Population Viscosity and the Evolution of Altruism

18 Pages·2000·0.9 MB·English

J. theor. Biol. (2000) 204, 481}496
doi:10.1006/jtbi.2000.2007, available online at http://www.idealibrary.com on
Population Viscosity and the Evolution of Altruism
J M * D S W -
OSHUA ITTELDORF AND AVID LOAN ILSON
*Departmentof Biology, ‚eidy‚aboratory, ;niversityof 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)
Thetermpopulationviscosityreferstolimiteddispersal,whichincreasesthegeneticrelatedness
ofneighbors.Thise!ectbothsupportstheevolutionofaltruismbyfocusingthealtruists’gifts
onrelativesofthealtruist,andalsolimitstheextenttowhichaltruismmayemergebyexposing
clustersofaltruiststosti!erlocalcompetition.Previousanalyseshaveemphasizedthewayin
whichthesetwoe!ectscancancel,limitingtheviabilityofaltruism.Thesepaperswerebased
onmodelsinwhichtotalpopulationdensitywasheld"xed.Wepresenthereaclassofmodels
inwhichpopulationdensityispermittedto#uctuate,sothatpatchesofaltruistsaresupported
atahigherdensitythanpatchesofnon-altruists.Undertheseconditions,populationviscosity
can support the selection of both weak and strong altruism.
( 2000AcademicPress
1. Introduction paradigm seemed to indicate that it would
Evolutionary altruism is de"ned to comprise supportweak but not strongaltruism; butin the
traits carriedby an individualwhichconfera "t- present model, we see both weak and strong
ness advantage upon others. Strong altruism ac- altruism emerge. The new result depends on
tually imposes a "tness cost upon the bearer of a variable overall populationdensity; in particu-
thetrait,whileinweakaltruismthedisadvantage lar, the communal bene"t of altruism must be
is only experienced relative to others (Wilson, such that local populations of altruists are sup-
1980). The emergence of altruism is a funda- ported at a higher density than corresponding
mental problem in evolutionary biology. How populations of non-altruists.
cannatureselectagenethatpromotesthe"tness Anumberoftheoreticalframeworkshavebeen
of others, especially when it is at the expense of developedtodescribewaysinwhichaltruismcan
the bearer of the gene itself? evolve, including multilevel selection theory,
Inorderforanaltruistictraittobeselected,the inclusive "tness theory, and evolutionary game
bene"ts of the altruism must fall disproportion- theory. Sober & Wilson (1998) review these
atelyonotheraltruists.Thepresentworkapplies frameworksand their relationship to each other.
computer modeling on a Cartesian grid to Gametheoryemphasizesthewaysinwhichthe
explore one very general paradigm: altruistic bene"ts of altruism may be focused on other
bene"t is dispersed blindly to all occupants of altruists through reciprocal exchange of bene"t
a geographic neighborhood, while the limited that recognizes and excludes (or punishes) free-
rateofpopulationdi!usionservestoenhancethe loaders.The&&Prisoner’sDilemma’’gamebecame
proportion of relatives of the altruist within onestandardmodelforsimulatingtheevolution-
the bene"tted region. Previous work with this ary viability of cooperative strategies in various
0022}5193/00/120481#16$35.00/0 ( 2000AcademicPress
482 J.MITTELDORFAND D.S.WILSON
environments. Axelrod & Hamilton (1981) pion- receivethebene"tswithoutpayingthecosts.The
eered computer simulation in this area, sponsor- advantage of altruism is realized on a larger
ingacompetitionamongstrategiesthatanointed scale*groups of altruists are more "t than
a victor in &&Tit-for-Tat’’ (TFT), a strategy that groups of sel"sh individuals. MLS theory seeks
cooperates or betrays each player depending on to analyse the balance between the local and
that player’s most recent behavior toward the the global processes, and thus to predict circum-
protagonist.The result was greeted with encour- stances under which altruism may evolve. For
agement,inthatTFTisastrategythatallowsfor altruistic traits whose bene"ts are focused in a
the possibility of cooperation. Recent analysis geographic locality, it is necessary that altruists
(Nakamaruetal.,1997,1998)hascombinedgame cluster,sothatthebene"tsconferredbyanaltru-
theory with geographic e!ects to illuminate the ist are more likely to fall upon other altruists.
