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J. theor. Biol. (2000) 204, 481}496doi:10.1006/jtbi.2000.2007, available online at http://www.idealibrary.com on

Population Viscosity and the Evolution of Altruism

JOSHUA MITTELDORF* AND DAVID SLOAN WILSON-

*Department of Biology, ¸eidy ¸aboratory, ;niversity of Pennsylvania, Philadelphia, PA 19104, ;.S.A.and -Department of Biology, S;N> Binghamton, Box 6000, Binghamton N> 13902-6000, ;.S.A.

(Received on 2 June 1999, Accepted in revised form on 10 January 2000)

The term population viscosity refers to limited dispersal, which increases the genetic relatednessof neighbors. This e!ect both supports the evolution of altruism by focusing the altruists' giftson relatives of the altruist, and also limits the extent to which altruism may emerge by exposingclusters of altruists to sti!er local competition. Previous analyses have emphasized the way inwhich these two e!ects can cancel, limiting the viability of altruism. These papers were basedon models in which total population density was held "xed. We present here a class of modelsin which population density is permitted to #uctuate, so that patches of altruists are supportedat a higher density than patches of non-altruists. Under these conditions, population viscositycan support the selection of both weak and strong altruism.

( 2000 Academic Press

1. Introduction

Evolutionary altruism is de"ned to comprisetraits carried by an individual which confer a "t-ness advantage upon others. Strong altruism ac-tually imposes a "tness cost upon the bearer ofthe trait, while in weak altruism the disadvantageis only experienced relative to others (Wilson,1980). The emergence of altruism is a funda-mental problem in evolutionary biology. Howcan nature select a gene that promotes the "tnessof others, especially when it is at the expense ofthe bearer of the gene itself ?

In order for an altruistic trait to be selected, thebene"ts of the altruism must fall disproportion-ately on other altruists. The present work appliescomputer modeling on a Cartesian grid toexplore one very general paradigm: altruisticbene"t is dispersed blindly to all occupants ofa geographic neighborhood, while the limitedrate of population di!usion serves to enhance theproportion of relatives of the altruist withinthe bene"tted region. Previous work with this

0022}5193/00/120481#16 $35.00/0

paradigm seemed to indicate that it wouldsupport weak but not strong altruism; but in thepresent model, we see both weak and strongaltruism emerge. The new result depends ona variable overall population density; in particu-lar, the communal bene"t of altruism must besuch that local populations of altruists are sup-ported at a higher density than correspondingpopulations of non-altruists.

A number of theoretical frameworks have beendeveloped to describe ways in which altruism canevolve, including multilevel selection theory,inclusive "tness theory, and evolutionary gametheory. Sober & Wilson (1998) review theseframeworks and their relationship to each other.

Game theory emphasizes the ways in which thebene"ts of altruism may be focused on otheraltruists through reciprocal exchange of bene"tthat recognizes and excludes (or punishes) free-loaders. The &&Prisoner's Dilemma'' game becameone standard model for simulating the evolution-ary viability of cooperative strategies in various

( 2000 Academic Press

482 J. MITTELDORF AND D. S. WILSON

environments. Axelrod & Hamilton (1981) pion-eered computer simulation in this area, sponsor-ing a competition among strategies that anointeda victor in &&Tit-for-Tat'' (TFT), a strategy thatcooperates or betrays each player depending onthat player's most recent behavior toward theprotagonist. The result was greeted with encour-agement, in that TFT is a strategy that allows forthe possibility of cooperation. Recent analysis(Nakamaru et al., 1997, 1998) has combined gametheory with geographic e!ects to illuminate thedistinction between bene"ts to survival and tofertility from cooperation. The game theory para-digm is limited in applicability to higher animalsin which the brain is developed su$ciently tosupport the recognition of individual others.

Inclusive "tness theory is most often applied tosituations in which altruists and the recipients ofaltruism are genetically related in a known way.In some circumstances, Hamilton's (1964) ruleprovides an exact measure of the degree to whichaltruism can evolve. The rule states that themaximum degree of altruism that can evolve isdirectly proportional to the coe$cient of related-ness among the altruist and the recipient of itsbehavior (b/c'1/r, where b is the bene"t torecipient, c the cost to altruist, and r the coe$c-ient of relatedness).

A related analytic approach borrows a tech-nique from statistical analysis to describe cluster-ing in terms of correlation coe$cients for pairs,triples, and higher-order geometric combina-tions. This method, "rst applied by Matsuda(1987) and Matsuda et al. (1987) is most usefulwhen the lowest terms can be shown to be anadequate approximation to the behavior of thefull system. Van Baalen & Rand (1998) are able toplace weak limits on the evolution of altruism byfocusing on the pair correlation alone. But in thestrongly clustered environments typical both ofthe bioshphere and its models, the pair correla-tion alone is of limited utility.

Multilevel selection (MLS) theory treats natu-ral selection as a hierarchical process, in whichthe relative advantages of altruism and sel"sh-ness occur at di!erent levels. For many kinds ofaltruistic traits, the levels may be associated withspatial scales. The advantage of sel"shness islocal*sel"sh individuals are more "t than altru-ists in their immediate vicinity because they

receive the bene"ts without paying the costs. Theadvantage of altruism is realized on a largerscale*groups of altruists are more "t thangroups of sel"sh individuals. MLS theory seeksto analyse the balance between the local andthe global processes, and thus to predict circum-stances under which altruism may evolve. Foraltruistic traits whose bene"ts are focused in ageographic locality, it is necessary that altruistscluster, so that the bene"ts conferred by an altru-ist are more likely to fall upon other altruists.

Analysis is simpli"ed by the assumption ofdiscrete and spatially separated groups; then theconditions favoring the evolution of altruismmay be de"ned. First, the groups must vary intheir frequency of altruists: the more variationthe better. Second, the groups must be competingagainst one another, with some groups growingin size and others shrinking or vanishing.The relative strength of selection within groups(favoring sel"shness) and between groups (favor-ing altruism) is determined by the relative time-scales for extinction of groups to take place andfor sel"shness to evolve to "xation within groups.Third, the altruistic groups must be able to ex-port their genes to the remainder of the globalpopulation. In the absence of a global dispersalmechanism the abundant progeny of altruisticgroups remain in the same locality to competeonly against one other. One ideal populationstructure for the evolution of altruism invokesperiods of isolation, during which groups of al-truists share their mutual bene"t, alternatingwith periods of dispersal, allowing for the exportof the altruistic groups' superior productivity.This scenario, which may be called &&alternatingviscosity'', is explicitly assumed by trait groupmodels in multilevel selection theory, and impli-citly assumed by most game theory and inclusive"tness theory models (Sober and Wilson, 1998and references therein).

