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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

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

The term population viscosity refers to limited dispersal, which increases the genetic relatedness

of neighbors. This e!ect both supports the evolution of altruism by focusing the altruists'gifts

on relatives of the altruist, and also limits the extent to which altruism may emerge by exposing

clusters of altruists to sti!er local competition. Previous analyses have emphasized the way in

which these two e!ects can cancel, limiting the viability of altruism. These papers were based

on models in which total population density was held "xed. We present here a class of models

in which population density is permitted to #uctuate, so that patches of altruists are supported

at a higher density than patches of non-altruists. Under these conditions, population viscosity

can support the selection of both weak and strong altruism.

2000 Academic Press

1. Introduction

Evolutionary altruism is de"ned to comprise

traits carried by an individual which confer a "t-

ness advantage upon others. Strong altruism ac-

tually imposes a "tness cost upon the bearer of

the trait, while in weak altruism the disadvantage

is only experienced relative to others (Wilson,

1980). The emergence of altruism is a funda-

mental problem in evolutionary biology. How

can nature select a gene that promotes the "tness

of others, especially when it is at the expense of

the bearer of the gene itself ?

In order for an altruistic trait to be selected, the

bene"ts of the altruism must fall disproportion-

ately on other altruists. The present work applies

computer modeling on a Cartesian grid to

explore one very general paradigm: altruistic

bene"t is dispersed blindly to all occupants of

a geographic neighborhood, while the limited

rate of population di!usion serves to enhance the

proportion of relatives of the altruist within

the bene"tted region. Previous work with this

paradigm seemed to indicate that it would

support weak but not strong altruism; but in the

present model, we see both weak and strong

altruism emerge. The new result depends on

a variable overall population density; in particu-

lar, the communal bene"t of altruism must be

such that local populations of altruists are sup-

ported at a higher density than corresponding

populations of non-altruists.

A number of theoretical frameworks have been

developed to describe ways in which altruism can

evolve, including multilevel selection theory,

inclusive "tness theory, and evolutionary game

theory. Sober & Wilson (1998) review these

frameworks and their relationship to each other.

Game theory emphasizes the ways in which the

bene"ts of altruism may be focused on other

altruists through reciprocal exchange of bene"t

that recognizes and excludes (or punishes) free-

loaders. The &&Prisoner's Dilemma'' game became

one standard model for simulating the evolution-

ary viability of cooperative strategies in various

0022}5193/00/120481#16 $35.00/0 2000 Academic Press

environments. Axelrod & Hamilton (1981) pion-

eered computer simulation in this area, sponsor-

ing a competition among strategies that anointed

a victor in &&Tit-for-Tat'' (TFT), a strategy that

cooperates or betrays each player depending on

that player's most recent behavior toward the

protagonist. The result was greeted with encour-

agement, in that TFT is a strategy that allows for

the possibility of cooperation. Recent analysis

(Nakamaru et al., 1997, 1998) has combined game

theory with geographic e!ects to illuminate the

distinction between bene"ts to survival and to

fertility from cooperation. The game theory para-

digm is limited in applicability to higher animals

in which the brain is developed su$ciently to

support the recognition of individual others.

Inclusive "tness theory is most often applied to

situations in which altruists and the recipients of

altruism are genetically related in a known way.

In some circumstances, Hamilton's (1964) rule

provides an exact measure of the degree to which

altruism can evolve. The rule states that the

maximum degree of altruism that can evolve is

directly proportional to the coe$cient of related-

ness among the altruist and the recipient of its

behavior (b/c'1/r, where bis the bene"tto

recipient, cthe cost to altruist, and rthe 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 useful

