J. theor. Biol. (1992) 157, 447-461
Kin Selection in Density Regulated Populations JOHN K. KELLY t
Department of Ecology and Evolutionary Biology, University of Connecticut, Storrs, CT 06269, U.S.A. (Received on 2 July 1991, Accepted on 13 February 1992) The process of kin selection has both intra- and inter-group components (Hamilton, 1975, in: Biosocial Anthropology Wade, 1980). Group advantageous characteristics can evolve when inter-group differences in fertility are sufficiently great to overcome any within-group disadvantage of the trait. The potential magnitude of inter-group differences in fertility is determined largely by the way a population is regulated. Inter-group differences decrease as the spatial scale over which a population is regulated becomes increasingly localized. The present paper extends previous work by Boyd (1982, Anita. Behav. 30, 972-982) on the quantitative relation between kin selection and density regulation. A simple genetic model is employed to examine the conditions under which the interaction of local density regulation and kin selection can maintain a stable polymorphism. The ecological factors determining the spatial and temporal scale of density regulation are discussed. Finally, the results are applied to two biological cases in which local density regulation may be influencing the direction of phenotypic plasticity in group advantageous characters.
Introduction
The theory of kin selection describes gene frequency change resulting from fitness determining interactions that occur within small clusters of individuals referred to as trait groups (Wilson, 1975) or neighborhoods (Nunney, 1985). These clusters of interacters can vary in their total production of offspring whenever there is genetic variance between them. The fitness of an allele is thus determined by both its effect on the individual carrying it and its effect on the other individuals in its trait group. Gene frequency change, Ap, in such a spatially structured population, can be decomposed into within and between group components (Robertson, 1966; Price, 1970) Ap = Ap, + ap~
where Apt is the average change in gene frequency due to selection within groups and Ape is the frequency change resulting from the differential production of trait groups. The important feature of this decomposition is that it isolates all components of gene frequency change that depend on the differential production of trait groups. The magnitude of inter-group differences in productivity is determined largely by the way a population is regulated. The purpose of this paper is to examine the effects
t Current address: Department of Ecology and Evolution, University of Chicago, Chicago, IL 60637, U.S.A. 447
0022-5193/92/160447 + 15 $08.00/0
© 1992 Academic Press Limited
448
J.K.
KELLY
that population dynamic factors will have on the pattern of offspring production over the range of a population and thus evaluate the importance of these ecological forces to kin selection.
Density Regulation The potential magnitude of inter-group differences in reproductive output is limited by the scale over which the population is regulated. Kin selection models generally assume that population regulation occurs globally (local hard selection) and has no effect on inter-group selection. Ecological studies suggest however that density regulating interactions often occur over relatively small scales (Levin, 1974). If a population is regulated at the scale of trait groups (local soft selection), all groups make equivalent contributions to the next generation and inter-group selection is prevented (Wilson & Colwell, 1981 ; Wade, 1985). Clearly, both scenarios are idealizations. Most natural populations are limited by multiple ecological constraints which may operate over different spatial scales. Second, the simple hard and soft selection models assume that the response of regulating elements to changes in the size of the focal population is immediate. While this may be realistic in some circumstances (MacArthur, 1972), the effects of increased density will often manifest themselves over longer temporal scales. In addition, the population may experience densitydependent mortality at multiple stages in the life cycle. Finally, density regulation may be linked to other factors such as migration that are also influencing kin selection (Kelly, 1992; Wilson et al., 1992, submitted to Evol. Ecol.). There are several ways to incorporate the effects of varying spatial or temporal scale of density regulation into kin selection. The approach to be used in this paper was introduced by Boyd (1982). He constructed a model that allows some fraction of density regulation to occur locally with a second stage of population reduction occurring globally. He demonstrates, using an ESS model, that optimal behavioral strategies will depend on the nature of local density-dependent mortality. Specifically, Boyd finds that altruism should be directed only towards individuals whose relatedness to the altruist exceeds a specific threshold value. This threshold value is determined by the relation between local density and group fitness. The present paper employs a simple genetic model to examine the evolutionary consequences of local density regulation for kin selection and specifically, the conditions for stable polymorphism. The Model Consider a population of haploid organisms with discrete generations. Assume that this population is subdivided into a very large number of local trait groups in which natural selection and local density regulation occur. At the end of each generation, all individuals congeal into a global mating pool. The progeny then resettle into distinct trait groups, each of size N. The population consists of two types, A and B, at frequencies p and q (q = 1 - p ) respectively. The A types perform a group advantageous behavior and increase the viability of all other members of their trait group
KIN
SELECTION
IN POPULATIONS
449
by an increment r (r > 0) and affect themselves by increment d. Thus, the net effect on the viability of each type prior to density regulation in the ith trait group is sAi=d+r(piN - l),
sB~=rpiN
where sA~ is the direct effect of group advantageous actions on A types, sn~ is the direct effect on B types, and p~ is the frequency of A in the ith trait group. These are the basic fitness functions in Wilson (I980). We shall treat density regulation with a "logistic" formulation, in which the magnitude of the direct effects (sA~and sin) are reduced by a density regulating factor. The absolute fitness of A and B types after density regulation in the ith trait group, wA~and wn~is given by the following functions wA~= l +s~(1 ---~),
w~;= 1 + s,~(1 -)!~).
(1)
M is a constant that determines the severity of local density regulation. Low values of M indicate severe local regulation. R~ is a measure of "resource consumption" in the ith trait group. The value of R~ is proportional to pre-regulatory survivorship in the ith trait group Ri = N[ 1 +pism + (l -p~)s~i]. Finally, the fitness functions [eqn (1)] include 1 as a basal fitness value (Wilson, 1990). Gene frequency change is given by p~w ~
p' -
(2)
~.piwa, + E (I -p,)wn,
where the summations indicate an average taken over all trait groups. The expression resulting from incorporating eqn (1) into eqn (2) is complex (see Appendix), but a tractable approximation is given by p[1 + SA(1 - RA/M)]
p'
~p[l- + SA(I
-- R A / M ) ] +
(1 -p)[1 + Ss(1 -
(3) Rs/M)]
where SA = d + r ( p A N - 1), Ss =psrN,
RA=N[I +pA(d+ r ( N - 1))] Rs = N[ 1 +pS(d+ r ( N - 1))]
and 0 .2
pA=p+ -,
p
0- 2
ps=p+ - .
l-p
~r2 is the variance in gene frequency between trait groups, pa and pB have been termed subjective frequencies of occurrence by Wilson (1977). They represent the average frequency that each type interacts with A types. The expressions in brackets in eqn (3) (defined as WA and WB respectively) approximate the global fitness of each type. The A type will increase in frequency whenever Wa > Ws. Note the similarity between
450
J . K . KELLY
the global fitness for each type and the corresponding individual fitness expressions [eqn (1)]. SA and Ss are equivalent to sA~ and sn~ with the local gene frequency replaced by pA and pn. R~ and RB are given by the resource utilization of a single trait group, R;, with pi replaced by pA and ps. The accuracy of the approximate system [eqn (3)] is demonstrated in the Appendix. The strength of inter-group selection depends on the amount of inter-group variance. Wright's fixation constant 0 -2 F =
-
p(l --p)
is a useful index. Using this as a frequency-independent measure of the amount of inter-group variance, pA and pn are PA =P(I - F ) + F,
P s = P ( 1 - F).
The magnitude of F is determined by how trait groups are formed at the beginning of each generation. We shall consider two extreme cases. (1) Trait groups are formed by random sampling from the progeny of all parents. (2) Trait groups are composed of a sample of progeny from two specific parents. This is the haploid equivalent of full sibling trait groups. The exact equation for gene frequency change is cubic in p~ and thus depends on the third-order moments of these two sampling schemes. These are derived in the Appendix. The approximate system [eqn (3)] requires only secondorder moments, however. For random sampling (case 1), F = l / N . In sibling groups (case 2), F = ½+ ½N. When local density regulation is absent (M is very large), the condition for the increase of A types reduces to d > r( 1 - F N ) .
(4)
This result is presented in a different form in Wilson (1977). It is equivalent to d > 0 when groups are formed randomly and to d > r(l - N ) / 2 when groups are composed of siblings. In neither case is the group advantageous character disfavored by increasing N. This result is somewhat surprising because inter-group variance tends to decline with larger N as sampling error is reduced. The fitness of A types is unhindered (in the absense of local density regulation) because wAiis a linearly increasing function of group size. In the presence of local density regulation however, this is no longer true. Ri also increases with N. Local regulation will have a greater effect on A types than B types (RA > Rs). Because per capita production is increasingly inhibited as R; gets closer to M, larger trait groups will no longer favor group advantageous characters if local regulation is sufficiently severe.