But nature provides abundant examples ofpopulation structures that do not alternate be-tween isolation and dispersal, and for which nei-ther explicit kin selection nor reciprocal exchangeare natural models. In order to extend our under-standing to encompass these phenomena, it isdesirable to avoid explicit assumptions about re-latedness, grouping and the timing of dispersal,and to allow all these concepts to emerge as

POPULATION VISCOSITY EVOLUTION ALTRUISM 483

a consequence of a general geographic structure.In our models, population viscosity &&groups'' areloose associations, where patches tend to bedominated for a time by one variety or anotheronly because of the limited speed at which sib-lings disperse. These models resist analytic treat-ment. Application of Hamilton's rule encounterstwo di$culties: "rst, there is no easy way togauge local relatedness; second, a global "tnessmeasure becomes elusive when reproductive suc-cess depends on the strength of local competition.Computer simulation may be an appropriate toolfor approaching this problem's irreducible com-plexity; in any case, simulation results can serveas a stimulus to analytic thought, and as one testof its verisimilitude.

A study by Wilson et al. (1992) (Wilson, Pollockand Dugatkin, WPD) indicated that computermodels of population viscosity could support theselection of weak altruism, but that strong altru-ism was not competitive in this environment.Taylor (1992) reached a similar conclusion with-out computer simulation, analysing a model inwhich discrete groups approximate the e!ects ofviscosity. Independently, Nowak & May (1992,1993) described an early grid model. Though theyfocused on the mathematical properties of theirmodel to the exclusion of biological implications,their results may also be interpreted to permitweak but not strong altruism to evolve. LikeWPD, they considered only "xed total popula-tion densities; however, their cost/bene"t schemewas structured in a somewhat di!erent way, en-hancing the prospects of altruists at low densities.

In the present paper, the WPD model is takenas a starting point. Altruism is modeled as havinga cost c and a bene"t b that contribute linearly to"tness. The cost is borne by the altruist alone,and the bene"t shared equally by the altruist andits four lattice neighbors. Competition for eachlattice site in each (non-overlapping) generationtakes place among a group of "ve neighborshaving a similar geometry. For the purpose ofcalculating altruistic contributions, each indi-vidual is counted as the center of its own neigh-borhood; thus every lattice site is surrounded bya local gene pool consisting of itself and fourneighboring sites, with sites further a"eld a!ect-ing the competition indirectly via their in#uenceon the "tness of the four neighbors.

WPD derive a version of Hamilton's rule ap-propriate to this model: the "tness of the averagealtruist in the global population exceeds that ofthe average non-altruist as long as

b/c'1/<, (1)

where < is a statistical analog of Hamilton'srelatedness variable, r. Speci"cally, < is theaverage of the pair correlation coe$cient overthe 5 recipients of the altruist's bene"t. One ofthese is the self, with correlation 1, and 4 arelattice neighbors, with correlation R: hence

<"(1#4R)/5. (2)

(The pair correlation R may be de"ned by theusual statistical prescription, and can be shownto equal the di!erence between the probabilitiesof "nding an altruist when looking adjacent toa random altruist and adjacent to a randomnon-altruist. This identity is detailed inAppendix A.)

WPD found that the altruistic trait did not fareso well in their simulation as this version ofHamilton's rule would have predicted. They ex-plained this result with reference to the latticeviscosity: the same limited dispersal that concen-trated altruists in patches also impeded their abil-ity to export their o!spring to distant parts of thegrid. The result may also be understood in termsof overlap between the group sharing an altruist'sbene"t and the group which competes directly foreach site in the daughter generation: When thetwo groups are identical, the altruistic bene"tapplies to all competitors equally, thus it hasexactly no e!ect. When there is no overlap be-tween these two groups, the above version ofHamilton's rule predicts accurately the condi-tions under which altruism may prevail. In thepresent model, there is partial overlap, and thusaltruism has some selective value, but not the fullpower predicted by Hamilton's rule. (This argu-ment is made quantitative in Appendix B.)

In the present work, we replicate the 1992results of WPD and broaden their analysis byrelaxing one key assumption: the requirement ofconstant population density. This modi"cationaddresses the issue of the altruists' excess produc-tivity having no place to go, and does so in a way


that is biologically reasonable, at least for someexpressions of altruism. In principle, the numberof occupants of each lattice site could be anyinteger, a free variable of the model; but we re-serve this extension for future investigation. Inour model, each site may be empty, or it may beoccupied by a single altruist or non-altruist. Vari-able population densities may be observed overareas su$ciently large that the concept of densityas a continuous variable has some meaning, butsu$ciently small that they are approximatelyuniform in composition. The original rules of theWPD model have been extended in three di!er-ent ways in order to support the presence of someempty cells in the long run, without causing theentire population to vanish.

With each of these three variations, we "ndthat there are ranges of the parameters for whichaltruism is a viable competitor to sel"shness, in-cluding some in which altruism evolves to "x-ation. Both strong and weak altruism may besupported under a broad set of assumptions. Infact, we "nd that the strong/weak distinction haslittle place in our understanding of the forces thatdetermine whether altruism can prevail. We con-clude that if the bene"t of an altruistic trait en-ables a larger population density to support itselfin a given environment, then with appropriateratios of bene"t to cost, that trait may evolvebased upon population viscosity alone.

Similar conclusions to the above were reachedvia a very di!erent path by Ingvarsson (1999).

2. The WPD Model

Two types of asexual individuals exist in thepopulation, di!ering only in their altruistic (A) vs.sel"sh (S) traits. The populations are arrayed atlattice intersections of a two-dimensional Car-tesian grid, 200]200 sites in most runs, withopposite edges connected to eliminate boundarye!ects. Each point on the grid represents a spatiallocation that can be occupied by a single indi-vidual, either A or S. (In the 1992 WPD model,vacancies were not permitted.) Interactions arelocal, such that the absolute "tness of anindividual depends on its neighborhood of "vegrid points, which includes its own locationplus the four neighboring points occupying ad-jacent north, east, south and west position on

the grid:

=S""tness of altruist"1!c#N

Ab/5,

=A""tness of non-altruist"1#N

Ab/5, (3)

where NA/5 is the average proportion of altruistic

neighbors (the plus-shaped neighborhood of "veincludes the self ).

Each altruist contributes b/5 "tness units toeveryone in the neighborhood, including itself, ata personal cost of c. Sel"sh individuals enjoy thebene"ts conferred by their altruistic neighborswithout paying the cost. Altruists decrease theirrelative "tness within their neighborhood whenc'0 (weak altruism) and decrease their absolute"tness when c'b/5 (strong altruism).

The model proceeds in "xed, non-overlappinggenerations. Every site is vacated at the end ofeach generation, and a new occupant is chosen asa clone of one of "ve individuals in the neighbor-hood comprising the central site itself and neigh-bors to the north, east, south, and west. These "veindividuals compete for the site with probabilitiesproportional to their "tnesses as computed ineqn (1). Note that "ve overlapping neighbor-hoods are used to compute contributions to the"tnesses of the "ve participants in the lottery thatdetermines which will seed the central site in thenew generation.

The structure of this model was suggested bya population of annual plants. A given area willsupport a "nite number of such plants in eachyear, and all the plants in a neighborhood com-pete to seed the site at the center of that neighbor-hood in the coming year. The "tness values ascomputed in eqn (1) correspond in this picture tothe quantity of seeds produced by each variety ofplant. All the seeds produced by plants in a givenneighborhood form a local gene pool, from whichonly a single plant will survive to occupy a givensite. The hypothetical altruistic trait permits allplants in a neighborhood around the altruistplant to generate more seeds.

