#### Basic model

Our analysis is based on the infinite island model analysed by Taylor (1992a); see also Wright (1931). A summary of the model notation used in this article is provided in Table 1. We consider an infinite population of asexual haploid individuals organized into patches of *n* individuals, and initially we assume that no social interaction takes place. Individuals produce a large number *K* >> 1 of offspring, then die, and each offspring disperses to another, randomly chosen, patch with probability *m* or else, with probability 1 − *m*, remains in its patch of origin. After dispersal, *n* offspring are chosen on each patch to mature to adulthood, whereas the rest die, and this returns the population to its original size for the beginning of the next lifecycle.

Table 1. A summary of symbols used in the analysis. Symbol | Definition |
---|

*a* | Fecundity cost of harming for harmer |

*A* | Personal cost function for harming |

*b* | Fecundity benefit of helping for recipients |

*B* | Recipient benefit function for helping |

| Fitness effect of social behaviour on recipient |

*c* | Fecundity cost of helping for helper |

*C* | Personal cost function for helping |

| Fitness effect of social behaviour on actor |

*c*_{X} | Reproductive value of class X |

*d* | Fecundity cost of harming for victims |

*D* | Victim cost function for harming |

D | Disperser |

*F* | Fecundity of a focal individual |

| Average fecundity of focal patch |

*g* | Genetic ‘breeding’ value for social behaviour |

*i* | Number of mutants among a focal individual’s patch mates |

*K* | Average number of offspring per individual in the population |

*m* | Dispersal rate |

M | Mutant |

*n* | Number of breeding spaces per patch |

N | Nondisperser |

*q*_{X} | Abundance of class X |

*r* | Coefficient of relatedness between patch mates (i.e. notincluding self) |

*R* | Coefficient of relatedness within a patch (i.e. including self) |

*t* | Generation number |

*u* | Generic nondisperser class |

*U* | Set of all nondisperser classes |

*v* | Generic disperser class |

*V* | Set of all disperser classes |

*w* | Darwinian fitness |

*x* | Helping strategy |

| Mutant helping strategy |

| Average helping strategy |

*y* | Harming strategy |

| Mutant harming strategy |

| Average harming strategy |

Assuming that all genetic variation in the population is neutral with respect to reproductive success, we can use the recursion approach of Taylor (1992a) to determine the relatedness structure of the population at equilibrium. We note that the expected relatedness of any individual in generation *t* to a randomly chosen member of her own patch (including herself) is *R*_{t} = 1/*n* + ((*n* − 1)/*n*)*r*_{t}, where 1/*n* is the probability of choosing herself, in which case the relatedness is 1, and (*n* − 1)/*n* is the probability of choosing a different individual, in which case the relatedness is *r*_{t}, the relatedness of two different individuals in the same patch. The only way for two different individuals from the same patch to be related is if neither dispersed, which occurs with probability (1 − *m*)^{2}. In this case, the two individuals will have an expected relatedness equal to the relatedness of two individuals chosen at random from that same patch in the previous generation, *R*_{t−1}. This allows us to write down a recursion:

- (1)

which can be solved for equilibrium (*R*_{t−1} = *R*_{t} = *R*) to yield an equilibrium relatedness of an individual to her group (including herself) of:

- (2)

as determined by Taylor (1992a). Substituting this into the expression *R* = 1/*n* + ((*n* − 1)/*n*)*r* and solving for *r* obtains the equilibrium relatedness between different individuals within the same patch:

- (3)

i.e. the *F*_{ST} of Wright (1943). Note that both these ‘whole-patch’ and ‘others-only’ relatedness coefficients (*R* and *r*; Pepper 2000) are expressed as averages over all individuals, with dispersers and nondispersers taken together. However, dispersing individuals always find themselves in a patch containing no relatives other than themselves, and so restricting attention to dispersers only, the equilibrium relatedness is *R*_{D} = 1/*n* to the whole patch and *r*_{D} = 0 to the other individuals in the patch. Noting that *R* = *mR*_{D} + (1 − *m*)*R*_{N} and *r* = *mr*_{D} + (1 − *m*)*r*_{N}, where *R*_{N} and *r*_{N} are the whole-patch and others-only relatedness for nondispersing individuals, we can solve to find that:

- (4)

and:

- (5)

for nondispersers. Note that for 0 < *m* < 1, we have *R*_{D} < *R* < *R*_{N} and *r*_{D} < *r* < *r*_{N}, i.e. dispersers experience a lower relatedness to their social partners, and nondispersers a higher relatedness to their social partners, than average (Fig. 1). Taylor (1992a) found that unconditional helping behaviour cannot be favoured if it incurs a net fecundity cost for the actor; the result also extends to unconditional harming behaviour, revealing that this is also disfavoured if the actor incurs a net fecundity cost. In the next two sections, we explore whether and when the increased relatedness experienced by nondispersers favours helping behaviour that is expressed only by nondispersers, and whether and when the decreased relatedness experienced by dispersers favours harming behaviour that is expressed only by dispersers.