Conditions for Polymorphism Local density regulation generates non-linear individual fitness functions. This allows the maintenance of both types in the population in some circumstances
KIN SELECTION
IN P O P U L A T I O N S
451
(Fig. 1). If a group advantageous character is favored when rare, it will increase in frequency. As the survivorship in trait groups increases so does consumption of resources (RA increases with p). Because within group fitness is independent of local regulation, selection within trait groups is: unhindered by an increase in the frequency
-
-
B
-
-
P FIG. I. The fitness of each type is plotted as a function of gene frequency. The verticle lines denotes the global frequency of A and the dotted lines and arrows describe the "frequency shift" that each type experiences due to the structuring of the population into trait groups (PA >P >Ps)- The magnitude of the frequency shift is proportional to the amount of inter-group variance. The circled points are the resulting global fitness of each type. Note that when the fitness functions are linear, (a), as they are in the absence of local regulation, the global fitness of one type is greater than the entire range of p. When fitness functions are non-linear, (b), the relative global fitness of the two types will change over p allowing the opportunity for polymorphism.
452
J.K.
KELLY
of A while inter-group selection is curtailed. The relative strength of conflicting selection at two levels is thus frequency dependent which can lead to a polymorphism. To determine the conditions under which a polymorphism will exist, we examine the stability of the fixation states. Assuming eqn (4) is satisfied and evaluating WA and WB at p = 0, A will increase in frequency when rare if M N
- - > 1 + F [ d + r ( N - 1)1.
(5)
Evaluating at p = 1, it can be shown that A will be fixed if
-->l+[d+r(N-l)] N
1+ r N F ( I - F ) d + r ( f N - 1) "
(6)
Numerical iterations of eqn (3) have shown that for parameter values satisfying eqn (5) but not eqn (6), the approximate system converges to a stable interior equilibrium. At binomial variance ( F = 1/N), there is a large region of parameter space in which polymorphism is predicted [Fig. 2(a)]. Very limited local regulation can maintain the selfish type in the population (M may be quite large). As expected, large trait group size favors the selfish type while high inter-group variance decreases the likelihood of its maintenance in the population [Fig. 2(b)]. It is quite unexpected however, that increased relatedness can also make protection of the altruistic type less likely [note the positive slope of the lower line if Fig. 2(b)]. When relatedness is high, all members of the rare type will be clustered into a few trait groups. Clustering inhibits the initial increase of B types because these individuals cannot "hide" in groups composed primarily of ,4 types, enjoying the benefits without any cost. Clustering of A types inhibits their initial increase when local regulation is severe. The few trait groups containing ,4 will be less fit than others in the population because the action of the group advantageous character results in overexploitation of local resources (RA > M). When overexploitation occurs, increasing the value of the direct fitness effect (SA) actually results in reduced fitness. These results indicate that local density regulation can have surprisingly signifiant effects on kin selection. Even in populations that are regulated primarily at a global scale (M/N large), the selfish type can be maintained in the population. Caution is necessary in interpreting the results presented above, however. The model is based on a rather specific system of density regulation in which some component of regulation occurs at precisely the scale of trait groups. In populations consisting of discrete local groups, such that the members of different trait groups do not interfere with one another directly, this seems a reasonable model. In other circumstances however, we should expect density regulation to occur at a scale independent of the particular fitness determining interactions that define the trait group. Such a model is presented elsewhere (Kelly, 1992, submitted to Theor. pop. Biol.) and indicates that the restriction of inter-group selection resulting from local density regulation tends to decline quite rapidly as the scale of regulation increases to include multiple trait groups. This model also suggests however, that regulation of local clusters of trait groups will significantly effect inter-group selection if a spatial structure exists in the distribution
KIN SELECTION
453
IN POPULATIONS
I0 (o)
,4 type fi
Polymor phism
B type fixed I 25
50
I0 (b)
,4 type fixed
B type fixed I 25
50
N
FIo. 2. The conditions under which a group advantageous character will evolve as a function of the severity of local density regulation and trait group size. Parameter values: d=0-01, r=0-05, N--10;
(a) F= I/N, (b) F='+.½N.
454
J . K . KELLY
of trait groups. Limited migration or localized mating will cause trait groups of similar composition to become bunched (Hamilton, 1964, 1975; Karlin, 1982). This "population viscosity" (sensu Hamilton, 1964, 1975) is generally believed to favor inter-group selection by increasing the average inter-group variance over the entire population. It is important to recognize, however, that if a population is regulated at some local scale, the variance between spatially proximal trait groups is more important than the total inter-group variance in determining gene frequency change. This is because the expected contribution of each locally regulated cluster (or "regulation group") to the next generation is equivalent. Thus, any differential group production will be restricted to between trait groups within a single regulation group. The strength of inter-group selection will be proportional to the average variance between trait groups within a regulation group. Local regulation significantly reduces the strength of inter-group selection because the local inter-trait group variance will be considerably less than the total inter-trait group variance in a "viscous" population. Both the models in Boyd (1982) and those presented here assume that changes in local density resulting from the expression of a group advantageous character have an immediate effect on the production of offspring. Regulatory response in many natural situations will be delayed however. Such a time lag can reduce the effect of local regulation on inter-group selection. Specifically, when response is delayed a generation, local variation in the severity of density regulation will be uncorrelated with local gene frequency unless progeny tend to live in the same habitats as their parents. For reasons of mathematical simplicity, the population structure assumed in deriving this model is characterized by a random mixing phase, this mixing will eliminate any correlation between group advantageous characters and the delayed regulatory responses they generate. In most real populations however, there is some limit on dispersal. As alluded to previously, this "population viscosity" is generally believed to favor inter-group selection by generating greater variance. An additional consequence of restricted migration, however, is an increased tendency for offspring to subsist on the same resource base as their parents. This effect will inhibit the evolution of group advantageous characters whenever the current "resource level" in a patch is determined by the density of its inhabitants in previous generations. If local density regulation is relatively insignificant, the advantage of increased relatedness will be more important than the disadvantage described above and population viscosity should favor intergroup selection. When local regulation is severe however, the "inheritance" of local resource scarcity by A types will overcome the advantage of increased relatedness. Indeed, under restrictive circumstances, increased relatedness can inhibit the initial increase of group advantageous characters even when the severity of local density regulation is constant [M is fixed, see Fig. 2(b)]. Clearly, the effect of population viscosity on kin selection is not as clear cut as it is generally perceived to be (see also Kelly, 1992; Wilson et al., 1992 submitted to Evol. Ecol.). In general, the magnitude of inter-group differences in productivity will depend on the spatial and temporal scale over which regulating effects manifest themselves and on the migration structure of the evolving species.
KIN SELECTION IN POPULATIONS
455
Determinants of the Spatial and Temporal Scales of Density Regulation The importance of spatial and temporal scale in describing the interactions between species has recently become a major focus of community ecology (Diamond & Case, 1986; Hastings, 1986; Roughgarden et aL, 1989). The spatial and temporal scale over which density regulating interactions occur is a critical factor in determining the diversity (Levin, 1974) and patterms of energy flow (Okubo, 1986) in communities, in addition to the evolutionary implications described here. As it pertains to kin selection, the spatial scale of density regulation describes the extent that the members of different trait groups draw on the same resource base or are regulated by common predators. Extensive overlap between trait groups indicates that the spatial scale is large. A diversity of ecological factors can influence the scale over which a population is regulated but the mobility and range of the pertinent species is likely to be a critical factor. Relatively immobile species, such as plants and sessile invertebrates, will often draw limiting resources only from nearby. Competitive effects will thus be quite localized and the scale of density regulation will be small (Horn, t979; Holsinger & Pacala, 1990). Conversely, highly mobile consumers draw resources from over a large spatial area and competitive effects are dispersed to a large number of conspecifics. This entails a high degree of overlap in resource use between the members of different trait groups. In populations limited by predation, the range of the predator is critical. If the predator is territorial, the degree to which individual prey share regulators will be smaller and the regulation of numbers more localized, than if pedators overlap in their ranges. The temporal scale over which a population is regulated depends to a considerable extent on the relative life span of the regulators and the regulated. In models describing food limited populations, it is often assumed that the dynamics of the resource occur over a substantially faster rate than the dynamics of the consumer (MacArthur, 1972). This dictates that the level of resource in any local patch will be determined largely by the current composition/density of consumers in that patch. These are the conditions under which we expect that local density regulation wilt have its most significant effects on kin selection. If consumer and resource dynamics occur over comensurate time scales, then any correlation between gene frequency and the severity of local density regulation will be reduced. In predation limited systems, the temporal scale of regulation will be determined by the rate at which the predator adjusts to changes in prey density within its range. Admittedly, these are idealized cases, but they serve to illustrate the importance of specific ecological features in determining the fitness of group advantageous characters.