Here is a numerical illustration based on Fig. 1:the position in the center is occupied by an A,which has two A and two S individuals as neigh-bors. If c"0.2 and b"0.5, then, from eqn (1), thefocal individual has a "tness of 1!0.2#0.3"1.1. The "tness of the four neighbors can also

FIG. 1. A sample segment of a saturated grid. A's are sitesoccupied by altruists and S's are non-altruists. The base"tness of each S is 1, and the base "tness of each A is (1!c).To this base is added a multiple of b contributed by thecount of A's in the 5-site neighborhood (shaped like &&#'')centered on the target site.


be calculated from eqns (1), remembering thateach individual occupies the center of its ownneighborhood. Adding the "tnesses of A's and S'sseparately, we "nd 3.2 "tness units for A and 2.6"tness units for S. These "tness units may standfor seeds in the neighborhood, or it is also conve-nient to think of them as tickets for a lottery.Since A's hold 3.2 lottery tickets to 2.6 for S's, theprobability that the central site will be occupiedby an A in the next generation is 3.2/5.8"0.552.Notice that the frequency of A among the o!-spring produced by the neighborhood (0.552) isa decline from the parental value of 3/5"0.600,which illustrates the local advantage of sel"sh-ness. The simulation repeats this procedure forevery position of the grid to determine the popu-lation array for the next generation.

2.1. MODELS WITH VARIABLE POPULATION DENSITIES:

DISTURBANCE AND REGROWTH

Fitness of an organism has meaning only inrelation to a particular environment. Enhanced"tness may manifest as a greater exponentialgrowth rate in free population expansion, or asa greater carrying capacity in a saturated niche,or as a more robust response to any number ofchallenges. Our extensions of the WPD modelare designed to permit the representation of suche!ects with minimal changes in the model's rules.

In the original version, every lattice site wasoccupied in every generation by exactly one indi-vidual. A minimal modi"cation would permitsome sites to be vacant. The rules governing thelottery then need to be generalized to allow forvacant sites (V's) as well as A's and S's. The

simplest rule would be to assign a constant&&"tness'' g to the void; each V participating ina lottery would receive g tickets, independent ofits surrounding neighborhood. The language de-scribing V's as a separate species with its own"tness is a convenient mathematical "ction, akinto the symmetric treatment of negative electronsand positive holes in a solid-state physicist'sequations for a semiconductor. The rationale forthis prescription is simply that if a neighborhoodis less than fully occupied in the parent genera-tion, there is a "nite probability that reproduc-tion will fail to create any occupant for its centralsite in the daughter generation.

In two-way competition between S's and V's,the V's have the upper hand when g'1, and thegrid quickly empties out. For g"1, the popula-tion in two-way competition is marginally unsta-ble, and eventually the grid evolves to saturation(all S) or extinction (all V). With g(1, the popu-lation grows always toward saturation, whichwould reduce the model to an exact replica ofWPD. The "rst two possibilities did not appearpromising; however the third, with g(1 anda population density that always grows, can bethe basis for an interesting model, as long asa mechanism is added to replenish the popula-tion of void sites.

One way to maintain a population of V's onthe grid is to introduce periodic disturbanceevents. On a "xed schedule, a percentage of allgrid sites is vacated, as if by a disease or naturalcatastrophe, or a seasonal change that kills o!a high proportion of individuals in the winter andpermits regrowth in the summer. The geometryof the disturbance may be that a "xed proportionof sites is chosen at random for evacuation(&&uniform culling''); or at the opposite extreme, allthe sites in one area of the grid may be vacated,while the sites outside this region remainuntouched (&&culling in a compact swath''). Boththese cases have been explored, and resultsare reported below in Section 3.2 and 3.3,respectively.

For the uniform culling case, we choose para-meters such that a high proportion of thepopulation is culled, denuding the grid except forisolated individuals. This maximizes the foundere+ect ("rst described by Mayr, 1942), as regenera-tion takes place in patches that are pure A or


pure S, because they are descended from a singlesurvivor. As patches begin to regenerate, theyexpand into surrounding voids, and direct com-petition is delayed for a time. In the initial phaseof regrowth, it is the competition A vs. V and S vs.V that dominates; only later do A's and S's com-pete head-to-head. But A communities, by ourassumptions, can grow faster than S communi-ties; hence the A's may compile a substantialnumerical advantage before direct encounters (Aupon S) become commonplace. This process isillustrated in Fig. 2. Detailed results for this vari-ation are reported in Section 3.2.

The other possibility we have explored is cull-ing in a compact swath. A square centered ona random point and sized to include half thegrid's total area is completely denuded in eachdisturbance event. Regeneration takes place fromthe edges, and regions of the boundary that, bychance, are dominated by A's expand most rap-idly into the void. This process is illustrated inFig. 3. Although the founder e!ect is evoked lessexplicitly in this variation than in the uniformculling case above, culling in a swath neverthelessproves to be a surprisingly e$cient mechanismfor segregating the population into patches ofpure A and pure S. Detailed results for this vari-ation are reported in Section 3.3.

2.2. MODELS WITH VARIABLE POPULATION DENSITIES:

UNSATURATED STEADY STATES

In the above model variations, the "tness of thevoid g(1 was chosen in such a way that popula-tions of A or S would grow and eventually

&&&&&&&&&&&&&&&&&&&&&&&&&&&

FIG. 2. Red dots are altruists; green are non-altruists; white dmixed state, 1/3 each altruists, non-altruists and empty sitespopulation at random. Regrowth has just started, and isolated i(c) Growth into the voids continues, with patches of altruists gdisturbance and regrowth, altruists have come to dominate the laor non-altruists may prevail. Higher frequency of distubranhead-to-head confrontation, favoring altruists.

FIG. 3. Disturbances that obliterate large patches of the gridvarieties. (a) The grid is initialized in a thoroughly mixed state,disturbance is introduced, occupying half the grid area in whichalready been visible consolidation, with patches of altruistsre-colonized from the edges. Patches of altruists are able to grofrom the disturbance experience direct red-on-green competitionof just one cycle of disturbance and regrowth, the disturbancehelping to segregate into separate regions dominated by eachcentered at random locations on the grid, altruists may evolve

eliminate voids in the absence of disturbance;disturbance events were inserted in the model inorder to keep the result from degenerating intothe saturated-grid case, with constant populationdensity. These models correspond to populationstructures that are subject to seasonal cycling, ormay be culled periodically by disturbances. An-other case worth exploring avoids periodic cull-ing. Voids are distributed through the grid andcompete more equally with A's and S's, battlingto a steady-state distribution even in the absenceof disturbances. Communities of A's and S's mayhave di!erent e$ciencies in using environmentalresources, which is re#ected in a lower averagevacancy rate in areas dominated by A's than incorresponding areas dominated by S's. In orderto model such situations, it is necessary to ar-range parameters so that A, S and V can coexistin steady state. Figure 4 illustrates a competitionin which denser patches of A's are able to holdtheir own against sparser patches of S's.

