Reprinted from

Proc. NatI. Acad. Sci. USAVol. 75, No. 4, pp. 1909-1913, April 1978Genetics

Mathematical theory of group selection: Structure of group selectionin founder populations determined from convexityof the extinction operator*

(founder effect/extinction/population genetics)

SCOTT A. BOORMANDepartment of Sociology and Cowles Foundation for Research in Economics, Yale University, New Haven, Connecticut 06520

Communicated by Nathan Keyfitz, December 23, 1977

ABSTRACT Genetic analysis for group selection is devel-oped for the case of a biallelic locus (A,a) undergoing groupselection of founder populations only. By contrast to R. Levins'E = E(x) models, extinction now depends on genetics at thepropagule stage but acts uniformly on larger populations. Bio-logical evidence supports this hypothesis, which also allowsmathematical treatment at once simpler and biologically moregeneral than the Fokker-Planck partial differential equationformalism adopted by Levins. It is presently possible to handlecytogenetics of both diploid and haplodiploid type. The modelis set up as a quasideterministic recursion in the 5-simplex 25,collapsing both drift and mendelian selection effects into asingle parameter u, which is a Fisher-Kimura-Ohta fixationprobability. In the analysis, it is shown that the stability of thefixed points is determined by the convexity of the extinctionoperator acting on propagules, assumed to be of size 2.

Thus,polymorphism exists * unique and

E2 5< (El + E3)/2 stable, fixations unstablepolymorphism exists * unique and

unstable, fixations stable,in which El and E3 are extinction probabilities for phenotypi-cally uniform founder populations and E2 is the correspondingprobability for founder populations of mixed phenotpe. Furtherparameter regions are defined where fixation of e groupse-ected gene is globally stable, and this is still possible even whenextinction pressure acting on carrying capacity populationsbecomes weak relative to a fixed mendelian selectionstrength.

In the mathematical literature on group selection following themajor paper of Levins (1), group selection has come to be in-terpreted as differential extinction in a metapopulation (2-5).Extinction is modeled by postulating an operator E = E(x) re-lating the rate (or probability) of extinction to the frequencyx of the group-selected gene; in the formalism of Levins, thisquantity appears as a new dissipative term in the classical PDE(partial differential equation). Models developed in this wayneglect any further connection between E(x) and populationsize; it is generally implicit in the models that all demes arealready at carrying capacity at the time of colonization (see thehandling of the recolonization terms in ref. 1). Size variationis thus treated as a Boolean (dichotomous) variable (N = 0 orN = K), and all selection is K-selection in the sense of ecologicaltheory.A discrepant picture emerges from the empirical and phe-

nomenological literature. Ever since the island biogeographytheories of MacArthur and Wilson (6, 7) and the complemen-tary experimental studies of Simberloff and other investigators

The costs of publication of this article were defrayed in part by thepayment of page charges. This article must therefore be hereby marked"advertiement" in accordance with 18 U. S. C. §1734 solely to indicatethis fact.

1909

(8-10), there has been growing acceptance that the extinctionof demes is commonplace in the population biology of manyspecies. The group selection potential of these extinctions hasalso come to be recognized. However, there remains a majordisparity between the ascribed sources of extinction and the wayin which E(x) enters the mathematical formulations. With fewexceptions, the growing empirical evidence points to concen-tration of extinction events in extremely tiny populations-usually under a dozen organisms.t When dealing with numbersthis small, it is obvious that deme size and deme extinctioncannot be successfully decoupled. This is recognized in theoriginal Lewontin-Dunn Monte Carlo study of selection dy-namics at the T (Brachy, short tail) locus in Mus musculus (12).The point is also brought out in recent contributions of Goel andRichter-Dyn, which directly infer the chance of extinction fromstochastic birth and death rates, and show that there is a criticaldeme size Nit above which there is a vanishing probability thatextinction will result from these forces alone (13).