distinction between bene"ts to survival and to Analysis is simpli"ed by the assumption of
fertilityfromcooperation.Thegametheorypara- discrete and spatially separated groups; then the
digmis limitedinapplicabilityto higheranimals conditions favoring the evolution of altruism
in which the brain is developed su$ciently to may be de"ned. First, the groups must vary in
support the recognition of individual others. their frequency of altruists: the more variation
Inclusive"tnesstheoryismostoftenappliedto thebetter.Second,thegroupsmustbecompeting
situationsinwhich altruists and the recipientsof against one another, with some groups growing
altruism are genetically related in a known way. in size and others shrinking or vanishing.
In some circumstances, Hamilton’s (1964) rule The relative strength of selection within groups
providesanexactmeasureofthedegreetowhich (favoring sel"shness) and between groups (favor-
altruism can evolve. The rule states that the ing altruism) is determined by the relative time-
maximum degree of altruism that can evolve is scales for extinction of groups to take place and
directlyproportionaltothecoe$cientofrelated- forsel"shnesstoevolveto"xationwithingroups.
ness among the altruist and the recipient of its Third, the altruistic groups must be able to ex-
behavior (b/c’1/r, where b is the bene"t to port their genes to the remainder of the global
recipient, c the cost to altruist, and r the coe$c- population. In the absence of a global dispersal
ient of relatedness). mechanism the abundant progeny of altruistic
A related analytic approach borrows a tech- groups remain in the same locality to compete
niquefromstatisticalanalysistodescribecluster- only against one other. One ideal population
ing in terms of correlation coe$cients for pairs, structure for the evolution of altruism invokes
triples, and higher-order geometric combina- periods of isolation, during which groups of al-
tions. This method, "rst applied by Matsuda truists share their mutual bene"t, alternating
(1987) and Matsuda et al. (1987) is most useful with periodsof dispersal,allowingfor the export
when the lowest terms can be shown to be an of the altruistic groups’ superior productivity.
adequate approximation to the behavior of the This scenario, which may be called &&alternating
fullsystem.VanBaalen&Rand(1998)areableto viscosity’’, is explicitly assumed by trait group
placeweaklimitsontheevolutionofaltruismby models in multilevel selection theory, and impli-
focusingonthepair correlationalone.Butinthe citlyassumedbymostgametheoryandinclusive
strongly clustered environments typical both of "tness theory models (Sober and Wilson, 1998
the bioshphere and its models, the pair correla- and references therein).
tion alone is of limited utility. But nature provides abundant examples of
Multilevel selection (MLS) theory treats natu- population structures that do not alternate be-
ral selection as a hierarchical process, in which tween isolation and dispersal, and for which nei-
the relative advantages of altruism and sel"sh- therexplicitkinselectionnorreciprocalexchange
ness occur at di!erent levels. For many kinds of arenaturalmodels.Inordertoextendourunder-
altruistictraits,thelevelsmaybeassociatedwith standing to encompass these phenomena, it is
spatial scales. The advantage of sel"shness is desirabletoavoidexplicitassumptionsabout re-
local*sel"shindividualsaremore"tthanaltru- latedness, grouping and the timing of dispersal,
ists in their immediate vicinity because they and to allow all these concepts to emerge as
POPULATIONVISCOSITYEVOLUTIONALTRUISM 483
a consequenceof a general geographic structure. WPD derive a version of Hamilton’s rule ap-
Inourmodels,populationviscosity&&groups’’are propriatetothismodel:the"tnessoftheaverage
loose associations, where patches tend to be altruist in the global population exceeds that of
dominated for a time by one variety or another the average non-altruist as long as
only because of the limited speed at which sib-
lings disperse. Thesemodels resist analytic treat- b/c’1/<, (1)
ment. Application of Hamilton’s rule encounters
two di$culties: "rst, there is no easy way to where < is a statistical analog of Hamilton’s
gauge local relatedness; second, a global "tness relatedness variable, r. Speci"cally, < is the
measurebecomeselusivewhenreproductivesuc- average of the pair correlation coe$cient over
cessdependsonthestrengthoflocalcompetition. the 5 recipients of the altruist’s bene"t. One of
Computersimulationmaybeanappropriatetool these is the self, with correlation 1, and 4 are
for approaching this problem’s irreducible com- lattice neighbors, with correlation R: hence
plexity; in any case, simulation results can serve
asastimulustoanalyticthought,andasonetest <"(1#4R)/5. (2)
of its verisimilitude.