Indirect Effects The model presented here assumes that the severity of local density regulation is a fixed feature of the environment (M is constant). Clearly however, organisms may exert some influence over the factors that regulated their numbers. This introduces the possibility of selection on acting on characters that generate fitness differences by specifically altering the way a local population is regulated. These traits, which
456
J . K . KELLY
represent a kind of indirect fitness effect, have been examined extensively by Wilson (1980, 1986). An example presented in Wilson (1986) illustrative. Consider a predator that can exploit a number of prey species. The abundance of each individual prey species is limited to a considerable extent by inter-specific competition. Optimal foraging theory predicts that the predator will evolve to maximize individual caloric intake. Such a predator will only shift from its optimal target when it is individually advantageous to do so. A gene that induces a carrier to expand its diet to include lower quality may reduce individual caloric intake but it will also have the indirect effect of reducing the competitive load on the primary resurce. Thus, a group advantage may result in the form of an absolute increase in the level of prey available to local consumers. If we were to describe this type of character with a model of the form presented here, inter-group selection would result from a correlation of local gene frequency and the severity of local regulation (M). A critical difference between these indirect group advantageous characters and the standard direct fitness interactions is that the two depend on the spatial and temporal scale of density regulation is precisely the opposite way. The indirect effect model requires that the population be regulated locally.
Phenotypic Plasticity Many of the factors determining the fitness of a group advantageous character can vary in space or time. Clearly, this will increase the likelihood for a stable polymorphism. A second possibility however is that selection may favor a plastic phenotype. I shall suggest two cases in which variable local resource availability may be favoring variable expression of group advantageous characters. Many arthopod species display considerable variation in the phenotypes determining sex allocation. Two common features are female biased sex ratios and sex ratio plasticity based on the quality of the local mating site (Charnov, 1982; Werren & Simbolotti, 1989). Two important theories have been developed in explaining these patterns. Local mate competition models (Hamilton, 1967; Alexander & Sherman, 1977; Bulmer & Taylor, 1980) predict the conditions under which a female biased sex ratio is optimal. The second important concept is represented by host quality models (a term derived from parasitoid studies in which a host organism is the mating site). Given how the fitness of each sex is affected by differences in resource availability, host quality models predict the direction sex ratio plasticity in response to variable quality of mating sites. Female biased sex ratio is a group advantageous character (Colwell, 1981 ; Wilson & Colweli, 1981). While the fitness of a local mating group is an increasing function of the number of females, it is individually advantageous to be the minority sex (in this case male). Local mate competition models thus represents a form of kin selection. The results of this paper suggest that the adaptive value of such a character may vary with the availability of resources within the mating habitat. Specifically, the model predicts that group advantageous traits will be favored when local resource levels are high. Indeed, numerous species seem to exhibit greater female bias in high quality mating sites (Charnov, 1982). This trend has the interesting implication that
KIN
SELECTION
IN
POPULATIONS
457
the parameter values of host quality models (the relative fitness of each sex as a function of host quality) may be partly determined by the resource dependent properties of kin selection (local mate competition). A link between these two concepts should prove useful to workers attempting to construct unified theories of sex allocation (Werren, 1984; Frank, 1986). Consider also the ongoing debate regarding the determinants of social tolerance in rodents (Chitty, 1967; Krebs & Myers, 1974; Charnov & Finerty, 1980; Cockburn, 1988; Kawata, 1990). Recent attention has focused on the contention that increased aggression is observed primarily at high population densities (Madison & McShea, 1987). Charnov & Finerty (1980) suggested that the relatedness of local groups changes with population density and that social tolerance is a plastic character that depends on kin structure. Their model predicts that greater aggression should be observed at high population densities due to the break up of kin associations. A number of the premises of the Charnov-Finerty hypothesis have been subject to empirical examination. Recent findings suggest that factors beside relatedness can affect the level of social tolerance between individuals (Madison & McShea, 1987; Kawata, 1990) and that no simple relation exists between relatedness and population density (Jannett, 1978; Madison et al., 1984; Kawata, 1990). The results presented here suggest that changes in relatedness may not be necessary to produce density-dependent shifts in altruistic behavior. If rodent behavior is regulated locally, it may be adaptive to be altruistic towards kin only at low densities. To my knowledge, no studies have been made simultaneously examining kinship, density and behavior in rodents, but some indirect evidence is suggestive. A number of laboratory studies (Brown, 1953; Clark, 1965; DeKock & Rohn, 1972) have demonstrated that populations founded by related individuals grow faster than comparable populations founded by unrelated individuals. This data is contradictory to the results of several field studies (Kawata, 1987; Boonstra & Hogg, 1988). It is possible that this incongruence can be attributed to differences in the way that laboratory and field populations are regulated. While laboratory animals are usually maintained in such a way that their reproduction is not limited by density-dependent controls (food is plentiful and predation is absent), animals in field studies are enclosed by fencing which maximizes crowding. Densities within enclosures in studies by both Kawata (1987) and Boonstra & Hogg (1988) exceeded normal. Local resource scarcity should thus be a more important factor in field than laboratory studies and the adaptive benefit of acting altruistically towards kin may be reduced. It could be that many rodent species are able to respond plastically to changes in resource availability because the adaptive value of altruism is resource dependent. Given the population fluctuations observed in many rodent species, the ability to respond to changes in resource availability from one generation to the next is clearly adaptive.
The author is indebted to Kent Holsinger for extensive revisions of early drafts of this paper and many helpful comments. The paper also benefitted from reviews by D. S. Wilson and an anonymous reviewer and from conversations with C. Tripler, S. Pacala, G. Hurtt, and M. Amba. The author is supported by an NSF graduate fellowship.
458
J.K.