The simplest assumption about voids, g(1 org'1, fail to generate the steady state that we arelooking for. The case g"1 may work as a steadystate for a short time, but it is unstable: the gridevolves eventually either to saturation (no voids)or extinction (all voids). One simple way to engin-eer a stable steady state is by introduction ofa second parameter, which we call m. m representsan extra presence in every "ve-way lottery, anacknowledgement that there is always a "nitechance that seeding will fail and a site willbecome vacant, even if its neighborhood issaturated in the parent generation. Speci"cally,

&&&&&&&&&&&&&&&&&&&&&&&&&"ots are empty sites. (a) The grid is initialized in a thoroughly. (b) A disturbance is introduced, destroying 95% of thendividuals have created small patches of homogeneous type.rowing more rapidly than non-altruists. (d) After 5 cycles ofndscape. With other parameter combinations, either altruistsces leads to more time spent in free expansion and less

prove to be an e$cient mechanism for segregating the two1/3 each altruists, non-altruists and empty sites. (b) A squareall life is destroyed. Notice that after 5 time steps, there hasand non-altruists beginning to segregate. (c) The void isw more quickly than non-altruists. Meanwhile, areas away, in which altruists are at a disadvantage. (d) Toward the endhas had two e!ects: enhancing the fortunes of altruists, andvariety. If disturbances continue on a regular schedule butto "xation.

FIG. 3. (Caption opposite)

FIG. 2. (Caption opposite)

FIG. 4. This grid is initialized with 1/3 each altruists, non-altruists and empty sites, randomized pointwise so that there areno local concentrations. There are no large disturbances in this run, but rules of the 5-point lotteries have been modi"ed, sothat there is always a "nite minimum probability that the site will be empty in the coming generation. Some spontaneoussegregation is already visible in (b), allowing colonies of altruists to pursue their local advantage. By square (d), it is apparentthat altruists "ll their regions e$ciently (98.7%) while regions dominated by non-altruists are 50% void. By chance, voidsareas sometimes appear as &&mini-disturbances'' near the borders between regions dominated by altruists and non-altruists,and when they do, the altruists expand rapidly into the empty region. Even in border regions where altruist confrontnon-altruists directly, the altruists are arrayed along the border with twice the density of non-altruists, and this can vitiate theadvantage of the freeloaders. The altruists may come to dominate the competition. In this example, altruists will evolve to"xation in several hundred time steps.


the rules of the lottery are modi"ed so that inaddition to the "ve competitors comprising theA's, S's and V's in a given neighborhood, there isalways an extra measure of m lottery tickets as-signed to V. Intuitively, it is clear that V cannever become extinct, i.e. the grid will not becomesaturated, because each lottery has a guaranteedminimum probability of generating a V. But if g issmall enough, then A's and S's will still have anadvantage in each lottery su$cient to insure thatthey do not succumb to V in competition. In fact,we "nd that with proper choice of g and m,a stable steady state results, in which V's arealways distributed through the population of A'sand S's, and the populations avoid the extremesof extinction and saturation.

Exploration of this four-parameter system (b, c,g and m) is the subject of Section 3.4 below. Themodel supports the emergence of altruism in ap-propriate parameter ranges, and favors quasi-periodic population oscillations for other ranges.Overall, we have found it to be a rich lode ofsubtle and unexpected phenomena.

Although the full three-species system (S}A}V)displays such complex behavior, the two-com-ponent systems A}V and S}V are quite amenableto analysis. For a system with only non-altruistsand voids, the population tends to a steady statewith a density approximated by

oss"1!

m(5(1!g))

(4)

The corresponding system with just altruistsand voids seeks a steady-state density approxi-mated by the quadratic formula

oss"

!B#J B2!4AC2A

(5)

whereA"4b,

B"5(1!c!g)!3b,

C"!5(1!c!g)#m!b.

These formulae are derived in Appendix C.They have served us as guidelines for selection of

parameter combinations to explore; speci"callythe relative advantage of altruist communities inoss

has proven a good predictor of the success ofaltruism in the simulated competition.

The four parameters b, c, g and m permit ampleroom to observe an interesting range of vari-ations in the simulation results. We are mostconcerned here with strong altruism, c'b/5,where the constant-density WPD model founda consistent advantage for the non-altruists.When the average density o

ssfor patches of altru-

ists is signi"cantly higher than the correspondingdensity for patches of non-altruists, we "ndparameter ranges where strong altruism cancompete successfully. These results are detailed inSection 3.4.

3. Results

We have seen that much of the interestingdetail relevant to the success of altruism is con-tained in the part of parameter space close tob"5c, the boundary that divides weak fromstrong altruism. Only for b(5c does the indi-vidual pay a net cost for its altruism, lowering itsown absolute "tness in order to bene"t its neigh-bors. In order to explode the region of parameterspace near to b"5c for further study, we intro-duce the parameter c, de"ned as the quotient ofthe total bene"t to others by the net cost to theself :

c"4b/5

c!b/5"

4b(5c!b).

(6)

The numerator, 4b/5, is the part of each altru-ist's bene"t that is exported to neighbors, whilethe denominator, c!b/5, is the net cost, since 1/5of the bene"t devolves upon the self. c"1 is anabsolute lower limit in the sense that it is neverpossible for altruism to evolve when the net e!ecton "tness of self plus neighbors is negative. At theother extreme, the threshold of strong altruism is5c!b"0, corresponding to in"nite c. c is a con-venient measure of the hurdle which selectionmust leap in order to select strong altruism.

In the results tabulated below, we focus on&&Stalemate c'' as a measure of each model'shospitability to altruism. Stalemate c is located asthe end result of multiple trials, seeking that

FIG. 5. Gamma is a measure of the minimum bene"t/costratio required for strong altruism to be viable. The de"nitionis such that all c(R correspond to strong altruism, butlower gamma means that conditions are more favorable forevolution of altruism. These results show that even witha saturated grid and no culling, strong altruism can becomeviable as cost and bene"t are scaled up in tandem.


combination of b and c for which altruists andnon-altruists have equal prospects, and for whichthe two populations may persist for manysimulated generations. Alternatively, there weresome parameter combinations for which thecompetition proved unpredictable and the victorin individual runs not well determined; for thesecases, stalemate c was derived from a pair (b, c)for which A and S were equally likely to prevail.Typically, 5}10 trials were used to determineeach value of stalemate c, and for many casesa smart trial and error routine was used to adjustb automatically (for "xed c) depending onwhether A or S prevailed in previous trials.

3.1. REPLICATION OF WPD MODEL RESULTS

In Table 1, corresponding to Fig. 5, c was aninput parameter, held "xed while b was varieduntil a stalemate condition was found betweenA's and S's, and c was computed from b and c.The last column, labeled &&Neighbor R'' is theautocorrelation coe$cient for pairs of neighbor-ing sites, and is the same R from eqn (2) above.