AstudybyWilsonetal.(1992)(Wilson,Pollock (The pair correlation R may be de"ned by the
and Dugatkin, WPD) indicated that computer usual statistical prescription, and can be shown
modelsofpopulationviscositycouldsupportthe to equal the di!erence between the probabilities
selectionof weak altruism, but that strong altru- of "nding an altruist when looking adjacent to
ism was not competitive in this environment. a random altruist and adjacent to a random
Taylor (1992) reached a similar conclusion with- non-altruist. This identity is detailed in
out computer simulation, analysing a model in Appendix A.)
which discrete groups approximate the e!ects of WPDfoundthatthealtruistictraitdidnotfare
viscosity. Independently, Nowak & May (1992, so well in their simulation as this version of
1993)describedanearlygridmodel.Thoughthey Hamilton’s rule would have predicted. They ex-
focused on the mathematical properties of their plained this result with reference to the lattice
modeltotheexclusionofbiologicalimplications, viscosity:the samelimiteddispersalthat concen-
their results may also be interpreted to permit tratedaltruistsinpatchesalsoimpededtheirabil-
weak but not strong altruism to evolve. Like itytoexporttheiro!springtodistantpartsofthe
WPD, they considered only "xed total popula- grid.The resultmayalso beunderstoodinterms
tion densities;however,their cost/bene"tscheme ofoverlapbetweenthegroupsharinganaltruist’s
was structured in a somewhat di!erent way, en- bene"tandthegroupwhichcompetesdirectlyfor
hancingtheprospectsofaltruistsatlowdensities. each site in the daughter generation: When the
In the presentpaper, the WPDmodelis taken two groups are identical, the altruistic bene"t
asastartingpoint.Altruismismodeledashaving applies to all competitors equally, thus it has
acostcandabene"tbthatcontributelinearlyto exactly no e!ect. When there is no overlap be-
"tness. The cost is borne by the altruist alone, tween these two groups, the above version of
andthebene"tsharedequallybythealtruistand Hamilton’s rule predicts accurately the condi-
its four lattice neighbors. Competition for each tions under which altruism may prevail. In the
lattice site in each (non-overlapping) generation present model, there is partial overlap, and thus
takes place among a group of "ve neighbors altruismhassomeselectivevalue,butnotthefull
having a similar geometry. For the purpose of power predicted by Hamilton’s rule. (This argu-
calculating altruistic contributions, each indi- ment is made quantitative in Appendix B.)
vidual is counted as the center of its own neigh- In the present work, we replicate the 1992
borhood; thus every lattice site is surrounded by results of WPD and broaden their analysis by
a local gene pool consisting of itself and four relaxing one key assumption: the requirement of
neighboring sites, with sites further a"eld a!ect- constant population density. This modi"cation
ing the competition indirectly via their in#uence addressestheissueofthealtruists’excessproduc-
on the "tness of the four neighbors. tivityhavingnoplacetogo,anddoessoinaway
484 J.MITTELDORFAND D.S.WILSON
that is biologically reasonable, at least for some the grid:
expressions of altruism. In principle, the number
of occupants of each lattice site could be any =""tness of altruist"1!c#N b/5,
S A
integer, a free variable of the model; but we re-
serve this extension for future investigation. In =""tness of non-altruist"1#N b/5, (3)
A A
our model, each site may be empty, or it may be
occupiedbyasinglealtruistornon-altruist.Vari- whereN /5istheaverageproportionofaltruistic
A
able population densities may be observed over neighbors (the plus-shaped neighborhood of "ve
areassu$cientlylargethattheconceptofdensity includes the self).