KELLY
REFERENCES
ALEXANDER,R. D. & SHERMAN,P. W. (1977). Local mate competition and parental investment in social insects. Science 196, 494-500. BOONSTRA & HOGG (1988). Friends and strangers: a test of the Charnov-Finerty hypothesis. Oecologia 77, 95-100. BoYD, R. (1982). Density dependent mortality and the evolution of social behavior. Anita. Behav. 30, 972-982. BROWN, R. Z. (1953). Social behavior, reproduction, and changes in the house mouse. Ecol. Monographs 23, 000-000. BULMER, M. G. & TAYLOR, P. D. (1980). Sex ratio under the haystack model. J. theor. Biol. 86, 83-89. CHARNOV,E. L. (1982). The Theory of Sex Allocation. Princeton, N J: Princeton University Press. CHARNOV, E. L. & FINERTY (1980). Vole population cycles: a case for kin selection? Oecologia 45, I-2. CHrrrv, D. (1967). The natural selection of self-regulation behavior in natural populations. Proc. Ecol. Soc. ,,lust. 2, 51-78. CLARK, J. R. (1965). Influence of numbers on reproduction and survival in two experimental vole populations. Proc. R. Soc. Lond. 144, 68-85. COCKBURN, A. (1988). Social Behavior in Fluctuating Populations. New York, NY: Croom Helm. COLWELL, R. K. (1981). Group selection is implicated in the evolution of female biased sex ratios. Nature, Lond. 290, 401-404. DEKOCK, L. L. & ROHN, l. (1972). Intra-specific behavior during the upswing of groups of Norway lemmings, kept under seminatural conditions. Z. Tierpsychol 30, 405-415. DIAMOND, J. & CASE, T. (eds) (1986). Communi O, Ecology. New York, NY: Harper & Row. FELLER, W. (1950). An Introduction to Probability Theoo, and Its Applications, Vol. I. New York, NY: Wiley. FRANK, S. A. (1986). Hierarchical selection theory and sex ratios I. general solutions for structured populations. Theor. pop. Biol. 29, 312-342. HAMILTON, W. n. (1964). The genetical evolution of social behavior I. J. theol'. Biol. 7, 1-16. HAMILTON, W. n. (1967). Extraordinary sex ratios. Science 156, 477-488. HAMILTON, W. D. (1975). Innate social aptitudes of man, an approach from evolutionary genetics. In: Biosocial Anthropology (Fox, R., ed.) New York, NY: Wiley. HASTINGS, A. (ed.) (1986). Community Ecology, Lecture Notes in Biomathematics 77. Berlin: Springer Verlag. HOLSINGER, K. & PACALA, S. W. (1990). Multiple niche polymorphisms in plant populations. Am. Nat. 135, 301-309. HORN, H. S. (1979). Adaptation from the Perspectioes of Optimality in Topics in Plant Population Biology (Solbrig, O. T., Jain, S., Johnson, G. B. & Raven, P. H., eds) New York, NY: Columbia University Press. JANNEX'r, F. J. (1978). The density-dependent formation of extended maternal families of the montane vole. Behav. Ecol. Sociobiol. 3, 245-263. KARLtN, S. (1982). Classification of selection-migration structures and conditions for protected polymorphism. In: Eoohaionary Biology (Hecht, M. K., Wallace, B. & Prance, G. T., eds) New York, NY: Plenum Press. KAWATA (1987). The effect of kinship on spacing among female red-backed voles. Oecologia 72, ! 15-122. KAWATA (1990). Fluctuating populations and kin interaction in mammals. Tree 17-20. KELLY, J. K. (1992). Restricted migration and the evolution of altruism. Evolution, in press. KREaS, C. J. & MYERS, J. H. (1974). Population cycles in small mammals. Adv. Ecol. Res. 8, 267-299. LEVIN, S. A. (1974). Dispersion and population interactions. Am. Nat. 108, 207-228. MACARTHUR, R. H. (1972). Geographical Ecology. New York, NY: Harper & Row. MADISON, D. M., FITZGERALD, R. W. & MCSHEA, W. J. (1984). Dynamics of social nesting in overwintering meadow voles : possible consequences for population cycling. Behao. Ecol. Sociobiol. 15, 9-17. MADmON, D. M. & MCSXEA, W. J. (1987). Seasonal changes in reproductive tolerance, spacing, and social organization in meadow voles: a microtine model. Am. Zool. 27, 899-908. NUNNEY, L. (1985). Group selection, altruism, and structured deme models. Am. Nat. 126, 212-230. OKURO, A. (1986). Planktonic micro-communities in the sea: biofluid mechanical view. In: Community Ecology Lecture Notes in Biomathematics 77 (Hastings, A., ed.) pp. 13-24. Berlin: Springer Verlag. PRICE, G. R. (1970). Selection and Covarianee. Nature, Lond. 227, 520-521. ROt3ERTSON, A. (1966). A mathematical model of the culling process ion dairy cattle. Anita. Prod. 8, 95-108.
459
KIN S E L E C T I O N IN P O P U L A T I O N S
ROUGHGARDEN, J., LEVm, S. A. & MAY, R. M. (1989). Perspectivesin Ecological Theory. Princeton, NJ: Princeton University Press. WADE, M. J. (1980) Kin selection: its components. Science 210, 665-667. WADE, M. J. (1985). Soft selection, hard selection, kin selection, and group selection. Am. Nat. 125, 61-74. WERREN, J. H. (1984). A model for sex ratio selection in parasitoid wasps: local mate competition and host quality effects. Neth. J. Zool. 34, 81-96. WERREN, J. H. t~ SIMBOLO'I-I-I,G. (1989). Combined effects of host quality and local mate competition on sex allocation in Lariophagus distinguendus. Fool. Ecol. 3, 203-213. WTLSON, D. S. (1975). A theory of group selection. Proc. natn. Acad. Sci. 72, 142-146. WILSON, D. S. (1977). Structured demes and the evolution of group advantageous traits. Am. Nat. 111, 157-185. WILSON, D. S. (I 980). The Natural Selection of Populationsand Communities. Menlo Park, CA : Benjamin Cummings. W~LSON, D. S. (1986). Adaptive indirect effects. In: CommuniO, Ecology (Case, T., & Diamond, J., eds). New York, NY: Harper & Row. WILSON, D. S. (1990). Weak AIhuism, Strong Group Selection. Oikos 59, 135-140. WILSON, D. S. • COLVVELL,R. K. (1981). Evolution of sex ratio in structured demes. Evolution 35, 882-897.
APPENDIX T h e global fitness o f A types is a weighted average t a k e n over the entire p o p u l a t i o n
~,p,wAi=~p,[l+(d+t.(piN_l))(1
N(l+pi(d+r(N-l))))]M
which w h e n e x p a n d e d is e q u i v a l e n t to
Y, piwA,=p =p[
I+SA 1-
+r~(d+r(N-l))\
0.2-
+ e,].
In a similar w a y it can be s h o w n that
y.(l-p3w,,=(1-p)
l+S, l - ~ +,-~-(d+,'(N-1)) X
\(l-p) 2 l-p
- - 0 .2
= (l - p ) [ Ws+ E21 where SA, Ss, RA, Rs, WA and Ws are defined following eqn (3) and ~3 is the expected value o f ( p ; - p ) 3. The approximate system [eqn (3)] is attained by neglecting the terms subsumed by E~ and E2. Clearly, this will only be accurate when El and Ez are small relative to WA a n d Ws. T h e m a g n i t u d e o f these terms d e p e n d s o n the values o f 0.2 a n d tr 3 which, in t u r n , d e p e n d s o n the p o p u l a t i o n structure. It is thus necessary to d e t e r m i n e second a n d third o r d e r m o m e n t s for the two s a m p l i n g schemes considered.
460
J . K . KELLY
Case 1: Random Sampling of Progeny Let X be the number of A types in a trait group. The probability of a trait group receiving k A types, P[X= k], is given by the binomial distribution
P[X=k]=(Nk)Pk(l--p)N-k. The variance and higher moments of a distribution can be attained by evaluating the derivatives of its generating function, G(s), at 1 (Feller, 1950). The generating function of the binomial distribution is
G(s) = ~ P[X=k]s k. kffiO
The second and third central moments are E[(X - X') 21= G"(1 ) + G'( I ) - (G'(1)) 2 = p ( l - p ) N E[ (X-.,(") 31= G"'(1) + 311 - G'(1)I[G"(I) + G'(1 )l + 2G'(I)[(G'(1)) z - l] =Np(1 - p ) ( 1 - 2 p ) . From these, it is easily verified that
O.2 E[(X--~2] p(l --p) N2
N
o-3 E[(X-,,J()3] p(l -p)(1 -2p) '
N3
N2
The accuracy of the approximate system under random progeny sampling is determined by inserting o-2 and o-3 into El and E2 E, _ p ( l - p ) r [ d + r ( N M
1 ) ] ( N - 1),
p(1 - p ) r [ d + r ( N M
E2-
1 ) ] ( N - 1)
Because the two terms are equivalent, they will only effect the rate of gene frequency change and not its direction. Second, note that E~ and E2 contain the factor p(l - p ) which dictates that they disappear as p goes to 0 or 1. Thus, the approximate system accurately describes the direction of selection over the entire range of gene frequency and becomes exact at the boundaries.
Case 2: Haploid Sibling Groups The probability distribution for this scheme is
2
2
•
N
.
k
N-k
KIN SELECTION
IN P O P U L A T I O N S
461
wherej denotes the number of A types among the two parents. It can be shown that the generating function for of distribution is
J[1 and the resulting moments /l+ 1 ~ ° ' 2 = P ( I - P ) / 2 2-N)'
( 1 + 3_3_/"
cr3=p(1-p)(1-2P)\4 4N]
Unlike the previous case, the error terms attained for this scheme are only equivalent at intermediate allele frequency
E,=r--~[d+r(N-1)l p(l-p)
~-~-~ +(l-p) 4~r~ 4-N
N2
1
1
E2=r--~[d+r(N-1)][P2(412 1N)+P(1-P)(-4-N--~)]. This indicates that the boundary conditions defined by the approximate system [eqns (5) and (6)] are not exact for the haploid sibling case. Using Et and E2, the exact boundary conditions can be derived for this specific sampling scheme. For A to increase when rare M > I + [d+r(N-1)] N
IN+l+ 2N
r(N- 1) ]. 2N(2d+ r(N- 1)) I
For A to go to fixation
-->l+[d+r(N-1)] N
1+
r(N- 1) ]. 2(2d+ r(N- 1)) I
Comparing these equations with the approximate conditions reveals that polymorphism is less likely than predicted by eqns (5) and (6). The difference is very slight however and graphs of approximate and exact conditions are practically indistinguishable for realistic parameters values.