This paper seeks to bring the mathematical theory of groupselection into closer line with biological premises of the Mac-Arthur-Lewontin-Goel type. A fresh mathematical approachis developed which is based on permitting extinction rates tovary with deme size as well as with genetic composition ofdemes. It is now assumed that group selection acts only ondemes at the propagule stage; extinction does not reappear untilcarrying capacity has been reached, and this second extinction("K-extinction") occurs at a uniform rate that is genetics-in-dependent. On these assumptions, it is possible to avoid PDEapproximations, whose justification is always mathematicallydelicate even in the classical theory (14, 15), and to obtain anaggregative dynamical equation which is a quasideterministicrecursion in 25 (see Eq. 1). Derivation of this equation relies ona preliminary parameter aggregation involving the collapsingof drift as well as mendelian selection and dominance effectsinto a single parameter u. §

Specializing in terms descriptive of altruism, Eq. 1 describesa model in which deme altruism is favored by group selection,but only in populations that are newly founded. These are the

Abbreviation: PDE, partial differential equation.* This is paper no. 2 in a'series. Paper no. 1 is ref. 28.t Paradoxically, in view of the behavior genetics orientation of groupselection theory, the kind of situation most accurately describedthrough the E(x) formalism may be group selection of endoparasiticmicroorganisms, where extinction of large populations can be ef-fected by the death of the host. [Compare the documented case ofthe evolution toward avirulence of certain myxoma virus strainsoriginally lethal to the Australian rabbit Oryctolagus cuniculus (ref.11).]

§ This kind of hierarchical parameter specification (see Fig. 1) issomewhat novel in the mathematical genetics of one-locus systemsbut has long been used in mathematical psychology, e.g., ref. 16.

Proc. Natl. Acad. Sci. USA 75 (1978)

so-called "pioneer" traits whose evolutionary importance hasbeen proposed by Gadgil (17; see also ref. 18). However, analysisof the mathematical structure of selection for such traits in aclosed metapopulation is new.

Derivation of the modelPostulate a Levins metapopulation consisting of isolated com-ponent demes and consider the biallelic locus (A,a) where a isfavored by group selection but opposed by mendelian selection(19). It will be assumed that there are just two phenotypes: onephenotype A borne by all aa's and a second N borne by allAA's. For definiteness, refer to these phenotypes as altruist andnonaltruist, respectively, even though this interpretation is notthe only one possible (see Discussion).Time in the model is partitioned into discrete and nonover-

lapping periods. These periods are defined with reference tothe time required for a propagule to reach gene fixation atcarrying capacity (see Assumptions A2-A3) and will thereforealways extend over more than one generation. Within eachperiod, the following sequence of events occurs: (i) extinction,(ii) growth, and (iii) recolonization. At the outset of the period,all sites in the metapopulation are occupied (though not nec-essarily at carrying capacity). Immediately thereafter, extinc-tion wipes out certain demes. Surviving populations not yet atcarrying capacity then grow to carrying capacity, concurrentlybecoming homozygous at the locus for one or the other gene.Finally, recolonization occurs, with colonists being drawn frompopulations that are at carrying capacity at the end of the pe-riod. This completes the period; sites freshly colonized at theend of the period initially contain-only founder populations ofsize 2 (see Assumption Al), so that survival for an additionalperiod is required if these populations are to reach carryingcapacity.

This phenomenology generally follows Levins, with thecrucial exception that there is presently a fundamental dis-tinction between propagules and populations at carrying ca-pacity.The assumptions are now stated formally:Assumption Al. All propagules (founder populations) are

of size 2, comprising one male and one female. There are thusthree propagule phenotype classes: (1) N/N, (2) NN/A, and(3) A/A.

Assumption A2. Growth of newly founded populations israpid, so that if such a population escapes extinction at theoutset of its history, it requires only a single period to reachcarrying capacity (N = K).

Assumption A3. Individual selection pressures (includingdrift) are strong, so that any population reaching carryingcapacity in a period will have lost heterozygosity at the (A,a)locus by the end of this period (the mechanics are developedin A6).

Assumption A4. (i) A genetics-dependent extinctionprobability Ei > 0 (i = 1,2,3) acts on sites occupied byfounderpopulations at the beginning of a period. If emptied by ex-tinction, the site remains vacant for the remainder of theperiod and is recolonized at the end of the period by a pro-pagule. Propagules are constituted at random from the genepool of carrying capacity populations at the time of recolon-izing. (ii) Group selection favors the a gene, i.e., E3 < E2 <E,.

