Abstract
In a spatially structured population, limited dispersal gives rise to local relatedness, potentially favoring indiscriminate helping behavior. However, it also leads to local competition, which reduces the benefits of helping local kin. This tension has become the focus for a growing body of theoretical work. Existing models, however, have focused chiefly on the net impact of limited dispersal on cooperative or competitive effort in a homogeneous population. Here, I extend existing models of kin selection in a group-structured population to allow for asymmetries in expected fecundity and reproductive success among group members. I explore the consequent impact of limited dispersal on the evolution of helping and harming behavior, and on the degree of reproductive inequality or skew. I show that when individuals in a group differ in their expected fecundity, limited dispersal gives rise to kin selection for harming behavior on the part of more fecund individuals, and for helping behavior on the part of less fecund individuals. As a result, philopatry tends to exaggerate differences in reproductive success, and so promotes greater reproductive skew.
In a spatially structured population, limited dispersal gives rise to local relatedness among neighboring individuals. Hamilton (1964a,b) suggested that this might lead to kin selection for indiscriminate local altruism. However, he also observed that limited dispersal leads to local competition among relatives, which reduces the benefits to helping kin (Hamilton 1975). This tension between the kin-selected benefits of local altruism and the costs of local kin competition has become the focus for a growing body of theoretical work (Taylor 1992a,b; Wilson et al. 1992; van Baalen and Rand 1998; Mitteldorf and Wilson 2000; Taylor and Irwin 2000; Irwin and Taylor 2001; Le Galliard et al. 2003; Gardner and West 2006; Lehmann et al. 2006; Lehmann 2007; Johnstone and Cant 2008a), and some (although rather fewer) supporting empirical tests (West et al. 2001; Griffin et al. 2004).
Taylor (1992a,b) first showed that in a group-structured population, with nonoverlapping generations, the benefits of local helping and the costs of increased local competition cancel out, such that indiscriminate helping (behavior that confers fecundity gains on others, at a cost to the fecundity of the actor) will never be favored, regardless of the level of dispersal (see also Wilson et al. 1992). Subsequent analyses have, however, found that various factors, such as sex-biased or “budding” dispersal (Gardner and West 2006; Lehmann et al. 2006; Gardner et al. 2007; Johnstone and Cant 2008a), patch elasticity (Taylor 1992a; Lehmann et al. 2006), and overlapping generations (Taylor and Irwin 2000; Irwin and Taylor 2001) can give rise to selection for local helping and/or harming behavior.
Existing analyses of kin selection in structured populations have, however, focused chiefly on the mean level of cooperative (or competitive) effort to be expected in a homogeneous population (although see Ross-Gillespie et al. 2007 and Lehmann et al. 2008 for models of polymorphism). Here, by contrast, I extend Taylor's (1992a) simple model to allow for differences among group members in expected reproductive success, and explore the differential impact of limited dispersal on the behavior of more and less fecund individuals. This allows me to predict how limited dispersal will influence reproductive skew in a group-structured population. Reproductive skew refers to inequality in the distribution of breeding success among members of a group—in high-skew societies such as those of honey bees, Mexican jays or meerkats, one or a few individuals monopolize reproduction, whereas in low-skew societies, all individuals have similar opportunities to breed (see Vehrencamp 1983; Keller and Reeve 1994). The key question I address is whether helping or harming will evolve when individuals within a local group occupy distinct roles that differ in their expected reproductive success (reflecting differences in size, seniority or competitive ability), and whether the evolution of such behavior will tend to exacerbate or reduce these differences.
There already exists a substantial theoretical literature concerned with reproductive skew (for reviews see Keller and Reeve 1994; Johnstone 2000; Buston et al. 2007; Johnstone and Cant 2008b). Moreover, ecological constraints on dispersal and the degree of relatedness among group members are factors that have received much attention as possible influences on the partitioning of reproduction (Keller and Reeve 1994; Johnstone 2000; Buston et al. 2007; Johnstone and Cant 2008b, and for reviews of empirical studies see Keller and Reeve 1994; Reeve and Keller 2001; Magrath et al. 2004; Kutsukake and Nunn 2006; Liebert and Starks 2006). Surprisingly, however, none of the existing models of skew have taken into account the impact of local competition. All of these models implicitly assume that offspring mix at large within the population, and that competition occurs at a global scale; assumptions that are uncomfortably at odds with the emphasis of skew theory on ecologically constrained dispersal and cobreeding among relatives.