Kin Selection in Density Regulated Populations JOHN K. KELLY t
Department of Ecology and Evolutionary Biology, University of Connecticut, Storrs, CT 06269, U.S.A. (Received on 2 July 1991, Accepted on 13 February 1992) The process of kin selection has both intra- and inter-group components (Hamilton, 1975, in: Biosocial Anthropology Wade, 1980). Group advantageous characteristics can evolve when inter-group differences in fertility are sufficiently great to overcome any within-group disadvantage of the trait. The potential magnitude of inter-group differences in fertility is determined largely by the way a population is regulated. Inter-group differences decrease as the spatial scale over which a population is regulated becomes increasingly localized. The present paper extends previous work by Boyd (1982, Anita. Behav. 30, 972-982) on the quantitative relation between kin selection and density regulation. A simple genetic model is employed to examine the conditions under which the interaction of local density regulation and kin selection can maintain a stable polymorphism. The ecological factors determining the spatial and temporal scale of density regulation are discussed. Finally, the results are applied to two biological cases in which local density regulation may be influencing the direction of phenotypic plasticity in group advantageous characters.
Introduction
The theory of kin selection describes gene frequency change resulting from fitness determining interactions that occur within small clusters of individuals referred to as trait groups (Wilson, 1975) or neighborhoods (Nunney, 1985). These clusters of interacters can vary in their total production of offspring whenever there is genetic variance between them. The fitness of an allele is thus determined by both its effect on the individual carrying it and its effect on the other individuals in its trait group. Gene frequency change, Ap, in such a spatially structured population, can be decomposed into within and between group components (Robertson, 1966; Price, 1970) Ap = Ap, + ap~
where Apt is the average change in gene frequency due to selection within groups and Ape is the frequency change resulting from the differential production of trait groups. The important feature of this decomposition is that it isolates all components of gene frequency change that depend on the differential production of trait groups. The magnitude of inter-group differences in productivity is determined largely by the way a population is regulated. The purpose of this paper is to examine the effects
t Current address: Department of Ecology and Evolution, University of Chicago, Chicago, IL 60637, U.S.A. 447
0022-5193/92/160447 + 15 $08.00/0
© 1992 Academic Press Limited
448
J.K.
KELLY
that population dynamic factors will have on the pattern of offspring production over the range of a population and thus evaluate the importance of these ecological forces to kin selection.
Density Regulation The potential magnitude of inter-group differences in reproductive output is limited by the scale over which the population is regulated. Kin selection models generally assume that population regulation occurs globally (local hard selection) and has no effect on inter-group selection. Ecological studies suggest however that density regulating interactions often occur over relatively small scales (Levin, 1974). If a population is regulated at the scale of trait groups (local soft selection), all groups make equivalent contributions to the next generation and inter-group selection is prevented (Wilson & Colwell, 1981 ; Wade, 1985). Clearly, both scenarios are idealizations. Most natural populations are limited by multiple ecological constraints which may operate over different spatial scales. Second, the simple hard and soft selection models assume that the response of regulating elements to changes in the size of the focal population is immediate. While this may be realistic in some circumstances (MacArthur, 1972), the effects of increased density will often manifest themselves over longer temporal scales. In addition, the population may experience densitydependent mortality at multiple stages in the life cycle. Finally, density regulation may be linked to other factors such as migration that are also influencing kin selection (Kelly, 1992; Wilson et al., 1992, submitted to Evol. Ecol.). There are several ways to incorporate the effects of varying spatial or temporal scale of density regulation into kin selection. The approach to be used in this paper was introduced by Boyd (1982). He constructed a model that allows some fraction of density regulation to occur locally with a second stage of population reduction occurring globally. He demonstrates, using an ESS model, that optimal behavioral strategies will depend on the nature of local density-dependent mortality. Specifically, Boyd finds that altruism should be directed only towards individuals whose relatedness to the altruist exceeds a specific threshold value. This threshold value is determined by the relation between local density and group fitness. The present paper employs a simple genetic model to examine the evolutionary consequences of local density regulation for kin selection and specifically, the conditions for stable polymorphism. The Model Consider a population of haploid organisms with discrete generations. Assume that this population is subdivided into a very large number of local trait groups in which natural selection and local density regulation occur. At the end of each generation, all individuals congeal into a global mating pool. The progeny then resettle into distinct trait groups, each of size N. The population consists of two types, A and B, at frequencies p and q (q = 1 - p ) respectively. The A types perform a group advantageous behavior and increase the viability of all other members of their trait group
KIN
SELECTION
IN POPULATIONS
449
by an increment r (r > 0) and affect themselves by increment d. Thus, the net effect on the viability of each type prior to density regulation in the ith trait group is sAi=d+r(piN - l),
sB~=rpiN
where sA~ is the direct effect of group advantageous actions on A types, sn~ is the direct effect on B types, and p~ is the frequency of A in the ith trait group. These are the basic fitness functions in Wilson (I980). We shall treat density regulation with a "logistic" formulation, in which the magnitude of the direct effects (sA~and sin) are reduced by a density regulating factor. The absolute fitness of A and B types after density regulation in the ith trait group, wA~and wn~is given by the following functions wA~= l +s~(1 ---~),
w~;= 1 + s,~(1 -)!~).
(1)
M is a constant that determines the severity of local density regulation. Low values of M indicate severe local regulation. R~ is a measure of "resource consumption" in the ith trait group. The value of R~ is proportional to pre-regulatory survivorship in the ith trait group Ri = N[ 1 +pism + (l -p~)s~i]. Finally, the fitness functions [eqn (1)] include 1 as a basal fitness value (Wilson, 1990). Gene frequency change is given by p~w ~
p' -
(2)
~.piwa, + E (I -p,)wn,
where the summations indicate an average taken over all trait groups. The expression resulting from incorporating eqn (1) into eqn (2) is complex (see Appendix), but a tractable approximation is given by p[1 + SA(1 - RA/M)]
p'
~p[l- + SA(I
-- R A / M ) ] +
(1 -p)[1 + Ss(1 -
(3) Rs/M)]
where SA = d + r ( p A N - 1), Ss =psrN,
RA=N[I +pA(d+ r ( N - 1))] Rs = N[ 1 +pS(d+ r ( N - 1))]
and 0 .2
pA=p+ -,
p
0- 2
ps=p+ - .
l-p
~r2 is the variance in gene frequency between trait groups, pa and pB have been termed subjective frequencies of occurrence by Wilson (1977). They represent the average frequency that each type interacts with A types. The expressions in brackets in eqn (3) (defined as WA and WB respectively) approximate the global fitness of each type. The A type will increase in frequency whenever Wa > Ws. Note the similarity between
450
J . K . KELLY
the global fitness for each type and the corresponding individual fitness expressions [eqn (1)]. SA and Ss are equivalent to sA~ and sn~ with the local gene frequency replaced by pA and pn. R~ and RB are given by the resource utilization of a single trait group, R;, with pi replaced by pA and ps. The accuracy of the approximate system [eqn (3)] is demonstrated in the Appendix. The strength of inter-group selection depends on the amount of inter-group variance. Wright's fixation constant 0 -2 F =
-
p(l --p)
is a useful index. Using this as a frequency-independent measure of the amount of inter-group variance, pA and pn are PA =P(I - F ) + F,
P s = P ( 1 - F).
The magnitude of F is determined by how trait groups are formed at the beginning of each generation. We shall consider two extreme cases. (1) Trait groups are formed by random sampling from the progeny of all parents. (2) Trait groups are composed of a sample of progeny from two specific parents. This is the haploid equivalent of full sibling trait groups. The exact equation for gene frequency change is cubic in p~ and thus depends on the third-order moments of these two sampling schemes. These are derived in the Appendix. The approximate system [eqn (3)] requires only secondorder moments, however. For random sampling (case 1), F = l / N . In sibling groups (case 2), F = ½+ ½N. When local density regulation is absent (M is very large), the condition for the increase of A types reduces to d > r( 1 - F N ) .
(4)
This result is presented in a different form in Wilson (1977). It is equivalent to d > 0 when groups are formed randomly and to d > r(l - N ) / 2 when groups are composed of siblings. In neither case is the group advantageous character disfavored by increasing N. This result is somewhat surprising because inter-group variance tends to decline with larger N as sampling error is reduced. The fitness of A types is unhindered (in the absense of local density regulation) because wAiis a linearly increasing function of group size. In the presence of local density regulation however, this is no longer true. Ri also increases with N. Local regulation will have a greater effect on A types than B types (RA > Rs). Because per capita production is increasingly inhibited as R; gets closer to M, larger trait groups will no longer favor group advantageous characters if local regulation is sufficiently severe.