Assumption A5. A uniform extinction probability E >0 actson carrying capacity populations at the beginning of a period.In the event of extinction, recolonization takes place as inA4.For Assumption Al to hold, demes must be sufficiently iso-

lated to make multiple recolonizations rare (see ref. 2 for dis-cussion of observed recolonization dynamics in arthropodspecies). By A2, a founder population escaping extinction inperiod n arrives at carrying capacity by the outset of periodn + 1. From Assumption A3, a carrying capacity populationthatescapes extinction remains unchanged in both size and genotypecomposition in the next period. Observe that both AssumptionsA2 and A3 can be made definitionally correct by making eachperiod long enough (see ref. 20 for related time-scale estimates).However, such an assumption may create difficulties in thehypothesis that extinction acts only once in each period andnever hits populations in intermediate growth stages.¶

Given Assumptions Al-A5, assume that at the outset of theinitial period all demes fall into one of the following five classes:I, N/VN propagules; II, N/A propagules; III, A/lA propag-ules; IV, carrying capacity demes, homozygous for A; and V,carrying capacity demes, homozygous for a. Observe that thisclassification appears to be purely phenotypic. Nevertheless,it is not hard to show that classifications I-V in fact uniquelydetermine the genotype composition of each deme in each class.Moreover, the classification will continue to be exhaustive atall subsequent time periods, always taking observations at theoutset of each period before extinction acts.To see this, use Assumption A3 to note that all heterozygosity

in propagules will have been eliminated by the time thesepopulations are again censused at carrying capacity. Thus, allcarrying capacity populations are of either type IV or type V.From the way in which propagules are constituted (using As-sumptions A4 and A5), it follows that there can be no Aa indi-viduals in founder populations. Thus classes I-III are in factdeterminative of propagule genotypes as well as phenotypes.These classes are exhaustive of the possible propagules, usingAssumption Al. Finally, by Assumption A2 together with As-sumption A4 and AS, all populations as of the time of censusare either propagules or at carrying capacity, whence the ex-haustiveness of the above classification is established.

It remains to relate genetic composition at carrying capacityto that at the founder stage:

Assumption A6. Homozygous propagules of type I grow tocarrying capacity in class IV; similarly for III and V. Mixedpropagules of type II grow to class V with probability u andto class IV with probability (1 - u).The first half of this assumption is tantamount to neglecting

mutation, which is not a serious loss of generality. The secondpart introduces a new parameter u. This parameter is alreadyfamiliar in classical theory as the fixation probability of an allelestarting in a fixed frequency class, which in the present instanceis 1/2 for class II (AA/aa) propagules. In view of the small pop-ulation numbers during early growth stages, one would expectu to be influenced by drift and other small population effectsas well as by mendelian selection; drift is handled in the classicaltheory for a population of fixed size by solving the steady-stateequation Ltt = 0, in which Lt is adjoint to the Fokker-Planckoperator L in (t -L)k = 0 describing mendelian selectionwith drift (23). For present purposes, it is only necessary to notethat in the present model it is u, not the classical selection pa-rameters s and h, which constitutes the natural parametrizationof genetics at the individual level (see Fig. 1). For measurementand testing, there is the important further implication thatunraveling the full details of genetics at the (A,a) locus is un-necessary; only u requires estimation, and this can be done in

¶ It would be possible to try to generalize the formalism so that allprocesses take place continuously in real time, by analogy to em-bedding in Markov renewal processes (21), but the gain in realismis doubtful. Compare remarks in ref. 22, p. 183.

1910 Genetics: Boorman

Proc. Natl. Acad. Sci. USA 75 (1978) 1911

/~~ ~:1 <->$-;.

(El pE21E3) E u

s h (rK)FIG. 1. Parameter hierarchy in Eq. 1.8 = individual selection

strength, h = dominance, r = rate of increase, K = carrying capa-city.

principle at a population level by keeping track of successfulpropagules and observing their equilibrium composition aftermany generations.

