When does weak selection produce qualitatively different results from strong selection?

In evolutionary game theory, it is typical to model organisms as having a base fitness that is modified slightly by the game interaction. The ratio of the game effect versus the base fitness determines the strength of selection, with weak selection meaning the game modifies overall fitness only slightly, and strong selection meaning that the game payoff is a big part of the overall fitness of the organism.

Most analytic models like to assume weak selection because it allows the authors to Taylor expand the selection function and linearize it by dropping terms that are higher order in the stength of selection. However, it is unclear that results derived for weak selection would necessarily hold under strong selection. Are there mathematical (or computational) models where weak and strong selection produce qualitatively different results? If not, is there an argument as to why weak and strong selection should produce the same results?

I've also seen this discussion come up outside of evolutionary game theory, in a more general biological setting. In particular, with respect to inclusive fitness. The take away message I got was that in the regime of weak selection relatedness means what you expect it to mean, but for strong selection the concept becomes very slippery and counter-intuitive. What does inclusive fitness theory tell us about going from weak to strong selection? Again, I prefer arguments supported by clean mathematical or computational models over ones based on intuition and words alone.

-
I haven't had time to look closely at it, but you might find this paper interesting: Wu et al. 2010. Universality of weak selection. ArXiv. It seems like they try to bend the rules and study differences between models. – fileunderwater Jan 9 '14 at 21:33
@fileunderwater thanks, I found that paper shortly after I asked the question and it is extremely relevant, I've been meaning to give a self-answer based on it but have been too busy with other things. – Artem Kaznatcheev Jan 9 '14 at 22:52

It has been shown that predictions based on weak selection can be different from that on strong selection. In fact, for rare mutations, each strategy has stationary distribution at the mutation selection equilibrium. The rank of the component depicts the relative prevalence of the available strategies. Yet the rank of the components of this distribution can change with the selection intensity. Thus results obtained under weak selection can not generally carry over to strong selection.

For more details, please refer to Fig. 1 in [1] and Fig. 2 in the supplementary information of [2].

1 Julián García and Arne Traulsen, (2012), Journal of Theoretical Biology, Leaving the loners alone: Evolution of cooperation in the presence of antisocial punishment

2 Christoph Hauert, Arne Traulsen, Hannelore Brandt, Martin A. Nowak and Karl Sigmund, (2007), Science, Via freedom to coercion: the emergence of costly punishment

-