weak selection - senecio7の日記

Traulsen A 2010 Mathematics of kin and group selection: formally equivalent? Evolution 64:316-323.

the underlying mathematics of game theory is fundamentally different from the approach of inclusive fitness

does a single cooperator have a fixation probability larger than neutral in the system?
this would mean that cooperation can be called advantageous in a finite population
the thresholds based on fixation probabilities are global variables that take into account all possible frequencies of cooperators
these thresholds typically characterize the probabilities of the stochastic process, with each realization taking a different trajectory
the direction of selection is a deterministic local statement valid for a particular frequency and all trajectories
sometimes, such deterministic statements describe the average of a stochastic process
but in general this is not the case

evolutionary game theory is a general way to model frequency-dependent selection
in its modern finite population formulation (which is closely related to theoretical population genetics), stochastic effects are taken into account
recently, several authors have drawn insights from this new formulation
these models do not only consider group selection, but also selection in network structured populations or cooperation based on similarity
proponents of kin selection theory have argued that the results of these papers can equivalently be obtained from inclusive fitness methods (Lehmann and Keller 2006; Lehmann et al. 2007a,b; West et al. 2007b)
here, it is argued that this equivalence is in fact only found in special cases and that we are not facing a purely semantic question

many authors argue that the Price equation is the fundamental basis of evolutionary dynamics and provides the formal foundation for both kin selection and group selection
others have argued that this central role of the Price equation is misleading

to calculate the change of E(p), we have to know Cov(p, f)
this implies that we also need an equation that describes the change of Cov(p, f), involving terms such as Cov(fp, f)
the equation in Cov(fp, f) will again depend on higher moments of the distributions for f and p
only if such higher order moments can be expressed in terms of the lower moments, the system of equations becomes dynamically sufficient

the Price equation only allows to make a local statement on the direction and speed of evolution
this makes it impossible to calculate important evolutionary quantities, such as the probability of fixation
when selection is weak and fitness effects are additive, the direction and intensity of selection are constant
fixation probabilities can be assessed for finite populations with inclusive fitness methods

frequency-dependent selection per se is not a problem for inclusive fitness
it typically considers a special case of linear games in which the fitness differences are constant
with nonlinearities, inclusive fitness arguments can become wrong (van Veelen 2009)

for frequency-dependent selection, the seemingly small difference in the definition of weak selection has decisive mathematical consequences (Wild and Traulsen 2007)
Nowak et al. (2004) obtain interesting new results for nonadditive games under weak selection, such as the so-called one-third rule
the underlying definition of weak selection differs such that the conclusions are altered
when the phenotype space is continuous, the weak selection approach of inclusive fitness is appropriate
in discrete phenotype spaces, the game-theoretic approach can be more meaningful

why weak selection has to be invoked at all?
weak selection is invoked in kin selection approaches because the relatedness can usually be calculated only for neutral systems
under weak selection, the term describing selection factorizes into terms that depend on the payoff values and terms that invoke averages such as the probability to choose a pair of different individuals
the latter terms correspond to the relatedness, which can therefore be calculated under neutrality to assess first-order effects
combining this neutral relatedness with almost neutral weak selection allows to derive consequences of the evolutionary dynamics
such calculations are nontrivial and would become extremely challenging in the presence of selection
this is very similar to coalescence theory, which also focuses on the neutral case
for strong selection, we would have to depart from the reference case of neutral selection, because even a high order Taylor expansion is local and can only describe the dynamics close to the reference point
typically, one would assume that under strong selection some aspects of the dynamics become deterministic

in contrast, theoretical population genetics usually only requires weak selection to simplify equations, for example, by diffusion approximation
for some population structures, weak selection does not have to be invoked at all to make exact analytical statements
this is the case for the multilevel selection model discussed by Traulsen and Nowak (2006)
it is also true for evolutionary games on cycles, where players are arranged on a line (Ohtsuki and Nowak 2006a)
although weak selection seems to be relevant for genetic systems, when it comes to cultural evolution it may not be an appropriate approximation