It was the primary conclusion of WPD thatb"5c (c"R) was a line of demarcation, atwhich altruism was marginally viable againstsel"sh behavior; for larger b the altruists wouldevolve to "xation, and for smaller b, they wouldperish. Taylor's (1992) analytic model found thesame boundary b"5c to be critical, and hisderivation relies upon the assumption that b andc are both small. With larger grids and longerruns, we are able both to con"rm that the rule

TABLE 1Stalemate gamma is a measure of the viability ofstrong altruism. As cost and bene,t rise in tandem,strong altruism begins to become competitive, even

in this saturated grid model

Stalemate Stalemate Neighborc b c R

0.01 &0.05 Large* 0.540.03 &0.15 Large* 0.590.10 &0.50 Large* 0.560.30 1.475 240 0.551.00 4.78 91 0.513.00 13.6 43 0.49

10.00 43.9 33 0.47

*Error bars on c are wide for these results.

holds for small b, c and to explore some of theregion where the rule fails. When b and c are bothsmall compared to 1, the minimum c to supportaltruism approaches in"nity. But for larger b andc, altruism may evolve to "xation for "nite c (evenin a saturated grid). For c"1, altruism will pre-vail if b'4.78, corresponding to c'91. But ifc"10, then b needs only be '43.9, correspond-ing to c'33.

Figure 5 shows that the boundary betweenweak and strong altruism is not as fundamentalas suggested by Taylor and WPD. In fact, weargue in Section 4 below that the demarcation atb"5c is an artifact of the geometry chosen byWPD, and has no deep signi"cance.

3.2. PERIODIC UNIFORM CULLING

In these runs, a high percentage of the popula-tion was culled at random on a regular schedule,such that much of the action consisted of re-growth from isolated founders into empty sur-roundings. No m variable is required for thismodel (remember that m was introduced in orderto engineer a stable steady-state populationdensity less than unity). However, the &&"tnessof the void'' g was an essential parameter of themodel, and g was varied from 0.5 to 0.92. Ninety-"ve percent of the occupied sites were randomlyvoided every 50, 100 or 200 generations; in

FIG. 7. Dispersed culling results. A "xed proportion ofsites is selected at random and vacated on a regular sched-ule. More frequent culling creates a more favorable environ-ment for evolution of altruism: 200 gen's ( ); 100 gen's ( );50 gen's ( ).


separate trials, 50, 75, 90, 95, 96.5, and 98% wereculled every 100 generations.

In runs with deep population cullings, re-growth takes place from small, isolated patchesthat tend to be dominated by one variety or theother (the founder e!ect). Much of the competi-tion is not directly A against S at their commonborder, but rather is a race for free expansion intounpopulated regions. Higher values of g implyslower growth rates for both varieties, but asg approaches 1, altruists are a!ected relativelyless; this is because at g"1, non-altruist popula-tions do not grow at all, but altruist populationsmay still grow for somewhat higher values (de-pending on parameters b and c). Therefore, high-er g and deeper, more frequent culling are theconditions more favorable to the emergenceof altruism, consistent with our model results(Figs 6}8).

Three trends apparent in these charts are read-ily explained. First, increasing g while holdingother parameters constant has the e!ect of in-creasing the viability of altruism, i.e. decreasingthe stalemate c. This is because higher g makesthe void a more formidable competitor, increas-ing the importance of the altruists' competitivegrowth advantage. Second, with g held constant,

FIG. 6. Dispersed culling results. 95% of sites are selectedat random and vacated every 100 generations. Patches ofaltruists are able to regrow at a greater rate than non-altruist patches. The parameter g controls the speed ofregrowth. The slower the regrowth, the more e!ective is thedisturbance for promoting the viability of altruism: g"0.50( ); g"0.60 ( ); g"0.80 ( ); g"0.92 ( ).

FIG. 8. Dispersed culling results. In this "gure, regrowthrate and frequency of culling are held "xed while proportionculled is varied from 75 to 98%. More severe culling createsa more favorable environment for evolution of altruism.

increasing the frequency of culling also makesaltruism more viable. For the entire range offrequency values in the chart, there is ample timeto regrow and "ll the grid to saturation betweencullings; however, the runs with least frequentculling (200 generations) spend more than 3/4 ofeach cycle in a saturated, direct-competitionphase, while those with the most frequent cullings(50 generations) spend almost the entire cycle ina growth phase. In direct competition, altruistsare at a disadvantage, but during the growthphase, regions with high concentrations of

FIG. 9. Block culling results. A square swath correspond-ing to half the total grid area is vacated every 100 genera-tions. Results demonstrate the same trend as Fig 6. Notethat much deeper culling is required in the dispersed case inorder to have comparable e!ect; this is because regrowthinto scattered voids is much more rapid than into large,vacant blocks. Conversely, culling in a compact block ismuch more e!ective for promotiing the evolution of altru-ism: g"0.60 ( )); g"0.80 ( )); g"0.90 ( ).

FIG. 10. Block culling results. The same trend appears asin Fig. 7 more frequent culling creates a more favorableenvironment for evolution of altruism: 150 gen's ( ); 100gen's ( ).


altruists advance and spread faster. Thirdly, andfor similar reasons, increasing the culling percent-age has a salutary e!ect on the viability of altru-ism. Thus, stalemate c decreases monotonicallywith decreasing culling factor (the percentagethat remain unculled) (see Figs 9 and 10).

A fourth trend presents more challenge: com-paring stalemate c for high and low c, we "ndsome upward trends with c and some downward.(These are adjacent lines of data all through thetables.) Where viability of altruism is high, ittends to be higher when c is lower; however, whenviability of altruism is already low, it tends to belowest for low values of c. The reason for theformer trend is that increasing c and b in parallelincreases the growth rate for both A's and S's(b always overpowers c); hence more of each cycleis spent in the saturated phase. The latter trend isjust what we observed in Section 3.1: for directcompetition on a saturated grid, higher c andb makes altruism more viable. The reason for thisremains unclear.

3.3. POPULATION CULLING IN A COMPACT SWATH

In a variation on the population culling modelin Section 3.2 above, we speci"ed that cullings cut

a square swath across the grid in which all occu-pants are removed. The swath had area equal tohalf the total grid, but was randomly placed eachtime. (The swath was also permitted to straddlethe grid's periodic boundaries, leaving behinda cross rather than a frame of occupied sites.)This mechanism corresponds in nature to dam-age from an extreme weather event, the outbreakof a parasite, or any other catastrophe which maydevastate a compact geographic region, leavingoutlying areas una!ected.

This scheme was found to be an e$cient seg-regating force, creating within a few culling cyclesvery clearly de"ned regions "lled densely withhomogeneous populations of type A or S (Fig. 3).Competition takes place both at boundaries be-tween A and S regions, and also in the rate atwhich each group regrows into the voided swath.As in Section 3.2, the frequency of culling deter-mines the balance between these two forces.Because of the segregation, these runs were sur-prisingly hospitable to the evolution of altruism.

Note that, compared with Section 3.2, we "nda much lower stalemate c (greater hospitability tostrong altruism) in runs with 50% compact cull-ing than with 50% distributed culling. In fact,experiments with a compact culling factor of50% are even more hospitable to altruism thanare uniform culling models in which only 5% areleft standing. Growth into a vacated region from

FIG. 11. In Section 3.4, evolution of altruism is madepossible by the greater population density in patches ofaltruists. Since each site is occupied by either 1 or 0 indi-viduals, there is room for this only to the extent that para-meters allow for a steady-state vacancy rate in patches ofnon-altruists. This "gure demonstrates that the populationelasticity is an important determinant of the hospitability ofany parameter set to the evolution of altruism. The y-axis,on a log scale, is &&stalemate c'', the ratio of communal bene"tto individual cost. Low values of c indicate that altruismmay emerge more easily. The x-axis is grid vacancy rate atstalemate, which represents the potential for populationelasticity.


the edges takes much longer than "lling in anequivalent area of smaller, distributed voids, andhence is a more sensitive test of a population'sability to expand.