It thus appears that the two bodies of work referred to above, one dealing with kin selection in structured populations and one dealing with reproductive skew, suffer from complementary shortcomings. On the one hand, demographically explicit models of kin selection in structured populations have largely ignored the issue of reproductive skew; on the other, models of skew have ignored the issue of local competition. By combining the two approaches, I hope to address both of these problems at the same time.
Below, I first adopt the simple, inclusive fitness approach of Taylor (1992a) to show that variation among group members in expected reproductive success has striking consequences for kin selection in a structured population. I then develop a more detailed model of reproductive competition, based on the tug-of-war game of Reeve et al. (1998), which illustrates how, as a result, reproductive skew can be expected to change with the level of dispersal.
The Model
I focus on an asexual, haploid, “infinite-island” population, with nonoverlapping generations. The population comprises an infinitely large number of patches, each of which contains two individuals, one “dominant” or “senior” individual, and one “subordinate” or “junior.” In each generation, both individuals on a patch produce large numbers of young and then die. We suppose that the dominant typically enjoys some competitive advantage over the subordinate, or controls a better territory or larger share of resources within the patch, or is larger or otherwise more fecund, and thus can expect to produce more offspring. The ratio of the subordinate's fecundity to that of the dominant will be denoted s (≤1, where in the extreme, s= 1 implies that both group members have equal expected numbers of young). Following reproduction (and death of the adults), a fraction d of the offspring on a patch disperses; of these, a fraction 1 −k succeeds in finding a new patch; each is equally likely to arrive (independent of the others) at any patch in the population. Finally, the offspring on a patch, both native and immigrant, compete on an equal basis for dominant and subordinate breeding positions. Those who fail to find a position die, and the cycle then repeats. For simplicity, we use the composite parameter h= (1 −d)/(1 −kd), which gives the probability that an offspring is native to the patch on which it competes for a breeding spot. Note that the offspring of a dominant parent enjoy no competitive advantage in the struggle for breeding positions, and are no more likely than are the offspring of a subordinate parent to achieve either breeding status or dominance in their turn (the advantage enjoyed by a dominant individual over a subordinate is limited to its greater fecundity).
Note that helping behavior, when favored, constitutes a form of altruism (because it entails a reduction in the direct fitness of the actor), whereas harming behavior, when favored, constitutes a form a selfishness (because it entails an increase in the direct fitness of the actor).
Figure 1 shows the strength of selection for harming and for helping on the part of the dominant and subordinate, respectively, as a function of s, the subordinate's fecundity relative to that of the dominant, and of h, the level of philopatry. Strength of selection here refers to the critical ratio of c (fecundity cost incurred by the actor) to |b| (magnitude of the impact of the act on the recipient's fecundity, whether positive or negative) below which the act will be favored. The graphs reveal that selection for both helping and harming intensifies as philopatry increases. Selection for helping on the part of the subordinate is generally stronger than selection for harming on the part of the dominant, and may be substantial when there is a marked discrepancy in expected fecundity between the two. Lastly although selection for helping on the part of the subordinate increases as its fecundity declines relative to that of the dominant (i.e., as s decreases), selection for harming on the part of the dominant peaks at intermediate levels of disparity in fecundity (i.e., at intermediate values of s).
Upper graph shows the strength of selection for harming behaviour on the part of the dominant, measured as the critical ratio c/| b| (the fecundity cost to the actor relative to the fecundity impact on the recipient) below which a harming act will be favored, as a function of s, the subordinate's fecundity relative to that of the dominant, and h, the probability that an adult breeder is native to its patch. The lower graph shows the strength of selection for helping behaviour on the part of the subordinate, measured in the same way.
An Application
The above analysis shows that when individuals within a patch differ in their fecundity, limited dispersal can give rise to kin-selection for helping and harming. Moreover, kin selection will act differently on more and less fecund individuals, potentially favoring harming on the part of the former and helping on the part of the latter. However, helping and harming behavior will, in turn, influence the expected fecundity of both individuals and the equilibrium level of relatedness, and will thus modify the action of kin selection. To determine the stable outcome of evolution, we thus need a more explicit model of how the behavior of both dominant and subordinate interact to determine their fecundity.
and 
denote the mean levels of dominant and subordinate competitive effort in the population. In each case, fitness is equal to fecundity, multiplied by expected offspring success (the probability that an offspring claims a dominant or subordinate breeding position multiplied by its reproductive value if it does so). The expression for offspring success represents the weighted sum of two terms; the first, expected success for the fraction (1 −d) of offspring that remain on their native patch, the second, expected success for the fraction d(1 −k) of offspring that disperse and survive to compete in other patches. In each term, the value 2 appears in the numerator because there are two breeding positions in each patch for which offspring can compete (with average reproductive value of 1); the denominator represents the number of offspring competing for these breeding positions in the patch in question.
