S1 and S2 we reproduce figure 1 in the main text with different parameter values. This
shows that qualitatively the results shown in figure 1 are robust to these changes in parameter
values. In figure S3 we change the scale of the vertical axis to show the full range of the
potential for helping.
Figure S1
Figure S1. Parameter values: a = 0.5; r = 0.1; s1 = 0.1; s2 = 0.9.
2
Figure S2
Figure S2. Parameter values: a = 0.1; r = 0.5; s1 = 0.1; s2 = 0.9.
Figure S3.
Figure S3. Parameter values: a = 0.5; r = 0.5; s1 = 0.1; s2 = 0.9.
3
References
1. Taylor, P.D. & Frank, S.A. 1996. How to make a kin selection model. J. Theor. Biol. 180, 27–
37.
2. Frank, S.A. 1998. Foundations of social evolution. Princeton University Press, Princeton, NJ.
3. Rodrigues, A.M.M. & Gardner, A. 2015. Simultaneous failure of two sex-allocation
invariants: implications for sex-ratio variation within and between populations. Proc. R.
Soc. B 282, 20150570.
4