Two trends visible in Section 3.2 above may bedetected here as well, and the same reasoningapplies: "rst, increasing g with everything elseheld constant creates an environment more hos-pitable to altruism because the growth rate ofnon-altruist patches is reduced while the growthrate for altruist patches is less a!ected. Second,increasing the frequency of culling also enhancesthe viability of altruism, by increasing thefraction of the time that growth is taking placeunimpeded into void regimes.

Another clear trend noted in our results is thatthe outcome of individual runs is less predictablein this section (with culling across a swath). Inother words, the range of c values for which A orS may randomly prevail in a given run is largestwith this paradigm. We speculate that the cor-relation between maximum e!ectiveness of groupselection and minimum predictability is a broadtrend. The problem of group selection may bestated: how can individual selection, which ismuch quicker and more e$cient, be forestalledlong enough for intergroup di!erences to showtheir e!ect? This formulation suggests that anystochastic e!ect, making the short-term outcomeless certain, is likely to decrease the importance ofindividual selection relative to group selection.

3.4. ELASTIC POPULATION DENSITIES

IN STEADY STATE

A trait with positive selective value may eitherincrease the rate of free population expansion, orelse it may augment the steady-state density atwhich a population may be supported in a givenenvironment. Models in this section allow forvariable population densities, with patches ofaltruists supported at a higher density thanpatches of non-altruists, but the population is notcontinually expanding as in the above variation.Rules of the lottery have been modi"ed as de-scribed in Section 2.2 above so that voids (V)co-exist with A's and S's in steady state. Altruistswill always lose in head-to-head competitionwith their sel"sh fellows but it is easy to arrangefor altruists to have a greater steady-state "lling

factor, so that regions dominated by altruism aremore densely populated than regions in whichnon-altruists prevail (see Fig. 11).

In Table 2, the left three columns are inputparameters: g and m control the &&"tness of voids''.c is the gross cost of altruism, before the altruist'sbene"t to himself, b/5, is deducted. The criticalb tabulated in the next column is the result ofmany computer runs searching for a b valuewhich leads to stalemate, i.e. altruists and non-altruists may coexist for many generations or,alternatively, each type may prevail in the com-petition equally often. The next two columnshold theoretical steady-state "lling values for al-truists and non-altruists, as computed by eqns (4)and (5) above. The ratio of these two is the driv-ing force for the evolution of altruism. In the nextcolumn, c is the bene"t : cost ratio correspondingto c and the critical b, as computed in eqn (6). Thelast column is the observed grid "lling factor atstalemate, the ratio of "lled sites to all grid sites.

Two trends are clearly visible: as g rises, voidsare relatively more competitive, and the advant-age of altruism becomes more important. Thiscan be seen by comparing each group of "ve lines(in which g is held constant) with the other two

TABLE 2Results from the variable density model. ¸ower c indicates greater viability foraltruists. Higher m increases the dynamic range of grid vacancy, leading to lowerc values.=ithin each set of ,ve trials, b and c rise in tandem. ¸ower values of b,c are associated with lower values of c, in contrast to the corresponding results for

the saturated grid in ¹able 1

Stalemate Theoretical Stalemate Observed

g m c b Altr oss

Self-oss

c o

0.80 0.20 0.01 0.0395 0.821 0.80 15.0 0.730.80 0.20 0.03 0.1225 0.856 0.80 17.8 0.760.80 0.20 0.10 0.428 0.920 0.80 24 0.820.80 0.20 0.30 1.35 0.967 0.80 36 0.970.80 0.20 1.00 4.61 0.989 0.80 47 0.97

0.90 0.20 0.01 0.0295 0.640 0.60 5.8 0.350.90 0.20 0.03 0.0880 0.709 0.60 5.7 0.380.90 0.20 0.10 0.3465 0.871 0.60 9.0 0.600.90 0.20 0.30 1.21 0.959 0.60 16.7 0.680.90 0.20 1.00 4.47 0.989 0.60 34 0.88

0.95 0.10 0.01 0.027 0.665 0.60 4.7 0.310.95 0.10 0.03 0.084 0.775 0.60 5.1 0.400.95 0.10 0.10 0.389 0.937 0.60 14.0 0.750.95 0.10 0.30 1.36 0.982 0.60 39 0.890.95 0.10 1.00 4.51 0.994 0.60 37 0.88


groups. Altruism can evolve at a lower c when g ishigher, because the voids are a bigger factor inthe competition.

The second trend is that within each group,critical c increases when both c and b are raised.The reason for this is less transparent, but wepropose that it can also be understood in terms ofthe relative competitive position of the void. Lar-ger c leads to larger balancing b, which increasesthe competitive advantage of both A and S withrespect to V. The result is that the stalematedbattle which de"nes critical c is fought in a gridwith a higher "lling factor.

Our results show generally that the addition tothe model of elastic population density createsa hospitable environment for the evolution ofaltruism, even without major disturbance events.We "nd that altruism can prevail with c ratios aslow as 4.7 (for comparison, the WPD adaptationof Hamilton's rule in this situation would call forc"2, while the WPD result for a "xed-densitygrid corresponds to in"nite c). The prospects ofthe altruists improve with decreasing steady-stateoccupancy of non-altruists. Figure 11 chartsstalemate c as a function of the measured grid

vacancy rate. The roughly exponential decline ofc with increasing vacancy rate suggests that thisvariable, which is really a surrogate for popula-tion elasticity, is the primary factor responsiblefor the variance of stalemate c.

The original reason for including voids in thegrid was to allow the modeling of elastic popula-tion densities. The population elasticity hasproven to be a critical factor; indeed Fig. 5 sug-gests that population elasticity explains most ofthe variation in the viability of altruism withinour model. Nevertheless, we have only begun toexplore the population elasticity variable, witha dynamic range of only a factor 3: the presentmodel allows each site to be occupied by at mostone individual, so that maximum populationdensity is 100%; minimum population density isabout 30% because when parameters are ad-justed for a density lower than about 30% thepopulation is too fragmented to survive, as itis vulnerable to random extinction events. Theminimum c"4.7 that we observe may well besurpassed in model variations that allow fora population variable at each grid site. This sug-gests a line of investigation for future research.


4. Discussion of Results

We have sought to construct models for altru-ism in which population structure emerges purelyfrom localized interactions. We began with "xed-density models after WPD, where the "nding wasthat weak but not strong altruism could be sup-ported, so long as b and c were both small. It istempting to seek general meaning in this "nding,and to seek some correspondence to the well-known rule that weak altruism but not strong mayevolve in a model where the altruist's bene"t isscattered randomly on a large population.