Conditions for Polymorphism Local density regulation generates non-linear individual fitness functions. This allows the maintenance of both types in the population in some circumstances
KIN SELECTION
IN P O P U L A T I O N S
451
(Fig. 1). If a group advantageous character is favored when rare, it will increase in frequency. As the survivorship in trait groups increases so does consumption of resources (RA increases with p). Because within group fitness is independent of local regulation, selection within trait groups is: unhindered by an increase in the frequency
-
-
B
-
-
P FIG. I. The fitness of each type is plotted as a function of gene frequency. The verticle lines denotes the global frequency of A and the dotted lines and arrows describe the "frequency shift" that each type experiences due to the structuring of the population into trait groups (PA >P >Ps)- The magnitude of the frequency shift is proportional to the amount of inter-group variance. The circled points are the resulting global fitness of each type. Note that when the fitness functions are linear, (a), as they are in the absence of local regulation, the global fitness of one type is greater than the entire range of p. When fitness functions are non-linear, (b), the relative global fitness of the two types will change over p allowing the opportunity for polymorphism.
452
J.K.
KELLY
of A while inter-group selection is curtailed. The relative strength of conflicting selection at two levels is thus frequency dependent which can lead to a polymorphism. To determine the conditions under which a polymorphism will exist, we examine the stability of the fixation states. Assuming eqn (4) is satisfied and evaluating WA and WB at p = 0, A will increase in frequency when rare if M N
- - > 1 + F [ d + r ( N - 1)1.
(5)
Evaluating at p = 1, it can be shown that A will be fixed if
-->l+[d+r(N-l)] N
1+ r N F ( I - F ) d + r ( f N - 1) "
(6)
Numerical iterations of eqn (3) have shown that for parameter values satisfying eqn (5) but not eqn (6), the approximate system converges to a stable interior equilibrium. At binomial variance ( F = 1/N), there is a large region of parameter space in which polymorphism is predicted [Fig. 2(a)]. Very limited local regulation can maintain the selfish type in the population (M may be quite large). As expected, large trait group size favors the selfish type while high inter-group variance decreases the likelihood of its maintenance in the population [Fig. 2(b)]. It is quite unexpected however, that increased relatedness can also make protection of the altruistic type less likely [note the positive slope of the lower line if Fig. 2(b)]. When relatedness is high, all members of the rare type will be clustered into a few trait groups. Clustering inhibits the initial increase of B types because these individuals cannot "hide" in groups composed primarily of ,4 types, enjoying the benefits without any cost. Clustering of A types inhibits their initial increase when local regulation is severe. The few trait groups containing ,4 will be less fit than others in the population because the action of the group advantageous character results in overexploitation of local resources (RA > M). When overexploitation occurs, increasing the value of the direct fitness effect (SA) actually results in reduced fitness. These results indicate that local density regulation can have surprisingly signifiant effects on kin selection. Even in populations that are regulated primarily at a global scale (M/N large), the selfish type can be maintained in the population. Caution is necessary in interpreting the results presented above, however. The model is based on a rather specific system of density regulation in which some component of regulation occurs at precisely the scale of trait groups. In populations consisting of discrete local groups, such that the members of different trait groups do not interfere with one another directly, this seems a reasonable model. In other circumstances however, we should expect density regulation to occur at a scale independent of the particular fitness determining interactions that define the trait group. Such a model is presented elsewhere (Kelly, 1992, submitted to Theor. pop. Biol.) and indicates that the restriction of inter-group selection resulting from local density regulation tends to decline quite rapidly as the scale of regulation increases to include multiple trait groups. This model also suggests however, that regulation of local clusters of trait groups will significantly effect inter-group selection if a spatial structure exists in the distribution
KIN SELECTION
453
IN POPULATIONS
I0 (o)
,4 type fi
Polymor phism
B type fixed I 25
50
I0 (b)
,4 type fixed
B type fixed I 25
50
N
FIo. 2. The conditions under which a group advantageous character will evolve as a function of the severity of local density regulation and trait group size. Parameter values: d=0-01, r=0-05, N--10;
(a) F= I/N, (b) F='+.½N.
454
J . K . KELLY
of trait groups. Limited migration or localized mating will cause trait groups of similar composition to become bunched (Hamilton, 1964, 1975; Karlin, 1982). This "population viscosity" (sensu Hamilton, 1964, 1975) is generally believed to favor inter-group selection by increasing the average inter-group variance over the entire population. It is important to recognize, however, that if a population is regulated at some local scale, the variance between spatially proximal trait groups is more important than the total inter-group variance in determining gene frequency change. This is because the expected contribution of each locally regulated cluster (or "regulation group") to the next generation is equivalent. Thus, any differential group production will be restricted to between trait groups within a single regulation group. The strength of inter-group selection will be proportional to the average variance between trait groups within a regulation group. Local regulation significantly reduces the strength of inter-group selection because the local inter-trait group variance will be considerably less than the total inter-trait group variance in a "viscous" population. Both the models in Boyd (1982) and those presented here assume that changes in local density resulting from the expression of a group advantageous character have an immediate effect on the production of offspring. Regulatory response in many natural situations will be delayed however. Such a time lag can reduce the effect of local regulation on inter-group selection. Specifically, when response is delayed a generation, local variation in the severity of density regulation will be uncorrelated with local gene frequency unless progeny tend to live in the same habitats as their parents. For reasons of mathematical simplicity, the population structure assumed in deriving this model is characterized by a random mixing phase, this mixing will eliminate any correlation between group advantageous characters and the delayed regulatory responses they generate. In most real populations however, there is some limit on dispersal. As alluded to previously, this "population viscosity" is generally believed to favor inter-group selection by generating greater variance. An additional consequence of restricted migration, however, is an increased tendency for offspring to subsist on the same resource base as their parents. This effect will inhibit the evolution of group advantageous characters whenever the current "resource level" in a patch is determined by the density of its inhabitants in previous generations. If local density regulation is relatively insignificant, the advantage of increased relatedness will be more important than the disadvantage described above and population viscosity should favor intergroup selection. When local regulation is severe however, the "inheritance" of local resource scarcity by A types will overcome the advantage of increased relatedness. Indeed, under restrictive circumstances, increased relatedness can inhibit the initial increase of group advantageous characters even when the severity of local density regulation is constant [M is fixed, see Fig. 2(b)]. Clearly, the effect of population viscosity on kin selection is not as clear cut as it is generally perceived to be (see also Kelly, 1992; Wilson et al., 1992 submitted to Evol. Ecol.). In general, the magnitude of inter-group differences in productivity will depend on the spatial and temporal scale over which regulating effects manifest themselves and on the migration structure of the evolving species.
KIN SELECTION IN POPULATIONS
455
Determinants of the Spatial and Temporal Scales of Density Regulation The importance of spatial and temporal scale in describing the interactions between species has recently become a major focus of community ecology (Diamond & Case, 1986; Hastings, 1986; Roughgarden et aL, 1989). The spatial and temporal scale over which density regulating interactions occur is a critical factor in determining the diversity (Levin, 1974) and patterms of energy flow (Okubo, 1986) in communities, in addition to the evolutionary implications described here. As it pertains to kin selection, the spatial scale of density regulation describes the extent that the members of different trait groups draw on the same resource base or are regulated by common predators. Extensive overlap between trait groups indicates that the spatial scale is large. A diversity of ecological factors can influence the scale over which a population is regulated but the mobility and range of the pertinent species is likely to be a critical factor. Relatively immobile species, such as plants and sessile invertebrates, will often draw limiting resources only from nearby. Competitive effects will thus be quite localized and the scale of density regulation will be small (Horn, t979; Holsinger & Pacala, 1990). Conversely, highly mobile consumers draw resources from over a large spatial area and competitive effects are dispersed to a large number of conspecifics. This entails a high degree of overlap in resource use between the members of different trait groups. In populations limited by predation, the range of the predator is critical. If the predator is territorial, the degree to which individual prey share regulators will be smaller and the regulation of numbers more localized, than if pedators overlap in their ranges. The temporal scale over which a population is regulated depends to a considerable extent on the relative life span of the regulators and the regulated. In models describing food limited populations, it is often assumed that the dynamics of the resource occur over a substantially faster rate than the dynamics of the consumer (MacArthur, 1972). This dictates that the level of resource in any local patch will be determined largely by the current composition/density of consumers in that patch. These are the conditions under which we expect that local density regulation wilt have its most significant effects on kin selection. If consumer and resource dynamics occur over comensurate time scales, then any correlation between gene frequency and the severity of local density regulation will be reduced. In predation limited systems, the temporal scale of regulation will be determined by the rate at which the predator adjusts to changes in prey density within its range. Admittedly, these are idealized cases, but they serve to illustrate the importance of specific ecological features in determining the fitness of group advantageous characters.