#### Evolution of helping in nondispersers

We now consider a vanishingly rare genetic mutation that causes the bearer to express indiscriminate, others-only helping behaviour that incurs a relative personal fecundity cost *c* and gives a relative fecundity benefit *b* that is shared equally by her *n* − 1 patch mates; these fecundity effects are expressed relative to the baseline fecundity, which is set equal to 1, and we assume *b*, *c* << 1. Crucially, we assume that this helping behaviour is expressed conditionally with respect to the individual’s own dispersal status; in particular, we assume that only nondispersers, who have remained in their natal patch, exhibit helping behaviour.

In order for the mutation to be favoured by selection it must, on average, increase the expected fitness of its bearers, i.e. the number of surviving offspring that the individual produces over her lifetime (Price 1970). There are two routes by which a gene may impact on the fitness of its bearer: first, by having a direct impact on that individual’s personal reproductive success, and second, by having an indirect impact on the individual’s personal reproductive success due to its presence among relatives who socially interact with her (neighbour-modulated fitness; Hamilton 1963, 1964). There are neither direct nor indirect effects of the gene on carriers that have dispersed away from their patch of origin, as they do not express the helping behaviour and they do not encounter any other individuals carrying the mutant gene. So, the condition for the gene to be favoured is if it leads, through direct and/or indirect effects, to an increase in the expected number of surviving offspring (i.e. Darwinian fitness) of nondispersers (see Appendix for mathematical details). The average fitness in the population is 1, because the population size remains constant from generation to generation, and dispersers and nondispersers have the same expected fitness; so, a condition for the mutant gene to be favoured by natural selection is that the expected fitness of a disperser carrying the gene exceeds 1.

We now proceed to calculate the expected fitness of a nondisperser who is carrying the mutant gene, and this is a function of her fecundity and the intensity of competition experienced by her offspring for one of the *n* breeding spaces within whichever patch they find themselves in after the dispersal event. The relative (to the population average) fecundity of a mutant nondisperser that shares her patch with *i* other mutant nondispersers is:

- (6)

Thus, the focal individual produces a total of *KF*_{MN,i} offspring, of which *mKF*_{MN,i} disperse, whereas (1 − *m*)*KF*_{MN,i} remain in the natal patch.

We consider first the fate of the *mKF*_{MN,i} dispersing offspring of the focal mutant individual. Each arrives in a separate patch, where they compete with (1 − *m*)*nK* locals and *mnK* other dispersers. As there are *n* breeding spaces within the patch, the probability that a particular dispersing offspring of the focal mutant survives competition to find a breeding space is *n*/(1 + (1 − *m*)*nK* + *mnK*) ≈ 1/*K*. Thus, the expected number of survivors among the dispersing offspring of the focal mutant is *mKF*_{MN,i}/*K* *=* *mF*_{MN,i,}.

Again, there are *n* breeding spaces available, and so the expected number of successful nondispersing offspring due to the focal mutant individual is

which, to first order in *b* and *c*, is:

- (8)

By adding the expected number of surviving dispersing and nondispersing offspring we can find the expected fitness of the focal mutant nondisperser, which is:

- (9)

This expected fitness is conditional on there being *i* other (nondispersing) mutants in the focal individual’s patch. As outlined above, the condition for invasion of the mutant gene is *w*_{MN} > 1, where *w*_{MN} is the average fitness of all nondispersing mutant individuals, i.e. *w*_{MN} = *E*_{MN}(*w*_{MN,i}). This average fitness obtains:

- (10)

Note that *i*/(*n* − 1) is the proportion of (nondisperser) mutants among the social partners of the focal individual, and so the average of this quantity, taken over all nondispersing mutants in the population, is equal to *r*_{N}, the others-only relatedness experienced by nondispersers. This makes use of the assumption that the mutant is vanishingly rare in the population. Similarly, (*i* + 1)/*n* is the proportion of (nondisperser) mutants among all the individuals in the focal individual’s patch, including herself; so, averaging this quantity over all nondisperser mutants in the population obtains *R*_{N}, the whole-group relatedness experienced by nondispersers. Making this substitution, we obtain a condition, in the form of Hamilton’s rule, for when the helping mutant will invade from rarity:

- (11)

Because we have assumed weak selection, this condition for increase is frequency independent (Rousset 2004, p. 80), and therefore it also describes the progress of the mutant gene when it is no longer rare. Although condition 11 has been derived using a neighbour-modulated fitness approach to kin selection, it readily yields an inclusive fitness interpretation. Helping by nondispersers is favoured when the sum of three components, equal to the inclusive fitness effect of their helping, is positive. First, their help leads to a personal loss of *c* offspring. Second, their help leads to *b* extra offspring for the other individuals on the patch, and these are valued at *r*_{N}, the relatedness to patch mates. Third, the net increase of *b* − *c* offspring on the patch leads to enhanced competition for locals, to the extent that these extra offspring do not disperse and nor do the other locals [i.e. (1 − *m*)^{2}], and this leads to the competitive exclusion of local offspring (including own offspring) which are valued by *R*_{N}, the whole-group relatedness appropriate for nondispersers.