when the lowest terms can be shown to be an

adequate approximation to the behavior of the

full system. Van Baalen & Rand (1998) are able to

place weak limits on the evolution of altruism by

focusing on the pair correlation alone. But in the

strongly clustered environments typical both of

the 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 which

the relative advantages of altruism and sel"sh-

ness occur at di!erent levels. For many kinds of

altruistic traits, the levels may be associated with

spatial scales. The advantage of sel"shness is

local*sel"sh individuals are more "t than altru-

ists in their immediate vicinity because they

receive the bene"ts without paying the costs. The

advantage of altruism is realized on a larger

scale*groups of altruists are more "t than

groups of sel"sh individuals. MLS theory seeks

to analyse the balance between the local and

the global processes, and thus to predict circum-

stances under which altruism may evolve. For

altruistic traits whose bene"ts are focused in a

geographic locality, it is necessary that altruists

cluster, 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 of

discrete and spatially separated groups; then the

conditions favoring the evolution of altruism

may be de"ned. First, the groups must vary in

their frequency of altruists: the more variation

the better. Second, the groups must be competing

against one another, with some groups growing

in 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 and

for 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 global

population. In the absence of a global dispersal

mechanism the abundant progeny of altruistic

groups remain in the same locality to compete

only against one other. One ideal population

structure for the evolution of altruism invokes

periods of isolation, during which groups of al-

truists share their mutual bene"t, alternating

with periods of dispersal, allowing for the export

of the altruistic groups'superior productivity.

This scenario, which may be called &&alternating

viscosity'', is explicitly assumed by trait group

models in multilevel selection theory, and impli-

citly assumed by most game theory and inclusive

"tness theory models (Sober and Wilson, 1998

and references therein).

But nature provides abundant examples of

population structures that do not alternate be-

tween isolation and dispersal, and for which nei-

ther explicit kin selection nor reciprocal exchange

are natural models. In order to extend our under-

standing to encompass these phenomena, it is

desirable to avoid explicit assumptions about re-

latedness, grouping and the timing of dispersal,

and to allow all these concepts to emerge as

482 J. MITTELDORF AND D. S. WILSON

a consequence of a general geographic structure.

In our models, population viscosity &&groups'' are

loose associations, where patches tend to be

dominated for a time by one variety or another

only because of the limited speed at which sib-

lings disperse. These models resist analytic treat-

ment. Application of Hamilton's rule encounters

two di$culties: "rst, there is no easy way to

gauge local relatedness; second, a global "tness

measure becomes elusive when reproductive suc-

cess depends on the strength of local competition.

Computer simulation may be an appropriate tool

for approaching this problem's irreducible com-

plexity; in any case, simulation results can serve

as a stimulus to analytic thought, and as one test

of its verisimilitude.

A study by Wilson et al. (1992) (Wilson, Pollock

and Dugatkin, WPD) indicated that computer

models of population viscosity could support the

selection 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 in

which discrete groups approximate the e!ects of

viscosity. Independently, Nowak & May (1992,

1993) described an early grid model. Though they

focused on the mathematical properties of their

model to the exclusion of biological implications,

their results may also be interpreted to permit

weak but not strong altruism to evolve. Like

WPD, they considered only "xed total popula-

tion densities; however, their cost/bene"t scheme

was structured in a somewhat di!erent way, en-

hancing the prospects of altruists at low densities.

In the present paper, the WPD model is taken

as a starting point. Altruism is modeled as having

a cost cand a bene"tbthat contribute linearly to

"tness. The cost is borne by the altruist alone,

and the bene"t shared equally by the altruist and

its four lattice neighbors. Competition for each

lattice site in each (non-overlapping) generation

takes place among a group of "ve neighbors

having a similar geometry. For the purpose of

calculating altruistic contributions, each indi-

vidual is counted as the center of its own neigh-

borhood; thus every lattice site is surrounded by

a local gene pool consisting of itself and four

neighboring sites, with sites further a"eld a!ect-

ing the competition indirectly via their in#uence

on the "tness of the four neighbors.

WPD derive a version of Hamilton's rule ap-

propriate to this model: the "tness of the average

altruist in the global population exceeds that of

the average non-altruist as long as

b/c'1/<, (1)

where <is a statistical analog of Hamilton's

relatedness variable, r. Speci"cally, <is the

average of the pair correlation coe$cient over

the 5 recipients of the altruist's bene"t. One of

these is the self, with correlation 1, and 4 are

lattice neighbors, with correlation R: hence

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

(The pair correlation Rmay be de"ned by the

usual statistical prescription, and can be shown

to equal the di!erence between the probabilities

of "nding an altruist when looking adjacent to

a random altruist and adjacent to a random

non-altruist. This identity is detailed in

Appendix A.)