Indirect Effects The model presented here assumes that the severity of local density regulation is a fixed feature of the environment (M is constant). Clearly however, organisms may exert some influence over the factors that regulated their numbers. This introduces the possibility of selection on acting on characters that generate fitness differences by specifically altering the way a local population is regulated. These traits, which
456
J . K . KELLY
represent a kind of indirect fitness effect, have been examined extensively by Wilson (1980, 1986). An example presented in Wilson (1986) illustrative. Consider a predator that can exploit a number of prey species. The abundance of each individual prey species is limited to a considerable extent by inter-specific competition. Optimal foraging theory predicts that the predator will evolve to maximize individual caloric intake. Such a predator will only shift from its optimal target when it is individually advantageous to do so. A gene that induces a carrier to expand its diet to include lower quality may reduce individual caloric intake but it will also have the indirect effect of reducing the competitive load on the primary resurce. Thus, a group advantage may result in the form of an absolute increase in the level of prey available to local consumers. If we were to describe this type of character with a model of the form presented here, inter-group selection would result from a correlation of local gene frequency and the severity of local regulation (M). A critical difference between these indirect group advantageous characters and the standard direct fitness interactions is that the two depend on the spatial and temporal scale of density regulation is precisely the opposite way. The indirect effect model requires that the population be regulated locally.
Phenotypic Plasticity Many of the factors determining the fitness of a group advantageous character can vary in space or time. Clearly, this will increase the likelihood for a stable polymorphism. A second possibility however is that selection may favor a plastic phenotype. I shall suggest two cases in which variable local resource availability may be favoring variable expression of group advantageous characters. Many arthopod species display considerable variation in the phenotypes determining sex allocation. Two common features are female biased sex ratios and sex ratio plasticity based on the quality of the local mating site (Charnov, 1982; Werren & Simbolotti, 1989). Two important theories have been developed in explaining these patterns. Local mate competition models (Hamilton, 1967; Alexander & Sherman, 1977; Bulmer & Taylor, 1980) predict the conditions under which a female biased sex ratio is optimal. The second important concept is represented by host quality models (a term derived from parasitoid studies in which a host organism is the mating site). Given how the fitness of each sex is affected by differences in resource availability, host quality models predict the direction sex ratio plasticity in response to variable quality of mating sites. Female biased sex ratio is a group advantageous character (Colwell, 1981 ; Wilson & Colweli, 1981). While the fitness of a local mating group is an increasing function of the number of females, it is individually advantageous to be the minority sex (in this case male). Local mate competition models thus represents a form of kin selection. The results of this paper suggest that the adaptive value of such a character may vary with the availability of resources within the mating habitat. Specifically, the model predicts that group advantageous traits will be favored when local resource levels are high. Indeed, numerous species seem to exhibit greater female bias in high quality mating sites (Charnov, 1982). This trend has the interesting implication that
KIN
SELECTION
IN
POPULATIONS
457
the parameter values of host quality models (the relative fitness of each sex as a function of host quality) may be partly determined by the resource dependent properties of kin selection (local mate competition). A link between these two concepts should prove useful to workers attempting to construct unified theories of sex allocation (Werren, 1984; Frank, 1986). Consider also the ongoing debate regarding the determinants of social tolerance in rodents (Chitty, 1967; Krebs & Myers, 1974; Charnov & Finerty, 1980; Cockburn, 1988; Kawata, 1990). Recent attention has focused on the contention that increased aggression is observed primarily at high population densities (Madison & McShea, 1987). Charnov & Finerty (1980) suggested that the relatedness of local groups changes with population density and that social tolerance is a plastic character that depends on kin structure. Their model predicts that greater aggression should be observed at high population densities due to the break up of kin associations. A number of the premises of the Charnov-Finerty hypothesis have been subject to empirical examination. Recent findings suggest that factors beside relatedness can affect the level of social tolerance between individuals (Madison & McShea, 1987; Kawata, 1990) and that no simple relation exists between relatedness and population density (Jannett, 1978; Madison et al., 1984; Kawata, 1990). The results presented here suggest that changes in relatedness may not be necessary to produce density-dependent shifts in altruistic behavior. If rodent behavior is regulated locally, it may be adaptive to be altruistic towards kin only at low densities. To my knowledge, no studies have been made simultaneously examining kinship, density and behavior in rodents, but some indirect evidence is suggestive. A number of laboratory studies (Brown, 1953; Clark, 1965; DeKock & Rohn, 1972) have demonstrated that populations founded by related individuals grow faster than comparable populations founded by unrelated individuals. This data is contradictory to the results of several field studies (Kawata, 1987; Boonstra & Hogg, 1988). It is possible that this incongruence can be attributed to differences in the way that laboratory and field populations are regulated. While laboratory animals are usually maintained in such a way that their reproduction is not limited by density-dependent controls (food is plentiful and predation is absent), animals in field studies are enclosed by fencing which maximizes crowding. Densities within enclosures in studies by both Kawata (1987) and Boonstra & Hogg (1988) exceeded normal. Local resource scarcity should thus be a more important factor in field than laboratory studies and the adaptive benefit of acting altruistically towards kin may be reduced. It could be that many rodent species are able to respond plastically to changes in resource availability because the adaptive value of altruism is resource dependent. Given the population fluctuations observed in many rodent species, the ability to respond to changes in resource availability from one generation to the next is clearly adaptive.
The author is indebted to Kent Holsinger for extensive revisions of early drafts of this paper and many helpful comments. The paper also benefitted from reviews by D. S. Wilson and an anonymous reviewer and from conversations with C. Tripler, S. Pacala, G. Hurtt, and M. Amba. The author is supported by an NSF graduate fellowship.
458
J.K.