Note that the relatedness coefficients can be written in the form of eqns 4 and 5; making this substitution into inequality 11 obtains an invasion condition that is expressed wholly in terms of model parameters (*n*, *m*, *b* and *c*):

- (12)

The above analysis has determined when a mutation that incrementally increases (or decreases, if we change the sign of *b* and *c*) the level of indiscriminate helping exhibited by nondispersers will invade from rarity under the action of natural selection. The RHS of inequality 12, which we may denote *f*(*n*,*m*), presents a quantity that must be exceeded by the benefit/cost ratio of helping in order for nondispersing individuals to be favoured to indiscriminately help their patch mates. We find that as *n* increases, the RHS of the inequality becomes larger (∂*f*/∂*n* > 0), i.e. helping is less favoured as the number of individuals per patch increases (Fig. 2a). We find that helping is most readily favoured at intermediate migration rates (there is some threshold 0 < *m** < 1 such that ∂*f*/∂*m* < 0 for *m* < *m** and ∂*f*/∂*m* > 0 for *m* > *m**; Fig. 2a). At the extreme of a very viscous population, *m*0, the condition for helping to be favoured in nondispersers is *b*/*c* > 1 + (2*n* − 1)*n*. However, the selection pressure here is vanishingly small (LHS of eqn 110 as *m*0), and so helping is a nearly neutral character in extremely viscous populations and is not expected to be elaborated by natural selection. At the other extreme of a fully mixing population, costly helping is never favoured (*f*∞ as *m*1). Numerical investigation reveals that the optimum migration rate for helping (i.e. *m* = *m**) is less than 0.5, but converges upon 0.5 as *n*∞, and the condition for helping to evolve at this optimum is approximately *b*/*c* > 4*n* − 2.

Strictly, this condition applies only if the ESS takes a positive value. An explicit solution for the ESS level of helping *x** requires explicit functional forms for *C*(*x*) and *B*(*x*). Numerical solutions for an illustrative example are given in Fig. 2b.

#### The evolution of harming in dispersers

We now investigate the evolutionary success of a vanishingly rare mutant gene that causes bearers to express harming behaviour upon dispersal to a new patch, and which confers a relative personal fecundity cost *a* and a further relative fecundity cost *d* that is shared equally among the *n* − 1 other individuals on the patch. Again we express *a* and *d* relative to baseline fecundity, and we assume that *a*, . This mutant gene will spread if, on average, it increases fitness of its bearers (whether they are dispersers or nondispersers), either due to direct or indirect fitness effects. There are no direct or indirect effects of the mutant gene on nondispersing mutants as these do not express the gene nor do they encounter dispersing mutants who do express the harming behaviour. Therefore, the mutant gene can only invade from rarity if it increases the average fitness of mutant dispersers (see Appendix for mathematical details). The relative fecundity of a mutant disperser is simply:

- (14)

This focal individual produces a total of *KF*_{MD} offspring, of which *mKF*_{MD} disperse to a new patch and (1 − *m*)*KF*_{MD} remain in the natal patch. Using the same approach as in the previous section, the probability that any one of the dispersing offspring of the focal mutant will survive the competition and find a breeding space is 1/*K*. This means that the expected number of dispersing offspring due to the focal individual is *mF*_{MD} = *m*(1 − *a*).

Again, there are *n* breeding spaces available, and so the expected number of successful nondispersing offspring due to the focal mutant individual is which is, to first order in *a* and *d*, equal to:

- (16)

By adding the expected numbers of dispersing and nondispersing offspring together we can find the average fitness of a mutant disperser, and this is:

- (17)

Note that 1/*n* is equal to *R*_{D,} the relatedness coefficient for the whole patch the mutant is found in (for dispersing mutants, *r*_{D} is 0). In order for the mutant gene to invade from rarity, it must provide a fitness advantage, i.e. *w*_{MD} > 1. This gives an inequality in the form of Hamilton’s rule:

- (18)

Again, we can derive an inclusive fitness interpretation. Harming by dispersers is favoured when the sum of three components, equal to the inclusive fitness effect of harming, is positive. First, the harming act leads to a personal loss of *a* offspring. Second, the act will inflict a cost *d* on fellow patch mates; but, as these are related by *r*_{D} = 0, this component of inclusive fitness is zero. Third, the net decrease of *a*+*d* offspring on the patch leads to decreased competition for locals, to the extent that dispersal does not occur, and hence an effective increase in local offspring (including the mutants own offspring) that would otherwise have been competitively excluded. These are valued by *R*_{D} = 1/*n*, the whole-group relatedness for dispersing mutants.

Again, this applies strictly to ESSs that take intermediate values. An explicit solution for the ESS level of helping *y** requires explicit functional forms for *D*(*y*) and *A*(*y*). A numerical solution for an illustrative example is given in Fig. 3b. Note that, although we have modelled helping and harming behaviours as separate traits, an alternative (but equivalent) analysis could treat this social behaviour as a single trait varying from extreme conflict to extreme cooperation.