Results
Figure 2 shows s, the equilibrium fecundity and/or fitness of a subordinate relative to that of a dominant breeder, as a function of h (the level of philopatry), for four different values of g, the relative competitive ability of subordinates (because fitness is proportional to fecundity, the relative fitness of subordinates is equal to their relative fecundity). The graph shows that, unsurprisingly, less competitive subordinates obtain less reproductive success compared to dominant individuals. More interestingly, when there is a competitive asymmetry between dominant and subordinate (i.e., when g < 1), then the relative success of the subordinate declines as individuals become more philopatric. In other words, limited dispersal leads to greater reproductive skew.
Equilibrium fecundity and/or fitness of a subordinate relative to that of a dominant breeder (s) in the tug-of-war game, as a function of the level of philopatry (h), for four different values of g, the relative competitive ability of the subordinate. Successively lower curves correspond to successively lower values of g: 1 (implying no difference in competitive ability between group members), 1/2, 1/4, and 1/8.
Figure 3 shows the equilibrium levels of competitive effort invested by dominant and subordinate individuals, again as a function of h (the level of philopatry), for four different values of g (the relative competitive ability of the subordinate). The graph reveals that dominant individuals invest more effort in competition when the subordinate is stronger, and this pattern holds regardless of the level of philopatry. As philopatry increases, there may be a slight change in the level of effort that the dominant invests, but this effect is weak and inconsistent across different levels of relative competitive ability. By contrast, philopatry and local competition have a strong influence on the level of competitive effort invested by subordinates. In a well-mixed population, weaker subordinates invest more effort in competition. But as philopatry increases, subordinate effort declines, and it does so more rapidly when the asymmetry in competitive ability is greater. This leads ultimately to a reversal in the relationship between subordinate strength and competitive effort, such that (when there is little dispersal), weaker subordinates invest less effort in competition. It is the decline in subordinate effort with philopatry that chiefly explains the increase in reproductive skew as illustrated in Figure 2.
Equilibrium competitive effort of the dominant (x*, shown in the top graph) and of the subordinate (y*, shown in the bottom graph) in the tug-of-war game, as a function of the level of philopatry (h), for four different values of g, the relative competitive ability of the subordinate. In the top graph, successively lower curves correspond to successively lower values of g: 1 (implying no difference in competitive ability between group members), 1/2, 1/4, and 1/8. In the bottom graph, curves are labeled with the relevant values of g.
Discussion
The above analysis reveals that the kin-selected consequences of limited dispersal are strongly affected by asymmetries in reproductive success. Taylor (1992a) showed that in a group-structured population with nonoverlapping generations, the kin-selected costs of increased local competition precisely cancel out the kin-selected benefits of helping others. It is now clear, however, that this holds true only in the absence of any asymmetries in reproductive success among group members. Where such asymmetries exist, limited dispersal gives rise to kin selection for harming behavior on the part of more successful individuals, and for helping behavior on the part of less successful individuals. The result, as illustrated by the tug-of-war model, is that philopatry tends to exaggerate within-group differences in reproductive success, and so leads to higher skew.
Why does the impact of limited dispersal differ according to the relative fecundity or fitness of individuals in a group? The reason is that more successful breeders are more closely related (on average) to locally competing offspring. Because they contribute a greater share of the offspring that compete over breeding spots in their patch, their average relatedness to such offspring is higher. Consequently, they suffer greater kin-selected costs as a result of increased local competition. This tends to oppose selection for helping behavior, which leads to a net increase in local productivity and hence to more intense local competition. Instead, selection on more successful breeders favors harming behavior, which leads to a net decrease in local productivity, and hence to less intense local competition. Precisely the reverse argument applies to less successful breeders. They contribute a smaller share of locally competing offspring, and so suffer less kin-selected costs as a result of increased local competition, which tends to favor helping over harming.
Perhaps surprisingly, selection for harming on the part of more successful breeders peaks at intermediate levels of skew (Fig. 1). The reason is that as skew becomes more pronounced, the mean level of relatedness between the breeders in a group increases, because they are more likely to be offspring of the same parent. Consequently, although the kin-selected benefits of reduced local competition are greater for more successful breeders when skew is higher (because they then contribute a greater fraction of the offspring that compete in their patch), so too are the kin-selected costs of harming their competitors. At very high levels of skew, the latter effect tends to counteract the former until, as one approaches complete reproductive monopoly, the two cancel out and all selection for harming is lost. Selection for helping on the part of less successful breeders, by contrast, grows even stronger as skew increases, because higher relatedness between breeders amplifies the benefits of helping.