Outside of the determination of u, the mendelian dominanceat the locus (h) is immaterial in the rest of the model; this followsfrom the fact that heterozygotes Aa are not represented in anyof the classes I-V. Because of the way in which heterozygosityis compressed into u, the same formalism also handles haplo-diploidy and other types of nondiploid chromosome systems.However, Assumption AS requires that mixed propagules mustgo to fixation of one gene or the other; in the absence of muta-tion, whose effect must be excluded if carrying capacity pop-ulations are to be treated as homozygous, the simplest way ofguaranteeing this is to limit the model to bisexual species, thusexcluding cloning as well as thelytoky and other exotic cyto-genetics which occur occasionally in the parasitic Hymenoptera(24, 25). This means that we have a formalism capable of han-dling group selection in the Hymenoptera and other haplo-diploid groups, but not clonal colony development in Bryozoa,Siphonophora, tunicates, and other marine colonial inverte-brates (26).

The ntities, p and q, respectively, express the A and afrequencies among those populations that are at carrying ca-pacity at the end of period n, when recolonization takesplace.

There are two fixed points of Eqs. 1-3 corresponding to genefixation. These are, respectively,

1yA - (E,0,0,1 - E1,0) * [A fixation] [5]

1+E-Ela =

IE (0,0,E0,1 - E3) [a fixation]. [6]1+E-E3

To analyze additional (polymorphism) fixed points, sety(n + 1) = 7(n) = y in Eq. 1 and write explicitly:

Tp2 = 7'

2rpq = 72

Tq2 = 73

[7a][7b][7c]

(1 - E1)'y1 + (1 - E2)(1 - U)Y2 + (1 - E)Y4 = 7o4 [7d]

(1-E3)Y3 + (1 -E2)uY2 + (1 -E)y5 = y5 [7e]

with

,sE171 + E22 +E373+ E(74 + 75) [8]and p and q defined from Eq. 3. Observe that each of Eqs.7a-7c is nonlinear of third order when considered indepen-dently.

First define x = q/p and notice that 72/71 = 2x, from Eqs.7a and 7b, while 72/Y3 = 2/x, from Eqs. 7b and 7c. Thus,

72 2xyl [9]73 = X271. [10]

Substituting Eqs. 9 and 10 in Eqs. 7d and 7e and simpli-fying,

Analysis: Fixed points and stabilityIndex the classes I-V by the running subscript i = 1,2,.. .,5.Censusing as of the outset of each time period, denote the fre-quency of the ith class by 7t, bat = 1. The mathematicalstructure is now described by the following recursion in 5- e IR51Jy, > 0, 2;'y, = I}

7(n + 1) =7(n)M, [1]7(n) = (7y(n),Y2(n),Y3(n),-Y4(n),Y5(n))

M = M[7y(n)] =[ Elp2 2Eipq Elq2 1-E1 0 1E2p2 2E2pq E2q2 (1 -E2)(1 - u) (1 -E2)uE3p2 2E3pq E3q2 0 1 -E3 [2]Ep2 2Epq Eq2 1-E 0Ep2 2Epq Eq2 0 .1-E

p= 1 -q= [3]

yi(1 - E1) + 72(1- E2)(1 - u) + 74(1- E)yl(1- E1) + 72(1 - E2) + 73(1 - E3) + (74 + 75)(1 - E)

and the initial conditions (I.C.'s) are

70 = (Y1(O),Y2(0),73(0),Y4(0),Y5(0)). [4]

These dynamics have the form of a nonlinear Markov chain.Observe that no renormalization of Eq. 1 is necessary, sinceMcan be directly interpreted as a matrix of transition probabilities.

E7y5 = [(1 - E3)X2 + 2u(1 - E2)x]YlE74 = [(1 - E) + 2(1 - E2)(1 -u)x]71,

[11]

[12]

while from Eq. 3 together with the definition of x,

x[(l - E1) + 2x(1- E2)(1 -u) + W4(1- E)]= 2x(1 - E2)u + x2(1-E3) + W5(1- E), [13]

in which (W4,W,5) (Y4,Y5)171. Obtain (w4,W5) from Eqs. 11 and12 and substitute in Eq. 13 to obtain a quadratic equation in xalone. This equation has one trivial root x = 0, correspondingto A fixation (Eq. 5), and a second root

x = 2u(1- E2) -(1- E1)2(1-E2)(1- u)-(1-E3) [14]

Thus, the polymorphism (if admissible) may be written ex-plicitly

7 = - (1,2x,x2, i,4x)_ (1 -E3)x + 2u(1-E2)

E

[15]