It is a lesson of MLS theory that weak andstrong altruism, as parts of a continuum, can beunderstood with a single theory. The question ofwhether either one can evolve depends upon thebalance between within-group and between-group selection (Sober and Wilson, 1998). Ouranalysis reveals that the coincidence between theweak/strong altruism boundary and the min-imum b/c ratio for the emergence of altruism doesnot signal any more general relationship, andthat only the narrowest signi"cance should beattached to this result. Not only do vacanciesallow the evolution of strong altruism, but so alsodoes increasing the absolute value of b andc (Table 1). Furthermore, if the rules of the gridare altered such that corners are included in eachneighborhood, then the demarcation becomesapproximately b"13c, though the boundary be-tween weak and strong altruism is now at b"9c,for neighborhoods of size 9 (results not tabulatedhere). Thus, the signi"cance of the weak/strongboundary is not even independent of geometry.

We have generalized the WPD model to allowfor variable population density, and found thisindeed to be a crucial factor for the viability ofaltruism. We began with models in which groupsof altruists, though unable to compete head-to-head against non-altruists, were nevertheless ableto grow into a vacant habitat at a faster rate. Notsurprisingly, models in which the populationswere periodically culled, creating empty space tobe repopulated, constituted a friendly environ-ment for the emergence of altruism. We movedfrom there to model variations in which dispersedvacancies were built into the population, in sucha way that clusters of altruists could exist insteady state at densities up to 3 times the

corresponding population density for clusters ofnon-altruists. We saw that altruists were able tosucceed in this model through a mechanism thatwas somewhat less transparent than that of freeexpansion: "rst, viscosity supports partition ofthe lattice into patches dominated by one or theother variety. The "tness advantage which altru-ists confer upon their neighborhood permitsthose patches dominated by altruists to establishdenser populations. In the competition that takesplace at patch boundaries, the greater density ofaltruists permits them to counteract the "tnessadvantage of non-altruists in close proximity,and in some cases to prevail.

Population viscosity models, because they as-sume only a two-dimensional geography and lim-ited dispersal speed, are a formalism of greatgenerality. We have seen that altruism may besupported in these models quite generally, andthat the sorts of strong altruism that can besupported are such that the altruistic trait mustcontribute to a higher population density inpatches of altruists than exists in patches of non-altruists. It is not di$cult to think of examples oftraits that have this property, and to contrastthem with other kinds of altruistic traits that donot. A forest canopy is completely closed, andnew trees can only grow up as old ones die out.A new variety of tree that produces more seedsthan the old will take over such a forest viaindividual selection, assuming that each seed hasthe same chance of "nding a vacant site as anyother. However, an altruistic trait that enables alltrees in a neighborhood to produce more seedscannot prevail. But consider a trait that enablesa tree to make more e$cient use of availablelight, such that the same number of seeds isgenerated by a tree covering a smaller patch ofsky. A single tree of this sort has no advantageover other trees in its neighborhood, since itproduces no more seeds than they do. It cannotemerge in a grove via individual selection; how-ever, a grove of constant area will support moresuch trees, and our viscous model predicts thatthis variety will come to dominate a forestthrough a group selection process.

Our model avoids imposing assumptions aboutgenetic relatedness, population density and popula-tion structure; rather, these properties are allowedto emerge spontaneously from local interactions


and geometry. We introduce as a measure of theviability of strong altruism the parameter c, equalto the ratio of total communal bene"t to netindividual cost of altruism. The lower the value ofc at which altruism may sustain itself, the morehospitable we say a model is to the emergence ofaltruism. c is bounded from below at unity, sinceit is impossible under very general assumptionsfor altruism to prevail when its average impacton "tness of self and neighbors is negative. Thesmallest values of c to emerge from our model arebetween 3 and 4. These values are greater thanpredicted by Hamilton's rule (which ignores thevariation in local competition) but may be highlysigni"cant for the evolution of group-level ad-aptations, which do not always require extremeself-sacri"ce (Sober and Wilson, 1998).

Evolutionary theory during the 1960s and 1970swas informed largely by the great body of popula-tion genetics literature created and inspired byFisher (1930) in the middle part of this century. Theapproach which Fisher pioneered treats populationgrowth di!erentially, using continuous functions toapproximate their discrete analogs. Large popula-tion size is implicit to this framework, and randommating is frequently invoked. In the ensuing dec-ades, computers have become ubiquitous andlarge-scale modeling has become practical and gen-erally available. The words &&chaos'' and &&complex-ity'' have emerged into common parlance, and theconventional wisdom has gained respect for thedisparity that frequently arises between systems ofdiscrete, random events and the continuous modelsused to approximate such systems analytically(Wilensky & Reisman, 1998). One of our "ndings(Section 3.3) is that stochasticity itself is a factorfavoring the selection of altruism. If computer sys-tems to support models such as the present one hadbeen conveniently available in 1966, it is possiblethat they might have played a helpful role in thegroup selection debate, which otherwise was proneto abstraction. Perhaps the present availability ofcomputer models treating group selection is su$-cient reason to re-open that debate and re-examineits essential conclusions.

REFERENCES

AXELROD, R. & HAMILTON, W. D. (1981). The evolution ofcooperation. Science 211, 1490}1496.

FISHER, R. A. (1930). ¹he Genetical ¹heory of Natural Selec-tion. New York. Clarendon Press.

INGVARSSON, P. K. (1999). Group selection in density-regu-lated populations revisited. Evol. Ecol. Res. 1, 527}536.

MATSUDA, H. (1987). Conditions for the evolution of altru-ism. In: Animal Socieities: ¹heories and Facts (Ito, Y.,Brown, J. L. & Kikkawa, J., eds), pp. 154}161. New York:Springer-Verlag.

MATSUDA, H., TAMACHI, N., OGITA, N. & SASAKI, A. (1987).A lattice model for population biology. In: Mathematical¹opics in Biology: ¸ecture Notes in Biomathematics(Teramoto, E. & Yamaguti, M., eds), Vol. 71, pp. 154}161.New York: Springer-Verlag.

MAYR, E. (1942). Systematics and the Origin of Species. NewYork: Columbia Univerity Press.

NAKAMARU, H., MATSUDA, H. & IWASA, Y. (1997). Theevolution of cooperation in a lattice-structured popula-tion. J. theor. Biol. 184, 65}81.

NAKAMARU, H., NOGAMI, H. and IWASA, Y. (1998). Score-dependent fertility model for the evolution of cooperationin a lattice. J. theor Biol. 194, 101}124.

NOWAK, M. A. & MAY, R. M. (1992). Evolutionary gamesand spatial chaos. Nature 359, 826}829.

NOWAK, M. A. & MAY, R. M. (1993). The spatial dilemmasof evolution. Int. J. Bifurc. Chaos 3, 35}78.

SOBER, E. & WILSON, D. S. (1998). Unto Others. Cambridge:Harvard University Press.

TAYLOR, P. D. (1992). Altruism in viscous populations*aninclusive "tness model. Evol. Ecol. 6, 352}356.

VAN BAALEN, M. & RAND, D. A. (1998). The unit of selectionin viscous populations and the evolution of altruism. J.theor. biol. 193, 631}648.

WILENSKY, U. & REISMAN, K. (1998). Thinking like a wolf,a sheep, or a "re#y: learning biology through computa-tional theories. In Proc. 2nd Int. Conf. on ComplexitySciences, Nashua, NH.

WILLIAMS, G. (1966). Adaptation and Natural Selection.Princeton, NJ: Princeton University Press.