as a continuous variable has some meaning, but Each altruist contributes b/5 "tness units to
su$ciently small that they are approximately everyoneintheneighborhood,includingitself,at
uniformincomposition.Theoriginalrulesofthe a personal cost of c. Sel"sh individualsenjoy the
WPD model have been extended in three di!er- bene"ts conferred by their altruistic neighbors
entwaysinordertosupportthepresenceofsome without paying the cost. Altruists decrease their
empty cells in the long run, without causing the relative "tness within their neighborhood when
entire population to vanish. c’0(weakaltruism)anddecreasetheirabsolute
With each of these three variations, we "nd "tness when c’b/5 (strong altruism).
thattherearerangesoftheparametersforwhich The model proceeds in "xed, non-overlapping
altruism is a viable competitor to sel"shness, in- generations. Every site is vacated at the end of
cluding some in which altruism evolves to "x- eachgeneration,andanewoccupantischosenas
ation. Both strong and weak altruism may be acloneofoneof"veindividualsintheneighbor-
supported under a broad set of assumptions. In hoodcomprisingthecentralsiteitself andneigh-
fact,we"ndthatthestrong/weakdistinctionhas borstothenorth,east,south,andwest.These"ve
littleplaceinourunderstandingoftheforcesthat individualscompeteforthesitewithprobabilities
determinewhetheraltruismcanprevail.Wecon- proportional to their "tnesses as computed in
clude that if the bene"t of an altruistic trait en- eqn (1). Note that "ve overlapping neighbor-
ablesalargerpopulationdensitytosupportitself hoods are used to compute contributions to the
in a given environment, then with appropriate "tnessesofthe"veparticipantsinthelotterythat
ratios of bene"t to cost, that trait may evolve determines which will seed the central site in the
based upon population viscosity alone. new generation.
Similarconclusionstothe abovewerereached The structure of this model was suggested by
via a very di!erent path by Ingvarsson (1999). a population of annual plants. A given area will
support a "nite number of such plants in each
year, and all the plants in a neighborhood com-
2. The WPD Model
petetoseedthesiteatthecenterofthatneighbor-
Two types of asexual individuals exist in the hood in the coming year. The "tness values as
population,di!eringonlyintheiraltruistic(A)vs. computedineqn(1)correspondinthispictureto
sel"sh (S) traits. The populations are arrayed at thequantityofseedsproducedbyeachvarietyof
lattice intersections of a two-dimensional Car- plant.Alltheseedsproducedbyplantsinagiven
tesian grid, 200]200 sites in most runs, with neighborhoodformalocalgenepool,fromwhich
opposite edges connected to eliminate boundary onlyasingleplant willsurvivetooccupya given
e!ects.Eachpointonthegridrepresentsaspatial site. The hypothetical altruistic trait permits all
location that can be occupied by a single indi- plants in a neighborhood around the altruist
vidual, either A or S. (In the 1992 WPD model, plant to generate more seeds.
vacancies were not permitted.) Interactions are HereisanumericalillustrationbasedonFig.1:
local, such that the absolute "tness of an the position in the center is occupied by an A,
individual depends on its neighborhood of "ve whichhastwoA andtwoSindividualsasneigh-
grid points, which includes its own location bors.Ifc"0.2andb"0.5,then,fromeqn(1),the
plus the four neighboring points occupying ad- focal individual has a "tness of 1!0.2#0.3
jacent north, east, south and west position on "1.1. The "tness of the four neighbors can also
POPULATIONVISCOSITYEVOLUTIONALTRUISM 485
simplest rule would be to assign a constant
&&"tness’’ g to the void; each V participating in
a lottery would receive g tickets, independent of
its surrounding neighborhood.The language de-
scribing V’s as a separate species with its own
"tness is a convenientmathematical"ction, akin
to the symmetric treatment of negative electrons
FIG. 1. Asamplesegmentofasaturatedgrid.A’saresites
and positive holes in a solid-state physicist’s
occupied by altruists and S’s are non-altruists. The base
"tnessofeachSis1,andthebase"tnessofeachAis(1!c). equationsfora semiconductor.The rationale for
To this base is added a multiple of b contributed by the thisprescriptionissimplythatifaneighborhood
countof A’s in the 5-site neighborhood(shaped like&&#’’)
is less than fully occupied in the parent genera-
centeredonthe targetsite.