WPD found that the altruistic trait did not fare

so well in their simulation as this version of

Hamilton's rule would have predicted. They ex-

plained this result with reference to the lattice

viscosity: the same limited dispersal that concen-

trated altruists in patches also impeded their abil-

ity to export their o!spring to distant parts of the

grid. The result may also be understood in terms

of overlap between the group sharing an altruist's

bene"t and the group which competes directly for

each site in the daughter generation: When the

two groups are identical, the altruistic bene"t

applies to all competitors equally, thus it has

exactly no e!ect. When there is no overlap be-

tween these two groups, the above version of

Hamilton's rule predicts accurately the condi-

tions under which altruism may prevail. In the

present model, there is partial overlap, and thus

altruism has some selective value, but not the full

power predicted by Hamilton's rule. (This argu-

ment is made quantitative in Appendix B.)

In the present work, we replicate the 1992

results of WPD and broaden their analysis by

relaxing one key assumption: the requirement of

constant population density. This modi"cation

addresses the issue of the altruists'excess produc-

tivity having no place to go, and does so in a way

POPULATION VISCOSITY EVOLUTION ALTRUISM 483

that is biologically reasonable, at least for some

expressions of altruism. In principle, the number

of occupants of each lattice site could be any

integer, a free variable of the model; but we re-

serve this extension for future investigation. In

our model, each site may be empty, or it may be

occupied by a single altruist or non-altruist. Vari-

able population densities may be observed over

areas su$ciently large that the concept of density

as a continuous variable has some meaning, but

su$ciently small that they are approximately

uniform in composition. The original rules of the

WPD model have been extended in three di!er-

ent ways in order to support the presence of some

empty cells in the long run, without causing the

entire population to vanish.

With each of these three variations, we "nd

that there are ranges of the parameters for which

altruism is a viable competitor to sel"shness, in-

cluding some in which altruism evolves to "x-

ation. Both strong and weak altruism may be

supported under a broad set of assumptions. In

fact, we "nd that the strong/weak distinction has

little place in our understanding of the forces that

determine whether altruism can prevail. We con-

clude that if the bene"t of an altruistic trait en-

ables a larger population density to support itself

in a given environment, then with appropriate

ratios of bene"t to cost, that trait may evolve

based upon population viscosity alone.

Similar conclusions to the above were reached

via a very di!erent path by Ingvarsson (1999).

2. The WPD Model

Two types of asexual individuals exist in the

population, di!ering only in their altruistic (A) vs.

sel"sh (S) traits. The populations are arrayed at

lattice intersections of a two-dimensional Car-

tesian grid, 200;200 sites in most runs, with

opposite edges connected to eliminate boundary

e!ects. Each point on the grid represents a spatial

location that can be occupied by a single indi-

vidual, either Aor S. (In the 1992 WPD model,

vacancies were not permitted.) Interactions are

local, such that the absolute "tness of an

individual depends on its neighborhood of "ve

grid points, which includes its own location

plus the four neighboring points occupying ad-

jacent north, east, south and west position on

the grid:

=

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

b/5,

=

""tness of non-altruist"1#N

b/5, (3)

where N

/5 is the average proportion of altruistic

neighbors (the plus-shaped neighborhood of "ve

includes the self ).

Each altruist contributes b/5 "tness units to

everyone in the neighborhood, including itself, at

a personal cost of c. Sel"sh individuals enjoy the

bene"ts conferred by their altruistic neighbors

without paying the cost. Altruists decrease their

relative "tness within their neighborhood when

c'0 (weak altruism) and decrease their absolute

"tness when c'b/5 (strong altruism).

The model proceeds in "xed, non-overlapping

generations. Every site is vacated at the end of

each generation, and a new occupant is chosen as

a 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 "ve

individuals compete for the site with probabilities

proportional to their "tnesses as computed in

eqn (1). Note that "ve overlapping neighbor-

hoods are used to compute contributions to the

"tnesses of the "ve participants in the lottery that

determines which will seed the central site in the

new generation.

The structure of this model was suggested by

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-

pete to seed the site at the center of that neighbor-

hood in the coming year. The "tness values as

computed in eqn (1) correspond in this picture to

the quantity of seeds produced by each variety of

plant. All the seeds produced by plants in a given

neighborhood form a local gene pool, from which

only a single plant will survive to occupy a given

site. The hypothetical altruistic trait permits all

plants in a neighborhood around the altruist

plant 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 Aand two Sindividuals as neigh-

bors. If c"0.2 and b"0.5, then, from eqn (1), the

focal individual has a "tness of 1!0.2#0.3

"1.1. The "tness of the four neighbors can also

484 J. MITTELDORF AND D. S. WILSON

FIG. 1. A sample segment of a saturated grid. A's are sites

occupied by altruists and S's are non-altruists. The base

"tness of each Sis 1, and the base "tness of each Ais (1!c).