KELLY
REFERENCES
ALEXANDER,R. D. & SHERMAN,P. W. (1977). Local mate competition and parental investment in social insects. Science 196, 494-500. BOONSTRA & HOGG (1988). Friends and strangers: a test of the Charnov-Finerty hypothesis. Oecologia 77, 95-100. BoYD, R. (1982). Density dependent mortality and the evolution of social behavior. Anita. Behav. 30, 972-982. BROWN, R. Z. (1953). Social behavior, reproduction, and changes in the house mouse. Ecol. Monographs 23, 000-000. BULMER, M. G. & TAYLOR, P. D. (1980). Sex ratio under the haystack model. J. theor. Biol. 86, 83-89. CHARNOV,E. L. (1982). The Theory of Sex Allocation. Princeton, N J: Princeton University Press. CHARNOV, E. L. & FINERTY (1980). Vole population cycles: a case for kin selection? Oecologia 45, I-2. CHrrrv, D. (1967). The natural selection of self-regulation behavior in natural populations. Proc. Ecol. Soc. ,,lust. 2, 51-78. CLARK, J. R. (1965). Influence of numbers on reproduction and survival in two experimental vole populations. Proc. R. Soc. Lond. 144, 68-85. COCKBURN, A. (1988). Social Behavior in Fluctuating Populations. New York, NY: Croom Helm. COLWELL, R. K. (1981). Group selection is implicated in the evolution of female biased sex ratios. Nature, Lond. 290, 401-404. DEKOCK, L. L. & ROHN, l. (1972). Intra-specific behavior during the upswing of groups of Norway lemmings, kept under seminatural conditions. Z. Tierpsychol 30, 405-415. DIAMOND, J. & CASE, T. (eds) (1986). Communi O, Ecology. New York, NY: Harper & Row. FELLER, W. (1950). An Introduction to Probability Theoo, and Its Applications, Vol. I. New York, NY: Wiley. FRANK, S. A. (1986). Hierarchical selection theory and sex ratios I. general solutions for structured populations. Theor. pop. Biol. 29, 312-342. HAMILTON, W. n. (1964). The genetical evolution of social behavior I. J. theol'. Biol. 7, 1-16. HAMILTON, W. n. (1967). Extraordinary sex ratios. Science 156, 477-488. HAMILTON, W. D. (1975). Innate social aptitudes of man, an approach from evolutionary genetics. In: Biosocial Anthropology (Fox, R., ed.) New York, NY: Wiley. HASTINGS, A. (ed.) (1986). Community Ecology, Lecture Notes in Biomathematics 77. Berlin: Springer Verlag. HOLSINGER, K. & PACALA, S. W. (1990). Multiple niche polymorphisms in plant populations. Am. Nat. 135, 301-309. HORN, H. S. (1979). Adaptation from the Perspectioes of Optimality in Topics in Plant Population Biology (Solbrig, O. T., Jain, S., Johnson, G. B. & Raven, P. H., eds) New York, NY: Columbia University Press. JANNEX'r, F. J. (1978). The density-dependent formation of extended maternal families of the montane vole. Behav. Ecol. Sociobiol. 3, 245-263. KARLtN, S. (1982). Classification of selection-migration structures and conditions for protected polymorphism. In: Eoohaionary Biology (Hecht, M. K., Wallace, B. & Prance, G. T., eds) New York, NY: Plenum Press. KAWATA (1987). The effect of kinship on spacing among female red-backed voles. Oecologia 72, ! 15-122. KAWATA (1990). Fluctuating populations and kin interaction in mammals. Tree 17-20. KELLY, J. K. (1992). Restricted migration and the evolution of altruism. Evolution, in press. KREaS, C. J. & MYERS, J. H. (1974). Population cycles in small mammals. Adv. Ecol. Res. 8, 267-299. LEVIN, S. A. (1974). Dispersion and population interactions. Am. Nat. 108, 207-228. MACARTHUR, R. H. (1972). Geographical Ecology. New York, NY: Harper & Row. MADISON, D. M., FITZGERALD, R. W. & MCSHEA, W. J. (1984). Dynamics of social nesting in overwintering meadow voles : possible consequences for population cycling. Behao. Ecol. Sociobiol. 15, 9-17. MADmON, D. M. & MCSXEA, W. J. (1987). Seasonal changes in reproductive tolerance, spacing, and social organization in meadow voles: a microtine model. Am. Zool. 27, 899-908. NUNNEY, L. (1985). Group selection, altruism, and structured deme models. Am. Nat. 126, 212-230. OKURO, A. (1986). Planktonic micro-communities in the sea: biofluid mechanical view. In: Community Ecology Lecture Notes in Biomathematics 77 (Hastings, A., ed.) pp. 13-24. Berlin: Springer Verlag. PRICE, G. R. (1970). Selection and Covarianee. Nature, Lond. 227, 520-521. ROt3ERTSON, A. (1966). A mathematical model of the culling process ion dairy cattle. Anita. Prod. 8, 95-108.
459
KIN S E L E C T I O N IN P O P U L A T I O N S
ROUGHGARDEN, J., LEVm, S. A. & MAY, R. M. (1989). Perspectivesin Ecological Theory. Princeton, NJ: Princeton University Press. WADE, M. J. (1980) Kin selection: its components. Science 210, 665-667. WADE, M. J. (1985). Soft selection, hard selection, kin selection, and group selection. Am. Nat. 125, 61-74. WERREN, J. H. (1984). A model for sex ratio selection in parasitoid wasps: local mate competition and host quality effects. Neth. J. Zool. 34, 81-96. WERREN, J. H. t~ SIMBOLO'I-I-I,G. (1989). Combined effects of host quality and local mate competition on sex allocation in Lariophagus distinguendus. Fool. Ecol. 3, 203-213. WTLSON, D. S. (1975). A theory of group selection. Proc. natn. Acad. Sci. 72, 142-146. WILSON, D. S. (1977). Structured demes and the evolution of group advantageous traits. Am. Nat. 111, 157-185. WILSON, D. S. (I 980). The Natural Selection of Populationsand Communities. Menlo Park, CA : Benjamin Cummings. W~LSON, D. S. (1986). Adaptive indirect effects. In: CommuniO, Ecology (Case, T., & Diamond, J., eds). New York, NY: Harper & Row. WILSON, D. S. (1990). Weak AIhuism, Strong Group Selection. Oikos 59, 135-140. WILSON, D. S. • COLVVELL,R. K. (1981). Evolution of sex ratio in structured demes. Evolution 35, 882-897.
APPENDIX T h e global fitness o f A types is a weighted average t a k e n over the entire p o p u l a t i o n
~,p,wAi=~p,[l+(d+t.(piN_l))(1
N(l+pi(d+r(N-l))))]M
which w h e n e x p a n d e d is e q u i v a l e n t to
Y, piwA,=p =p[
I+SA 1-
+r~(d+r(N-l))\
0.2-
+ e,].
In a similar w a y it can be s h o w n that
y.(l-p3w,,=(1-p)
l+S, l - ~ +,-~-(d+,'(N-1)) X
\(l-p) 2 l-p
- - 0 .2
= (l - p ) [ Ws+ E21 where SA, Ss, RA, Rs, WA and Ws are defined following eqn (3) and ~3 is the expected value o f ( p ; - p ) 3. The approximate system [eqn (3)] is attained by neglecting the terms subsumed by E~ and E2. Clearly, this will only be accurate when El and Ez are small relative to WA a n d Ws. T h e m a g n i t u d e o f these terms d e p e n d s o n the values o f 0.2 a n d tr 3 which, in t u r n , d e p e n d s o n the p o p u l a t i o n structure. It is thus necessary to d e t e r m i n e second a n d third o r d e r m o m e n t s for the two s a m p l i n g schemes considered.
460
J . K . KELLY
Case 1: Random Sampling of Progeny Let X be the number of A types in a trait group. The probability of a trait group receiving k A types, P[X= k], is given by the binomial distribution
P[X=k]=(Nk)Pk(l--p)N-k. The variance and higher moments of a distribution can be attained by evaluating the derivatives of its generating function, G(s), at 1 (Feller, 1950). The generating function of the binomial distribution is
G(s) = ~ P[X=k]s k. kffiO
The second and third central moments are E[(X - X') 21= G"(1 ) + G'( I ) - (G'(1)) 2 = p ( l - p ) N E[ (X-.,(") 31= G"'(1) + 311 - G'(1)I[G"(I) + G'(1 )l + 2G'(I)[(G'(1)) z - l] =Np(1 - p ) ( 1 - 2 p ) . From these, it is easily verified that
O.2 E[(X--~2] p(l --p) N2
N
o-3 E[(X-,,J()3] p(l -p)(1 -2p) '
N3
N2
The accuracy of the approximate system under random progeny sampling is determined by inserting o-2 and o-3 into El and E2 E, _ p ( l - p ) r [ d + r ( N M
1 ) ] ( N - 1),
p(1 - p ) r [ d + r ( N M
E2-
1 ) ] ( N - 1)
Because the two terms are equivalent, they will only effect the rate of gene frequency change and not its direction. Second, note that E~ and E2 contain the factor p(l - p ) which dictates that they disappear as p goes to 0 or 1. Thus, the approximate system accurately describes the direction of selection over the entire range of gene frequency and becomes exact at the boundaries.
Case 2: Haploid Sibling Groups The probability distribution for this scheme is
2
2
•
N
.
k
N-k
KIN SELECTION
IN P O P U L A T I O N S
461
wherej denotes the number of A types among the two parents. It can be shown that the generating function for of distribution is
J[1 and the resulting moments /l+ 1 ~ ° ' 2 = P ( I - P ) / 2 2-N)'
( 1 + 3_3_/"
cr3=p(1-p)(1-2P)\4 4N]
Unlike the previous case, the error terms attained for this scheme are only equivalent at intermediate allele frequency
E,=r--~[d+r(N-1)l p(l-p)
~-~-~ +(l-p) 4~r~ 4-N
N2
1
1
E2=r--~[d+r(N-1)][P2(412 1N)+P(1-P)(-4-N--~)]. This indicates that the boundary conditions defined by the approximate system [eqns (5) and (6)] are not exact for the haploid sibling case. Using Et and E2, the exact boundary conditions can be derived for this specific sampling scheme. For A to increase when rare M > I + [d+r(N-1)] N
IN+l+ 2N
r(N- 1) ]. 2N(2d+ r(N- 1)) I
For A to go to fixation
-->l+[d+r(N-1)] N
1+
r(N- 1) ]. 2(2d+ r(N- 1)) I
Comparing these equations with the approximate conditions reveals that polymorphism is less likely than predicted by eqns (5) and (6). The difference is very slight however and graphs of approximate and exact conditions are practically indistinguishable for realistic parameters values.
