Table 1.

Closed-form results for Hamilton’s rule with three life stages. d₁, d₂, and d₃ are the scales of density regulation for life stages one, two, and three, respectively. α is the degree of kin competition. For clarity, both dispersal rates are set equal, m₁=m₂=m, and selection is assumed to occur in the first life stage. The rows are organized from most permissive to the spread of altruism at the top, to most restrictive at the bottom. When d₁=1, the values of d₂ and d₃ are irrelevant. Fitness effects are scaled for “whole-group” altruism (see Appendix A).

d1	d2	d3	α	Hamilton’s rule
0	0	0	0	$\frac{b}{N-(N-1){(1-m)}^4}-c>0$
0	0	1	(1−m)4	$\frac{b}{N}-c>0$
0	1	0	(1−m)2	$\frac{b}{N+(N-1){(1-m)}^2}-c>0$
0	1	1	(1−m)2+(1−m)4−(1−m)6	$\frac{b(1-{(1-m)}^2)}{N-{(1-m)}^2}-c>0$
1	n/a	n/a	1	−c>0