To this base is added a multiple of bcontributed by the

count of A's in the 5-site neighborhood (shaped like &&#'')

centered on the target site.

be calculated from eqns (1), remembering that

each individual occupies the center of its own

neighborhood. Adding the "tnesses of A's and S's

separately, we "nd 3.2 "tness units for Aand 2.6

"tness units for S. These "tness units may stand

for 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, the

probability that the central site will be occupied

by an Ain the next generation is 3.2/5.8"0.552.

Notice that the frequency of Aamong the o!-

spring produced by the neighborhood (0.552) is

a decline from the parental value of 3/5"0.600,

which illustrates the local advantage of sel"sh-

ness. The simulation repeats this procedure for

every 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 in

relation to a particular environment. Enhanced

"tness may manifest as a greater exponential

growth rate in free population expansion, or as

a greater carrying capacity in a saturated niche,

or as a more robust response to any number of

challenges. Our extensions of the WPD model

are designed to permit the representation of such

e!ects with minimal changes in the model's rules.

In the original version, every lattice site was

occupied in every generation by exactly one indi-

vidual. A minimal modi"cation would permit

some sites to be vacant. The rules governing the

lottery then need to be generalized to allow for

vacant sites (V's) as well as A's and S's. The

simplest rule would be to assign a constant

&&"tness'' gto the void; each Vparticipating in

a lottery would receive gtickets, independent of

its surrounding neighborhood. The language de-

scribing V's as a separate species with its own

"tness is a convenient mathematical "ction, akin

to the symmetric treatment of negative electrons

and positive holes in a solid-state physicist's

equations for a semiconductor. The rationale for

this prescription is simply that if a neighborhood

is 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 central

site 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 the

grid 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, which

would reduce the model to an exact replica of

WPD. The "rst two possibilities did not appear

promising; however the third, with g(1 and

a population density that always grows, can be

the basis for an interesting model, as long as

a mechanism is added to replenish the popula-

tion of void sites.

One way to maintain a population of V'son

the grid is to introduce periodic disturbance

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!

a high proportion of individuals in the winter and

permits regrowth in the summer. The geometry

of the disturbance may be that a "xed proportion

of sites is chosen at random for evacuation

(&&uniform culling''); or at the opposite extreme, all

the sites in one area of the grid may be vacated,

while the sites outside this region remain

untouched (&&culling in a compact swath''). Both

these cases have been explored, and results

are 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 the

population is culled, denuding the grid except for

isolated individuals. This maximizes the founder

e+ect ("rst described by Mayr, 1942), as regenera-

tion takes place in patches that are pure Aor

POPULATION VISCOSITY EVOLUTION ALTRUISM 485

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

FIG. 2. Red dots are altruists; green are non-altruists; white dots are empty sites. (a) The grid is initialized in a thoroughly

mixed state, 1/3 each altruists, non-altruists and empty sites. (b) A disturbance is introduced, destroying 95% of the

population at random. Regrowth has just started, and isolated individuals have created small patches of homogeneous type.

(c) Growth into the voids continues, with patches of altruists growing more rapidly than non-altruists. (d) After 5 cycles of

disturbance and regrowth, altruists have come to dominate the landscape. With other parameter combinations, either altruists

or non-altruists may prevail. Higher frequency of distubrances leads to more time spent in free expansion and less

head-to-head confrontation, favoring altruists.

FIG. 3. Disturbances that obliterate large patches of the grid prove to be an e$cient mechanism for segregating the two

varieties. (a) The grid is initialized in a thoroughly mixed state, 1/3 each altruists, non-altruists and empty sites. (b) A square

disturbance is introduced, occupying half the grid area in which all life is destroyed. Notice that after 5 time steps, there has

already been visible consolidation, with patches of altruists and non-altruists beginning to segregate. (c) The void is

re-colonized from the edges. Patches of altruists are able to grow more quickly than non-altruists. Meanwhile, areas away

from the disturbance experience direct red-on-green competition, in which altruists are at a disadvantage. (d) Toward the end

of just one cycle of disturbance and regrowth, the disturbance has had two e!ects: enhancing the fortunes of altruists, and

helping to segregate into separate regions dominated by each variety. If disturbances continue on a regular schedule but

centered at random locations on the grid, altruists may evolve to "xation.

pure S, because they are descended from a single

survivor. As patches begin to regenerate, they

expand into surrounding voids, and direct com-

petition is delayed for a time. In the initial phase

of regrowth, it is the competition Avs. Vand Svs.