I have referred to “dominant” and “subordinate” individuals, suggesting that differences in expected fecundity between the two are the result of differences in competitive ability. But such differences might arise for many other reasons too. To illustrate, the “headstart hypothesis” of Queller (1989, 1994) suggests that when offspring depend on continued care by adults, as in social insects, potential new breeders may do better to become helpers assisting at their mother's nest because they can reap immediate inclusive fitness gains via the “head start” of having sibs already close to maturity. In this model, the expected fecundity of a new breeder is lower than that of an established breeder simply because the former experiences an initial delay (with associated mortality risk) in bringing offspring to maturity (Queller 1989; Nonacs 1991; Queller 1994). Differences in expected fecundity might arise simply due to the timing of breeding, rather than as a result of differences in competitive ability or any other intrinsic characteristic. In this context, the present analysis suggests that limited dispersal is likely to favor the evolution of helping behavior, because of the differential impact of local competition on established and potential new breeders (see Pen and Weissing 2000; Lehmann et al. 2008).
Although there are many factors that might give rise to differences in fecundity or mating success among members of a group, not all will be “known” to the individuals in question. Pen and West (2008), for instance, consider the impact of reproductive skew in a model similar to the present analysis, but make the implicit assumption that differences in reproductive success arise by chance, so that individuals cannot “predict” their share of reproduction within a patch (although the level of skew itself is predictable). Formally, this is reflected in their assumption that all individuals have the same expected reproductive success, and adopt the same strategy; there are no “dominants” or “subordinates” in their analysis. Although such unpredictable skew can (as they show) modify the action of kin selection, the assumption of strategic homogeneity precludes any possibility that kin selection will in turn modify the level of skew, which in their model is externally specified. By contrast, I have focused on the differential impact of kin selection on the behavior of dominant and subordinate individuals, who differ predictably in their expected reproductive success. Only when such predictable differences exist, and individuals consequently occupy distinct strategic roles, will limited dispersal influence skew in the way I describe here.
What are the implications of my results for existing theories of reproductive skew? Previous skew models have, as stated in the Introduction, ignored the possible impact of local competition. They implicitly assume that offspring compete on a population-wide scale, even though they go on to breed in association with kin. Although this is not impossible, it seems very likely that when there are ecological constraints on dispersal, as most skew models assume, offspring will experience local competition (e.g., over their parents' breeding territory). As I have shown, this will have a different effect on dominant and subordinate breeders, leading to changes in the predicted level of skew.
Some of the effects of local competition, which existing models do not account for, bear directly on current attempts to test theories of skew. For instance, in the absence of local competition, the tug-of-war model of Reeve et al. (1998) predicts that the level of skew will be very insensitive to relatedness among members of a group. This contrasts with the predictions of “transactional” skew models (in which some individuals concede shares of reproduction to others to maintain group stability), which predict a strong influence of relatedness on skew (see Johnstone 2000). Consequently, the relationship between skew and relatedness has been a focus for empirical tests (e.g., Field et al. 1998; Queller et al. 2000; Reeve et al. 2000; Hannonen and Sundström 2003; Haydock and Koenig 2003; Charpentier et al. 2005). But when local competition is taken into account, the present analysis reveals that tight constraints on dispersal, which give rise to high levels of local relatedness, also favor high skew in the tug-of-war game. Across populations or species, therefore, the tug-of-war model (with local competition) predicts a positive association between skew and relatedness.
More generally, one attractive feature of transactional models has been that they link conflict resolution and skew within a group to external, ecological factors, such as the potential for independent breeding and the cost of dispersal. As the cost of dispersal decreases, for instance, subordinates have more to gain from leaving a group, and so may require a larger share of reproduction if they are to remain. By contrast, incomplete control models such as the tug-of-war game make no reference to a subordinate's outside options, and thus predict no impact of ecological constraints on the level of skew. The present analysis shows, however, that effective dispersal rates determine the intensity of local competition, and so are likely to influence conflict resolution and skew even when there is no threat of group break-up (or no possibility of transactional exchanges).
To sum up, I suggest that there is much to be gained by integrating the two bodies of theory to which I have referred. Models of skew need to incorporate the impact of local competition among offspring, because this can substantially modify the resolution of conflicts among breeders; at the same time, by incorporating asymmetries among group members, models of kin selection in structured populations can be extended to address a far wider range of questions.
Associate Editor: S. Otto
