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

Dividing the inclusive fitness effect by 1 − *Q*_{0}, using the formula for the stationary diversity within a deme for the Moran process (eqn 20) and taking the low mutation limit ( lim _{μ0}*S*/(1 − *Q*_{0})), which results in *Q*_{k} 1, shows that the inclusive fitness effect is proportional to

- (32)

which, when *m*_{k} = 1/*n*_{d} for all **k** (i.e. unstructured population), reduces to eqn 9. Hence, the mutant is selected for when

- (33)

##### 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

- (34)

Inserting this equation into eqn 15 and setting *s* = 0, we then have

- (35)

Dividing the inclusive fitness effect by 1 − *Q*_{0}, using the formula for the stationary diversity within a deme (eqn 26) and taking the low mutation limit ( lim _{μ0}*S*/(1 − *Q*_{0})), reveals that the inclusive fitness effect is proportional to

- (36)

which is always a net cost (e.g. Rousset, 2004, eqn 7.21) independent of the structure of the population! When migration is random (*m*_{k} = 1/*n*_{d} 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

- (37)

We now expand the double sum appearing in the second term of the right-hand side as

- (38)

because *m*_{−k} = *m*_{k} and

- (39)

which is a measure of average diversity. Substituting the last three equations into the inclusive fitness effect (eqn 15), dividing it by (1 − *Q*_{0}) and rearranging we obtain

- (40)

where *γ* ≡ (1 − *μ*)^{2}. Taking the low mutation limit ( lim _{μ0}*S*/(1 − *Q*_{0})) and using the stationary diversity within a deme for the Cannings process (eqn 28), we have

- (41)

and

- (42)

With these formulae in hand, the inclusive fitness effect finally becomes

- (43)

where

- (44)

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 *X*_{k}s 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**) ≡ ∑_{i}*m*_{i} e^{ıi·z} the characteristic function (the Fourier transform) of the dispersal distribution, where , and let (**z**) ≡ ∑_{i}*Q*_{i} e^{ıi·z} be the Fourier transform of the *Q*_{i}s. Fourier transforming eqn 27 and rearranging, we find that

- (45)

which, once solved for the characteristic function (**z**), yields

- (46)

where

- (47)

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 *Q*_{k} as

- (48)

where _{k}()≡(1/*n*_{d})∑_{j}(**z**(**j**))e^{−ık·z(j)} is the inverse Fourier transform of at distance **k**.

Noting that *m*_{k} = _{k}(*ψ*) and using the stationary *Q*_{k}, we can write

- (49)

Substituting eqn 47 into the last equation and taking the low mutation limit ( lim _{μ0}*X*_{k}), we obtain after simplification

- (50)

Now that the *X*_{k}s 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 *m*_{0} = (1 − *m*) and *m*_{i} = *mg*_{i}, and approximate the *X*_{k}s 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≠0}*g*_{i} e^{ıi·z}). Inserting this expression into [(1 − *s*)*ψ*/(1 + *s* + (1 − *s*)*ψ*)] and by Taylor expansion at *m* = 0, we get

- (51)

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

- (52)

except that

- (53)

These approximations are valid irrespective of the shape of the dispersal distribution, whenever the dispersal rate *m* is small.