[16]Q - (1 + x)[E(l + x) + 2u(1 - E2) + (1 - E3)x]/E, [17]

using Eqs. 9-12 together with bat = 1 to determine yIObserve that in the polymorphism Eq. 15 the frequencies

(71,72,73) occur in proportions similar to those classically as-sociated with Hardy-Weinberg equilibrium.The fixed point Eq. 15 can be shown to be unchanged if p

Genetics: Boorman

Proc. Natl. Acad. SC{. USA 75 (1978)

E1

E30 0.5 1

Altruist frequencyStable a fixation

t polymorphism | globallyq stable

A fixationgloballystable

0 ;AI V 1/2U

FIG. 2. Ei concave: E2 < (E1 + E3)/2. q from Eq. 3.

is given by 4/(7y4 + 75) rather than by Eq. 3. This reveals animportant robustness of the model, since the existence andlocation of polymorphism (as well as the stability behavior ofall fixed points as discussed below) is unaffected by whetherrecolonization takes place at the beginning or at the end of aperiod.From the solution Eqs. 15-17, any internal fixed point of Eqs.

1-3 will be unique when it exists. The solution Eqs. 15-17 willbe interior to 5 if and only if 0 < x <co. It is simple to verifythat (A,a) polymorphism occurs when and only when the so-lution is interior, so that a necessary and sufficient conditionfor existence of a polymorphic equilibrium is

<X oE3 + 1-2E2 < U < 1 El [18a]2(1- E2) 2(1 -E2)

or alternatively1-El < U < E3 + 1-2E2 [18b]20-E2) 2(1 -E2)

Let v - (E3 + 1 - 2E2)/2(1 - E2) and ,4 (1 - E1)/2(1 -E2);note that neither depends on E. It is obvious that 18a and 18bare mutually exclusive, since the first interval is (v,IA) while thesecond is (,vP). Note that max(uv) < 1/2 since E1 > E2 > E3, i.e.,group selection favors a (Assumption A4).

There is a fundamental disjunction between two cases, de-pending on whether

E2 ; (E1 + E3)/2, [19]

i.e., whether Ei is convex or concave as a function of the geno-type mix. First consider the concave case, < in 19. Then 18a isdisallowed and 18b is the applicable polymorphism criterion.Since ,u is always strictly positive, the u-interval of allowedpolymorphism is bounded away from 0. It is clear phe-nomenologically that u > v should correspond to a fixation,which is globally stable (except for the 9A given by Eq. 5), andsimilarly u < A should make A everywhere stable. The mostinteresting case is obviously 1/2 > u > v, since then the modelpredicts that group selection should carry its favored allele tofixation despite opposing selection at the individual level. Thesepredictions may be checked by expanding Eq. 1 about 7a andiyA, respectively, and analyzing the local stability behaviorabout fixation (e.g., whether any eigenvalue of the characteristicequation has modulus exceeding 1). This gives the stabilitycriteria

u < (1-FE1 4 A fixation is stable [20]2(1-E2)

U>E3 I 2E*= a fixation is stable. [21]

2(1 -F2)

(a) v > 0

q

(b) v < 0

q

E ,

E

0 0.5 1

1 _ Altruist frequency

AUfixatio a fixationpolymorph ism/ globallystable

A fixation /globally I

stable /o

! ~~~~~~~~~~~~~~~~~~~~~~I0 v A 1 /

11 a fixation

I / globallyI stableI Unstablepolymorphism

rAl I0 A 1/2

U -p

FIG. 3. Ei convex: E2 > (E1 + E3)/2.

which match exactly with 18b to complete the analysis. Onethen has the situation summarized in Fig. 2, which depicts thefixation global stability intervals along the u-axis, separated bya third interval in which there is a unique and globally stablepolymorphism. Numerical results on the comparative staticsof the polymorphism and the stability behavior of the systemwill be reported elsewhere.The structure when Ei is convex, > in 19, is essentially the

dual of that just described. Now 18a becomes the operativecriterion for existence of polymorphism and the roles of v andA are flipped, with ;i being the upper bound of the polymor-phism interval and v the lower.