tion, there is a "nite probability that reproduc-
tionwillfailtocreateanyoccupantforitscentral
site in the daughter generation.
be calculated from eqns (1), remembering that In two-way competition between S’s and V’s,
each individual occupies the center of its own theV’shavetheupperhandwheng’1,andthe
neighborhood.Addingthe"tnessesofA’sandS’s grid quickly empties out. For g"1, the popula-
separately, we "nd 3.2 "tness units for A and 2.6 tionintwo-waycompetitionismarginallyunsta-
"tness units for S. These "tness units may stand ble,andeventuallythegrid evolvesto saturation
forseedsintheneighborhood,oritisalsoconve- (allS)orextinction(allV).Withg(1,thepopu-
nient to think of them as tickets for a lottery. lation grows always toward saturation, which
SinceA’shold3.2lotteryticketsto2.6forS’s,the would reduce the model to an exact replica of
probability that the central site will be occupied WPD. The "rst two possibilities did not appear
byan Ain the nextgenerationis 3.2/5.8"0.552. promising; however the third, with g(1 and
Notice that the frequency of A among the o!- a population density that always grows, can be
spring produced by the neighborhood (0.552) is the basis for an interesting model, as long as
a decline from the parental value of 3/5"0.600, a mechanism is added to replenish the popula-
which illustrates the local advantage of sel"sh- tion of void sites.
ness. The simulation repeats this procedure for One way to maintain a population of V’s on
everypositionof thegridto determinethepopu- the grid is to introduce periodic disturbance
lation array for the next generation. events. On a "xed schedule, a percentage of all
grid sites is vacated, as if by a disease or natural
catastrophe, or a seasonal change that kills o!
2.1. MODELSWITHVARIABLEPOPULATIONDENSITIES: ahighproportionofindividualsinthewinterand
DISTURBANCEANDREGROWTH permits regrowth in the summer. The geometry
Fitness of an organism has meaning only in ofthedisturbancemaybethata"xedproportion
relation to a particular environment. Enhanced of sites is chosen at random for evacuation
"tness may manifest as a greater exponential (&&uniformculling’’);orattheoppositeextreme,all
growth rate in free population expansion, or as the sites in one area of the grid may be vacated,
a greater carrying capacity in a saturated niche, while the sites outside this region remain
or as a more robust response to any number of untouched (&&culling in a compact swath’’). Both
challenges. Our extensions of the WPD model these cases have been explored, and results
aredesignedtopermittherepresentationofsuch are reported below in Section 3.2 and 3.3,
e!ectswithminimalchangesinthemodel’srules. respectively.
In the original version, every lattice site was For the uniform culling case, we choose para-
occupiedineverygenerationbyexactlyoneindi- meters such that a high proportion of the
vidual. A minimal modi"cation would permit populationisculled,denudingthegridexceptfor
some sites to be vacant. The rules governing the isolated individuals. This maximizes the founder
lottery then need to be generalized to allow for e+ect("rstdescribedbyMayr,1942),asregenera-
vacant sites (V’s) as well as A’s and S’s. The tion takes place in patches that are pure A or
486 J.MITTELDORFAND D.S.WILSON
pureS, because they are descended from a single eliminate voids in the absence of disturbance;
survivor. As patches begin to regenerate, they disturbance events were inserted in the model in
expand into surrounding voids, and direct com- order to keep the result from degenerating into
petitionis delayedfor a time.In the initialphase thesaturated-gridcase,withconstantpopulation
ofregrowth,itisthecompetitionAvs.VandSvs. density. These models correspond to population
Vthat dominates;onlylaterdo A’s and S’s com- structuresthatare subjectto seasonalcycling,or
pete head-to-head. But A communities, by our may be culled periodically by disturbances. An-
assumptions, can grow faster than S communi- other case worth exploring avoids periodic cull-
ties; hence the A’s may compile a substantial ing. Voids are distributed through the grid and
numericaladvantage before direct encounters (A compete more equally with A’s and S’s, battling
upon S) become commonplace. This process is toasteady-statedistributioneveninthe absence
illustratedinFig.2.Detailedresultsforthisvari- ofdisturbances.CommunitiesofA’s andS’smay
ation are reported in Section 3.2. have di!erente$ciencies in using environmental
The other possibility we have explored is cull- resources, which is re#ected in a lower average
ing in a compact swath. A square centered on vacancy rate in areas dominated by A’s than in
a random point and sized to include half the corresponding areas dominated by S’s. In order
grid’s total area is completely denuded in each to model such situations, it is necessary to ar-
disturbanceevent.Regenerationtakesplacefrom range parameters so that A, S and V can coexist
the edges, and regions of the boundary that, by insteadystate.Figure4 illustratesa competition
chance, are dominated by A’s expand most rap- in which denser patches of A’s are able to hold
idly into the void. This process is illustrated in their own against sparser patches of S’s.