WILSON, D. S. (1980). ¹he Natural Selection of Populations& Communities. Menlo Park: Benjamin/Cummings.

WILSON, D. S., POLLOCK, G. B. & DUGATKIN, L. A. (1992).Can altruism evolve in purely viscous populations? Evol.Ecol. 6, 331}341.

APPENDIX A

Consider a grid fully occupied by NA

altruistsand N

Snon-altruists, for a total of N

A#N

S"N

occupants.Let the number of A}A neighbor pairs be de-

noted NAA

, and similarly for NAS

and NSS

. Thetotal number of neighbor pairs is N

AA#N

SS#

NAS

"2N, divided as follows:

NAA

#12

NAS

"2NA,

NSS#1

2NAS

"2NS.

Now de"ne o"NA/N as the proportion of A's

in the grid, and let k"12NAA

/NA

be the probability


of "nding an A in a random site adjacent toanother A.

Then the correlation coe$cient can be de"nedin the usual way,

R"

SxyT!SxTSyTSxxT!SxT2

,

where x is any two-valued function, say x"1at a site holding an altruist and x"0 for anon-altruist. y is then the same function evalu-ated at an adjacent site, so that SyT has thesame meaning as SxT, and SxyT is the product ofthe function with itself, averaged over all adjacentpairs.

The choice of values 0 and 1 simpli"es thecomputation, but the pair correlation so de"nedis independent of this choice. SxT"SyT"o fol-lows immediately from the de"nition, andSxyT"ko is derived from counting the A}Apairs. Similarly, SxxT"o because it evaluates to1 for each A and 0 for each S. Hence,

R"

k!o1!o

. (A.1)

Alternatively, R can be de"ned as the di!er-ence in probability of "nding an A when lookingnext to an A and looking next to an S. The formeris k by de"nition. The latter is

P(A D S)"12NAS

/(NSS#1

2NAS

)"NAS

/(4NS)

"o (1!k)/(1!o).

So this alternative de"nition of R is seen to be

R"k!o (1!k)(1!o)

"

k!o1!o

.

APPENDIX B

We modify the WPD model by randomizingthe grid locations of all individuals once in eachgeneration. This may be conceived as a zero-viscosity version of the model, in which thepopulation is thoroughly mixed. Another way tocharacterize the di!erence from the originalWPD model is that clustering of similar types has

been eliminated, so that the neighbor correlation(Hamilton's relatedness r) is zero.

This simpli"ed system may be approachedquasi-analytically. We assume that the grid holdsa random mixture of half A's and half S's, andask, for a given c, what value of b will lead toa lottery result which replicates the equal popula-tion proportions. With the random mixing ofeach generation as speci"ed, this must lead toa stalemate in the competition.

We note that there are just ten types of "ve-sitelotteries, and we analyse them exhaustively, thencombine them in statistical proportions. The tendi!erent possibilities are:

S S A A ASAS AAS AAS AAA AAAS S S S A

S A A A ASSS SSS ASS ASA ASAS S S S A

1 : 5 : 10 : 10 : 5 : 1

The numbers under each pair of diagram aretheir relative frequencies, taken from Pascal's tri-angle. Random shu%ing in each generation as-sures that these frequencies will be attained.Within each vertical pair, the proportions arefurther subdivided 1 : 4 or 4 : 6, re#ecting theprobability that an A or an S will appear at thecentral location. A full accounting of frequenciesfor the ten diagrams is

1 : 1 : 4 : 4 : 6 : 6 : 4 : 4 : 1 : 1 .

The sum of the proportion numbers is 32, sowe proceed using 32 as a denominator to seeka weighted average of the results of the ten lotte-ries. To determine the average "tness of the lot-tery participants, we surround each diagram withindividuals of type X, 50% of whom are A's. X'sare reckoned as contributing b/2 to the "tness ofeach neighboring individual. For example, con-sider the diagram

XXSX

XAASXXSXX


The central A has itself and 1 other A fora neighbor, so its "tness is 1#2b!c. The A inthe wing has 3X neighbors, one A and itself, fora "tness of 1#7b/2!c. The 3 S's each have"tness 1#5/b2 by the same reckoning. So theresult for this lottery is that the probability ofA being victorious is

(1#2b!c)#(1#7b/2!c)(1#2b!c)#(1#7b/2!c)#3(1#5/b2)

"

2#11b/2!2c5#13b!2c

. (B.1)

The above contribution is weighted by 4/32and combined with nine other expressions sim-ilarly derived to create an expression for theproportion of A's in the daughter generation. Thesum of nine fractions (one is zero) with di!erentdenominators is awkward to treat analytically,but a Newton's method solution proceeds with-out di$culty. The expression is set equal to 1/2,and the system may be solved (numerically) forb for a given value of c. The results are

c"0.01, b"0.01666616,

c"0.1, b"0.166376,

c"1, b"1.641464,

c"10, b"16.274024.

For small b and c, the sum of fractions can belinearized, neglecting quadratic and higher terms,to produce a simple expression in b and c, whichcan be solved completely by hand. The answer,

b"5/c3 for small b, c

is apparent from the numerical results.There is a short, convenient but less-rigorous

argument that leads to this same result: Theb that is exported by each altruist is partitionedas 2/5 that bene"ts direct competitors within thataltruist's own lotteries and 3/5 which is scatteredto other lotteries. (This comes from the fact that

1/5 of the altruists will be at the center of theirgroups, and all of their b go to direct competitors,while 4/5 are located in the wings, where only 1/4of their b support direct competitors.) In theaverage "vesome, there are 2.5 A's. Each receives1b from itself and 2b/5 from each of the 1.2 otherA's, for a total "tness of 1#8b/5!c. The 2.5 S'sin the "vesome each receive an average 2b/5 from2.5 A's in the group, for an average "tness of1#b. To specify that the lottery results in anA victory just half the time, we equate the averge"tness of A's and S's in the group: 1#b"1#8b/5!c reduces to b"(5/3)c.

APPENDIX C

Consider a grid "lled with S's and V's (holes),thoroughly homogenized. The proportion of S'sis o, and the proportion of V's is 1-o. The averagelottery will include 5o A's, each with "tness 1, and5(1!o) V's, each with "tness g. Additional lot-tery tickets numbering m are assigned to the V's. Ifthe grid is in steady state, the lottery will result inan S a fraction o of the time. Equating the lotteryodds to o, we have

o"5o/(5o#5(1!o)g#m). (C.1)

When this equation is solved for o, the result iseqn (4), o

ss"1!m/(5(1!g)).

For a binary population of A's and V's,the same logic applies, but with shared bene"tadding an extra ripple. The average A, sur-rounded by 4 cells, receives an altruistic contribu-tion to its "tness of 4ob in addition to the b whichit confers upon itself. As above, there are 5(1!o)V's, each with "tness g, plus a bonus of m. Thesteady-state equation corresponding to eqn (5)for altruists is

o"5o (1!c#(b#4ob)/5)

5o(1!c#(b#4ob)/5#5(1!o)g#m.

(C.2)

Clearing the denominator turns this intoa quadratic equation in o, whose solution iseqn (5) for o.

population viscosity and the evolution of altruism€¦ · evolutionary altruism is de"ned to...

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