Again it is clear phenomenologically that for u > i a fixationis stable and for u <v A fixation is stable. Now, however, thestability criterion (21) predicts that a fixation will be stable assoon as polymorphism exists at all, using 21, 18a, and 19 with>. Similarly, with respect to A fixation, 20 predicts stability upto the upper boundg of the polymorphism interval (v,,u). Thuswhen polymorphism exists it must be unstable, lying on amanifold separating the domains of attraction of the two stablefixations y and ya (see Fig. 3). This behavior is dual to thatfound in the concave case, with the stability characters of allthree fixed points being flipped by the shift in convexity. Thesymmetry breaks down in only one possible respect: in thepresent convex case, one may have v < 0 and therefore u = 0falling in the polymorphism interval, implying that this intervalis half-open rather than open and internal as before. For suchparameter cases in the convex case, A fixation is never globallystable for any u. Fig. 3b illustrates this situation, which is ob-viously favorable to the group-selected trait. Again, numericalresults will be reported elsewhere.

Observe that the linear case, E2 = (E1 + E3)/2, is a highlydegenerate one in the present theory [note that this is essentiallythe case emphasized in ref. 1, though the Levins model workswith E(x) rather than through the Et]. The degeneracy becomesclear if one chooses Es close to linear, E2 _(E1 + E3)/2, sincethen At and v will be numerically close and, regardless of whichis greater, small changes in u can drastically alter the out-come.

1912 Genetics: Boorman

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Proc. Natl. Acad. Scd. USA 75 (1978) 1913

DiscussionThe present model illustrates the thesis that the analysis of agroup selection theory, construed at the level of global meta-population dynamics, should be directly related back to simpleproperties of extinction, exemplified by 19.

Complementarities between Convex and Concave Cases.Consider two separate parameter sets (E1) and (Es), each pro-ducing the same polymorphism interval (a,#). Let Es be con-cave and Es convex, i.e., a = A(Ei) = v(Ej), (3= A(E) = v(Ej).For fixed u e (a,(), which convexity case is more advantageousfor the group-selected gene? This question cannot be answeredin general without reference to the initial conditions (Eq. 4),since in the convex case both fixations are locally stable. In thiscase the a gene will thus win completely-go to fixation-provided that it is initially common, but will lose completelyif initially rare. This is reminiscent of the behavior of the co-operative hunting model of ref. 27, though the principle gen-erating the threshold is presently quite different. The locationof the threshold will depend on u and the Es, but for any fixedparameters and initial conditions the outcome will be an all-or-nothing one.By contrast, in the concave case the system will go to the same

polymorphic equilibrium regardless of the initial conditions(except for yA and Hya). This is to be compared with the resultsof the companion paper (28). In flipping the convexity, the agene therefore faces what may be viewed as a strategic choice:the possibility of complete victory is traded for the certaintyof avoiding total loss.

Sources of Extinction. Note first that the convexity of theEt is opposite to the convexity of the complementary demesurvival probabilities Si = 1 - Es; thus,

E2 5 (El + E3)/2 S2 i (SI + S3)/2. [22]

It is these latter probabilities Si that are most directly inter-pretable. In cases where aa allies can undertake an advanta-geous division of labor, S3 >> S2 and Si will be concave, so Ejis convex. 11 Note that this is the case with the threshold ratherthan the stable polymorphism. Es will also be convex if S2 <min(SI,SA), which is a kind of group selection analog to classicalheterozygote inferiority giving rise to disruptive selection; a caselike this might arise if the nonaltruist ("asocial") member killedthe altruist ("social") member of mixed (class II) propagules,thereby eliminating the reproductive prospects of both (seeremarks of Schaller on intraspecific aggression in large carni-vores, ref. 29).The opposite occurs if 0 < S3 -S2 << LS < S2 as might be

expected in cases of a genetically controlled alarm call traitwhere multiple warnings contribute little more survival chancethan a single warning. In this case, Si will be convex and Esconcave so that polymorphism will be stable.