Fig. 3. Althoughthe founder e!ect is evokedless Thesimplestassumptionaboutvoids,g(1or
explicitly in this variation than in the uniform g’1,failtogeneratethesteadystatethatweare
cullingcaseabove,cullinginaswathnevertheless lookingfor.Thecaseg"1mayworkasasteady
proves to be a surprisingly e$cient mechanism state for a short time, but it is unstable: the grid
for segregating the population into patches of evolveseventuallyeither to saturation (no voids)
pure A and pure S. Detailed results for this vari- orextinction(allvoids).Onesimplewaytoengin-
ation are reported in Section 3.3. eer a stable steady state is by introduction of
asecondparameter,whichwecallm.mrepresents
2.2. MODELSWITHVARIABLEPOPULATIONDENSITIES: an extra presence in every "ve-way lottery, an
UNSATURATEDSTEADYSTATES acknowledgement that there is always a "nite
Intheabovemodelvariations,the"tnessofthe chance that seeding will fail and a site will
voidg(1waschoseninsuchawaythatpopula- become vacant, even if its neighborhood is
tions of A or S would grow and eventually saturated in the parent generation. Speci"cally,
&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&"
FIG. 2. Reddotsarealtruists;greenarenon-altruists;whitedotsareemptysites.(a)Thegridisinitializedinathoroughly
mixed state, 1/3 each altruists, non-altruists and empty sites. (b) A disturbance is introduced, destroying 95% of the
populationatrandom.Regrowthhasjuststarted,andisolatedindividualshavecreatedsmallpatchesofhomogeneoustype.
(c)Growthintothevoidscontinues,withpatchesofaltruistsgrowingmorerapidlythannon-altruists.(d)After5cyclesof
disturbanceandregrowth,altruistshavecometodominatethelandscape.Withotherparametercombinations,eitheraltruists
or non-altruists may prevail. Higher frequency of distubrances leads to more time spent in free expansion and less
head-to-headconfrontation,favoringaltruists.
FIG. 3. Disturbancesthatobliteratelargepatchesofthegridprovetobeane$cientmechanismforsegregatingthetwo
varieties.(a)Thegridisinitializedinathoroughlymixedstate,1/3eachaltruists,non-altruistsandemptysites.(b)Asquare
disturbanceisintroduced,occupyinghalfthegridareainwhichalllifeisdestroyed.Noticethatafter5timesteps,therehas
already been visible consolidation, with patches of altruists and non-altruists beginning to segregate. (c) The void is
re-colonizedfromtheedges.Patchesofaltruistsareableto growmorequicklythan non-altruists.Meanwhile,areasaway
fromthedisturbanceexperiencedirectred-on-greencompetition,inwhichaltruistsareatadisadvantage.(d)Towardtheend
ofjustonecycleofdisturbanceandregrowth,thedisturbancehashadtwoe!ects:enhancingthefortunesofaltruists,and
helping to segregate into separate regions dominated by each variety. If disturbances continue on a regular schedule but
centeredatrandomlocationsonthegrid,altruistsmayevolveto"xation.
