We now have the two ingredients, fitness effects and probabilities of identity, that allow us to evaluate the inclusive fitness effect explicitly.
Overlapping generations: Moran process
Nonoverlapping generations: Wright–Fisher process
The inclusive fitness effect when each individual of the population dies in each generation has been analysed by Taylor (1992b) and Rousset (2004, p. 124). Inserting eqn 25 into the right-hand side of eqn 29, we have
Inserting this equation into eqn 15 and setting s = 0, we then have
Dividing the inclusive fitness effect by 1 − Q0, using the formula for the stationary diversity within a deme (eqn 26) and taking the low mutation limit ( lim μ0S/(1 − Q0)), reveals that the inclusive fitness effect is proportional to
which is always a net cost (e.g. Rousset, 2004, eqn 7.21) independent of the structure of the population! When migration is random (mk = 1/nd for all k), eqn 32 becomes equivalent to eqn 36. Hence, the direction of selection on the mutant for weak selection is the same in panmictic populations, whether there is overlapping generations or not.
Overlapping generations: Cannings process
The inclusive fitness effect when each adult individual has a probability s of surviving per generation has been studied previously under more stringent life-cycle assumptions, that is, for the infinite island model of dispersal by Taylor & Irwin (2000), for the stepping-stone model of dispersal and interactions by Irwin & Taylor (2001) and for an arbitrarily dispersal distribution but with only local interactions by Lehmann & Keller (2006). Inserting eqn 27 into the right-hand side of eqn 29, we have
We now expand the double sum appearing in the second term of the right-hand side as
because m−k = mk and
which is a measure of average diversity. Substituting the last three equations into the inclusive fitness effect (eqn 15), dividing it by (1 − Q0) and rearranging we obtain
where γ ≡ (1 − μ)2. Taking the low mutation limit ( lim μ0S/(1 − Q0)) and using the stationary diversity within a deme for the Cannings process (eqn 28), we have
With these formulae in hand, the inclusive fitness effect finally becomes
which will depend on the shape of the dispersal distribution and must be evaluated in the low mutation limit (when μ 0). Hence, the inclusive fitness effect does not reduce to a simple form without further assumption on the life cycle.
We will now express the Xks in terms of the dispersal distribution to subsequently obtain a low migration approximation of the inclusive fitness effect (Nagylaki, 1982; Rousset, 2004). To that aim we use classical results on Fourier analysis and follow similar developments as presented in Rousset (2004, chapter 3). Call ψ(z) ≡ ∑imi eıi·z the characteristic function (the Fourier transform) of the dispersal distribution, where , and let (z) ≡ ∑iQi eıi·z be the Fourier transform of the Qis. Fourier transforming eqn 27 and rearranging, we find that
which, once solved for the characteristic function (z), yields
which, when s 0, makes direct contact with the standard formulae (e.g. Malécot, 1975; Nagylaki, 1976; Epperson, 1999; Rousset, 2004). From these equations we can unleash the stationary Qk as
where k()≡(1/nd)∑j(z(j))e−ık·z(j) is the inverse Fourier transform of at distance k.
Now that the Xks are expressed in terms of the characteristic function of the dispersal distribution, we can establish a low migration approximation for these functions. To that aim, we write m0 = (1 − m) and mi = mgi, and approximate the Xks by letting the migration rate m go to zero (Rousset, 2004, chapter 3). From these definitions, the characteristic function of dispersal can be expressed as ψ = 1 − m(1 − ∑i≠0gi eıi·z). Inserting this expression into [(1 − s)ψ/(1 + s + (1 − s)ψ)] and by Taylor expansion at m = 0, we get
By inverse Fourier transforming this expression and noting that the inverse transform of a constant a is k(a) = 0 except that 0(a) = a, we finally obtain
These approximations are valid irrespective of the shape of the dispersal distribution, whenever the dispersal rate m is small.