Vthat dominates; only later do A's and S's com-

pete head-to-head. But Acommunities, by our

assumptions, can grow faster than Scommuni-

ties; hence the A's may compile a substantial

numerical advantage before direct encounters (A

upon S) become commonplace. This process is

illustrated 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 on

a random point and sized to include half the

grid's total area is completely denuded in each

disturbance event. Regeneration takes place from

the edges, and regions of the boundary that, by

chance, are dominated by A's expand most rap-

idly into the void. This process is illustrated in

Fig. 3. Although the founder e!ect is evoked less

explicitly in this variation than in the uniform

culling case above, culling in a swath nevertheless

proves to be a surprisingly e$cient mechanism

for segregating the population into patches of

pure Aand 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 the

void g(1 was chosen in such a way that popula-

tions of Aor Swould grow and eventually

eliminate voids in the absence of disturbance;

disturbance events were inserted in the model in

order to keep the result from degenerating into

the saturated-grid case, with constant population

density. These models correspond to population

structures that are subject to seasonal cycling, or

may be culled periodically by disturbances. An-

other case worth exploring avoids periodic cull-

ing. Voids are distributed through the grid and

compete more equally with A's and S's, battling

to a steady-state distribution even in the absence

of disturbances. Communities of A's and S'smay

have di!erent e$ciencies in using environmental

resources, which is re#ected in a lower average

vacancy rate in areas dominated by A's than in

corresponding areas dominated by S's. In order

to model such situations, it is necessary to ar-

range parameters so that A,Sand Vcan coexist

in steady state. Figure 4 illustrates a competition

in which denser patches of A's are able to hold

their own against sparser patches of S's.

The simplest assumption about voids, g(1or

g'1, fail to generate the steady state that we are

looking for. The case g"1 may work as a steady

state for a short time, but it is unstable: the grid

evolves eventually either to saturation (no voids)

or extinction (all voids). One simple way to engin-

eer a stable steady state is by introduction of

a second parameter, which we call m.mrepresents

an extra presence in every "ve-way lottery, an

acknowledgement that there is always a "nite

chance that seeding will fail and a site will

become vacant, even if its neighborhood is

saturated in the parent generation. Speci"cally,

486 J. MITTELDORF AND D. S. WILSON

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 are

no local concentrations. There are no large disturbances in this run, but rules of the 5-point lotteries have been modi"ed, so

that there is always a "nite minimum probability that the site will be empty in the coming generation. Some spontaneous

segregation is already visible in (b), allowing colonies of altruists to pursue their local advantage. By square (d), it is apparent

that altruists "ll their regions e$ciently (98.7%) while regions dominated by non-altruists are 50% void. By chance, voids

areas 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 confront

non-altruists directly, the altruists are arrayed along the border with twice the density of non-altruists, and this can vitiate the

advantage 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 in

addition to the "ve competitors comprising the

A's, S's and V's in a given neighborhood, there is

always an extra measure of mlottery tickets as-

signed to V. Intuitively, it is clear that Vcan

never become extinct, i.e. the grid will not become

saturated, because each lottery has a guaranteed

minimum probability of generating a V. But if gis

small enough, then A's and S's will still have an

advantage in each lottery su$cient to insure that

they do not succumb to Vin competition. In fact,

we "nd that with proper choice of gand m,

a stable steady state results, in which V's are

always distributed through the population of A's

and S's, and the populations avoid the extremes

of extinction and saturation.

Exploration of this four-parameter system (b,c,

gand m) is the subject of Section 3.4 below. The

model 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 of

subtle and unexpected phenomena.