FIG. 2. (Captionopposite)
FIG. 3. (Captionopposite)
FIG. 4. Thisgridisinitializedwith1/3eachaltruists,non-altruistsandemptysites,randomizedpointwisesothatthereare
nolocalconcentrations.Therearenolargedisturbancesinthisrun,butrulesofthe5-pointlotterieshavebeenmodi"ed,so
thatthere is alwaysa"nite minimumprobabilitythatthesitewill beemptyin thecominggeneration.Somespontaneous
segregationisalreadyvisiblein(b),allowingcoloniesofaltruiststopursuetheirlocaladvantage.Bysquare(d),itisapparent
thataltruists "lltheirregionse$ciently(98.7%)whileregionsdominatedbynon-altruistsare50%void.Bychance,voids
areassometimesappearas&&mini-disturbances’’nearthebordersbetweenregionsdominatedbyaltruistsandnon-altruists,
and when they do, the altruists expand rapidly into the empty region. Even in border regions where altruist confront
non-altruistsdirectly,thealtruistsarearrayedalongtheborderwithtwicethedensityofnon-altruists,andthiscanvitiatethe
advantageofthefreeloaders.Thealtruistsmaycometodominatethecompetition.Inthisexample,altruistswillevolveto
"xationin severalhundredtimesteps.
POPULATIONVISCOSITYEVOLUTIONALTRUISM 487
the rules of the lottery are modi"ed so that in parameter combinations to explore; speci"cally
addition to the "ve competitors comprising the the relative advantageof altruist communitiesin
A’s,S’sandV’s inagivenneighborhood,thereis o has proven a goodpredictor of the success of
ss
always an extra measure of m lottery tickets as- altruism in the simulated competition.
signed to V. Intuitively, it is clear that V can Thefourparametersb,c,gandmpermitample
neverbecomeextinct,i.e.thegridwillnotbecome room to observe an interesting range of vari-
saturated, because each lottery has a guaranteed ations in the simulation results. We are most
minimumprobabilityofgeneratingaV.Butifgis concerned here with strong altruism, c’b/5,
small enough, then A’s and S’s will still have an where the constant-density WPD model found
advantageineachlotterysu$cienttoinsurethat a consistent advantage for the non-altruists.
theydonotsuccumbtoVincompetition.Infact, Whentheaveragedensityo forpatchesofaltru-
ss
we "nd that with proper choice of g and m, istsissigni"cantlyhigherthanthecorresponding
a stable steady state results, in which V’s are density for patches of non-altruists, we "nd
alwaysdistributedthroughthe populationofA’s parameter ranges where strong altruism can
and S’s, and the populations avoid the extremes competesuccessfully.Theseresultsaredetailedin
of extinction and saturation. Section 3.4.
Explorationofthisfour-parametersystem(b,c,
g and m) is the subject of Section 3.4 below. The
3. Results
modelsupports the emergence of altruism in ap-
propriate parameter ranges, and favors quasi- We have seen that much of the interesting
periodicpopulationoscillationsforotherranges. detail relevant to the success of altruism is con-
Overall, we have found it to be a rich lode of tained in the part of parameter space close to
subtle and unexpected phenomena. b"5c, the boundary that divides weak from
Althoughthefullthree-speciessystem(S}A}V) strong altruism. Only for b(5c does the indi-
displays such complex behavior, the two-com- vidualpayanetcostforitsaltruism,loweringits
ponentsystemsA}VandS}Varequiteamenable ownabsolute"tnessinordertobene"titsneigh-
to analysis. For a system with only non-altruists bors.Inordertoexplodetheregionofparameter
and voids,the populationtends to a steady state space near to b"5c for further study, we intro-
with a density approximated by duce the parameter c, de"ned as the quotient of
the total bene"t to others by the net cost to the
m self:
o "1! (4)
ss (5(1!g))
4b/5 4b
The corresponding system with just altruists c" " (6)
c!b/5 (5c!b).
and voids seeks a steady-state density approxi-
mated by the quadratic formula
The numerator, 4b/5, is the part of each altru-
ist’s bene"t that is exported to neighbors, while
!B#J B2!4AC thedenominator,c!b/5,isthenetcost,since1/5
o " (5)
ss 2A of the bene"t devolves upon the self. c"1 is an
absolute lower limit in the sense that it is never
possibleforaltruismtoevolvewhenthenete!ect
where
A"4b, on"tnessofselfplusneighborsisnegative.Atthe
otherextreme,thethresholdofstrongaltruismis
5c!b"0,correspondingtoin"nitec.cisacon-
B"5(1!c!g)!3b,
venient measure of the hurdle which selection
must leap in order to select strong altruism.