Notice that successful group selection in the present modelremains possible even in the face of very strong opposing in-dividual selection. Consider the limiting case u = 0. Then, ifEs is convex, the condition for local stability of a fixation is v< 0, i.e.,

1+E3<2E2 [23]

usng 21. Since then E2 > %/2, 23 is inconsistent with weak ex-tinction, Es << 1. Occurrence of high extinction rates in founderpopulations is not, however, nearly as strong an assumption asthe corresponding hypothesis when extinction acts on popula-tions at carrying capacity. Thus in the study reported in ref. 2,extinction rates were found which were as high as 0.1 deme/generation (compare table 1 in ref. 3), with almost all extinctiontaking place very shortly after recolonizing.Note also that E does not appear in 18a and 18b at all and

that E << 1 (weak extinction pressure on carrying capacitypopulations) is consistent with group selection cases where afixation is globally stable.

Iam indebted to Paul R. Levitt, Warren Ewens, and the referees forcritical comments on the present model. I also thank David M. Kelleyfor carrying out the numerical studies of the main recursion (Eq. 1)with particular reference to the case of weak selection. The presentresearch was funded through National Science Foundation GrantsSOC76-24512 and SOC76-24394 and Predecessor Grants SOC74-06395and SOC76-24219; the Cowles Foundation for Research in Economicsgenerously provided facilities in support of this research.

1. Levins, R. (1970) in Some Mathematical Questions in Biology(American Mathematical Society, Providence, RI), Vol. 2, pp.75-108.

2. Eshel, I. (1972) Theor. Popul. Biol. 3,258-277.3. Boorman, S. A. & Levitt, P. R. (1973) Theor. Popul. Biol. 4,

85-128.4. Levin, B. R. & Kilmer, W. L. (1975) Evolution 28,527-545.5. May, R. M. (1975) Nature 254,485.6. MacArthur, R. H. & Wilson, E. 0. (1967) The Theory of Island

Biogeography (Princeton Univ. Press, Princeton, NJ).7. MacArthur, R. H. (1972) Geographical Ecology (Harper and

Row, New York).8. Simberloff, D. S. & Wilson, E. 0. (1969) Ecology 50, 278-296.9. Wilson, E. 0. & Simberloff, D. S. (1969) Ecology 50,267-277.

10. Schoener, A. (1975) Am. Nat. 108,715-738.11. Fenner, F. & Ratliffe, F. N. (1965) Myxomatosis (Cambridge

Univ. Press, London).12. Lewontin, R. C. & Dunn, L. C. (1960) Genetics 45,705-722.13. Goel, N. S. & Richter-Dyn, N. (1974) Stochastic Models in

Biology (Academic, New York).14. Feller, W. (1952) Ann. Math. Stat. 55,468-519.15. Norman, M. F. (1975) SIAM J. Appl. Math. 29, 225-242.16. Atkinson, R. C., Bower, G. H. & Crothers, E. J. (1965) An Intro-

duction to Mathematical Learning Theory (Wiley, NewYork).

17. Gadgil, M. (1975) Proc. Natl. Acad. Sci. USA 72, 1199-1201.18. Eshel, I. (1977) Theor. Popul. Biol. 11, 410-424.19. Lewontin, R. C. (1970) Annu. Rev. Ecol. Syst. 1, 1-18.20. Kimura, M. & Ohta, T. (1969) Genetics 61,763-771.21. Dynkin, E. B. (1965) Markov Processes (Springer, Berlin) Vol.

I.22. White, H. C. (1970) Chains of Opportunity (Harvard Univ. Press,

Cambridge, MA).23. Ewens, W. J. (1969) Population Genetics (Methuen, London).24. White, M. J. D. (1973) Animal Cytology and Evolution (Cam-

bridge Univ. Press, London).25. Swanson, C. P. (1957) Cytology and Cytogenetics (Prentice-Hall,

Englewood Cliffs, NJ).26. Beklemishev, W. N. (1969) Principles of Comparative Anatomy

of Invertebrates: Promorphology (Oliver and Boyd, Edinburgh,Scotland) Vol. 1.

27. Boorman, S. A. & Levitt, P. R. (1973) Proc. Natl. Acad. Sci. USA70, 187-189.

28. Levitt, P. R. (1978) Theor. Popul. Biol., in press.29. Schaller, G. B. (1972) The Serengeti Lion (Univ. Chicago Press,

Chicago).

11 Examples are reviewed (emphasizing cooperative hunting) inBoorman, S. A. (1977) "Mathematical network models for cooperativehunting partnerships," Harvard-Yale Preprints in MathematicalSociology, No. 4.

Genetics: Boorman