C"!5(1!c!g)#m!b. In the results tabulated below, we focus on
&&Stalemate c’’ as a measure of each model’s
These formulae are derived in Appendix C. hospitabilitytoaltruism.Stalematecislocatedas
Theyhaveservedusasguidelinesforselectionof the end result of multiple trials, seeking that
488 J.MITTELDORFAND D.S.WILSON
combination of b and c for which altruists and
non-altruistshaveequalprospects,andforwhich
the two populations may persist for many
simulated generations. Alternatively, there were
some parameter combinations for which the
competitionprovedunpredictableandthevictor
in individual runs not well determined; for these
cases, 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 determine
each value of stalemate c, and for many cases
asmarttrialanderrorroutinewasusedtoadjust
b automatically (for "xed c) depending on
whether A or S prevailed in previous trials.
FIG. 5. Gammaisameasureoftheminimumbene"t/cost
ratiorequiredforstrongaltruismtobeviable.Thede"nition
is such that all c(R correspond to strong altruism, but
3.1. REPLICATIONOFWPDMODELRESULTS
lowergammameansthatconditionsaremorefavorablefor
In Table 1, corresponding to Fig. 5, c was an evolution of altruism. These results show that even with
asaturatedgridandnoculling,strongaltruismcanbecome
input parameter, held "xed while b was varied
viableascostandbene"tarescaledupintandem.
until a stalemate condition was found between
A’s and S’s, and c was computed from b and c.
holds for small b, c and to explore some of the
The last column, labeled &&Neighbor R’’ is the
regionwheretherulefails.Whenbandcareboth
autocorrelation coe$cient for pairs of neighbor-
small compared to 1, the minimum c to support
ing sites, and is the same R from eqn (2) above.
altruismapproachesin"nity.Butforlargerband
It was the primary conclusion of WPD that
b"5c (c"R) was a line of demarcation, at c,altruismmayevolveto"xationfor"nitec(even
in a saturated grid). For c"1, altruism will pre-
which altruism was marginally viable against
vail if b’4.78, corresponding to c’91. But if
sel"sh behavior; for larger b the altruists would
c"10,thenbneedsonlybe’43.9,correspond-
evolve to "xation, and for smaller b, they would
ing to c’33.
perish. Taylor’s (1992) analytic model found the
same boundary b"5c to be critical, and his Figure 5 shows that the boundary between
weak and strong altruism is not as fundamental
derivationreliesuponthe assumptionthatband
as suggested by Taylor and WPD. In fact, we
c are both small. With larger grids and longer
argueinSection4belowthatthedemarcationat
runs, we are able both to con"rm that the rule
b"5c is an artifact of the geometry chosen by
WPD, and has no deep signi"cance.
TABLE 1
Stalemate gamma is a measure of the viability of
strongaltruism.Ascostandbene,triseintandem,
3.2. PERIODICUNIFORMCULLING
strongaltruismbeginstobecomecompetitive,even
Intheseruns,ahighpercentageofthepopula-
in this saturated grid model
tionwas culledat randomon aregularschedule,
such that much of the action consisted of re-
Stalemate Stalemate Neighbor
growth from isolated founders into empty sur-
c b c R
roundings. No m variable is required for this
0.01 &0.05 Large* 0.54 model(rememberthatm wasintroducedinorder
0.03 &0.15 Large* 0.59
to engineer a stable steady-state population
0.10 &0.50 Large* 0.56
density less than unity). However, the &&"tness
0.30 1.475 240 0.55
1.00 4.78 91 0.51 of the void’’ g was an essential parameter of the
3.00 13.6 43 0.49 model,andgwasvariedfrom0.5to0.92.Ninety-
10.00 43.9 33 0.47
"ve percent of the occupied sites were randomly
*Errorbarsonc arewidefortheseresults. voided every 50, 100 or 200 generations; in