Although the full three-species system (S}A}V)

displays such complex behavior, the two-com-

ponent systems A}Vand S}Vare quite amenable

to analysis. For a system with only non-altruists

and voids, the population tends to a steady state

with a density approximated by

oQQ"1!m

(5(1!g)) (4)

The corresponding system with just altruists

and voids seeks a steady-state density approxi-

mated by the quadratic formula

oQQ"!B#(B!4AC

2A(5)

where

A"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"cally

the relative advantage of altruist communities in

oQQ has proven a good predictor of the success of

altruism in the simulated competition.

The four parameters b,c,gand mpermit ample

room to observe an interesting range of vari-

ations in the simulation results. We are most

concerned here with strong altruism, c'b/5,

where the constant-density WPD model found

a consistent advantage for the non-altruists.

When the average density oQQ for patches of altru-

ists is signi"cantly higher than the corresponding

density for patches of non-altruists, we "nd

parameter ranges where strong altruism can

compete successfully. These results are detailed in

Section 3.4.

3. Results

We have seen that much of the interesting

detail relevant to the success of altruism is con-

tained in the part of parameter space close to

b"5c, the boundary that divides weak from

strong altruism. Only for b(5cdoes the indi-

vidual pay a net cost for its altruism, lowering its

own absolute "tness in order to bene"t its neigh-

bors. In order to explode the region of parameter

space near to b"5cfor further study, we intro-

duce the parameter c,de"ned as the quotient of

the total bene"t to others by the net cost to the

self :

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, while

the denominator, c!b/5, is the net cost, since 1/5

of the bene"t devolves upon the self. c"1isan

absolute lower limit in the sense that it is never

possible for altruism to evolve when the net e!ect

on "tness of self plus neighbors is negative. At the

other extreme, the threshold of strong altruism is

5c!b"0, corresponding to in"nite c.cis a con-

venient measure of the hurdle which selection

must leap in order to select strong altruism.

In the results tabulated below, we focus on

&&Stalemate c'' as a measure of each model's

hospitability to altruism. Stalemate cis located as

the end result of multiple trials, seeking that

POPULATION VISCOSITY EVOLUTION ALTRUISM 487

TABLE 1

Stalemate gamma is a measure of the viability of

strong altruism.As cost and bene,t rise in tandem,

strong altruism begins to become competitive,even

in this saturated grid model

Stalemate Stalemate Neighbor

cbcR

0.01 &0.05 Large*0.54

0.03 &0.15 Large*0.59

0.10 &0.50 Large*0.56

0.30 1.475 240 0.55

1.00 4.78 91 0.51

3.00 13.6 43 0.49

10.00 43.9 33 0.47

*Error bars on care wide for these results.

FIG. 5. Gamma is a measure of the minimum bene"t/cost

ratio required for strong altruism to be viable. The de"nition

is such that all c(R correspond to strong altruism, but

lower gamma means that conditions are more favorable for

evolution of altruism. These results show that even with

a saturated grid and no culling, strong altruism can become

viable as cost and bene"t are scaled up in tandem.

combination of band cfor which altruists and

non-altruists have equal prospects, and for which

the two populations may persist for many

simulated generations. Alternatively, there were

some parameter combinations for which the

competition proved unpredictable and the victor

in individual runs not well determined; for these

cases, stalemate cwas derived from a pair (b,c)

for which Aand Swere equally likely to prevail.

Typically, 5}10 trials were used to determine

each value of stalemate c, and for many cases

a smart trial and error routine was used to adjust

bautomatically (for "xed c) depending on

whether Aor Sprevailed in previous trials.

3.1. REPLICATION OF WPD MODEL RESULTS

In Table 1, corresponding to Fig. 5, cwas an

input parameter, held "xed while bwas varied

until a stalemate condition was found between

A's and S's, and cwas computed from band c.

The last column, labeled &&Neighbor R'' is the

autocorrelation coe$cient for pairs of neighbor-

ing sites, and is the same Rfrom eqn (2) above.

It was the primary conclusion of WPD that

b"5c(c"R) was a line of demarcation, at

which altruism was marginally viable against

sel"sh behavior; for larger bthe altruists would

evolve to "xation, and for smaller b, they would

perish. Taylor's (1992) analytic model found the

same boundary b"5cto be critical, and his

derivation relies upon the assumption that band

care both small. With larger grids and longer

runs, we are able both to con"rm that the rule

holds for small b,cand to explore some of the

region where the rule fails. When band care both

small compared to 1, the minimum cto support

altruism approaches in"nity. But for larger band

c, altruism may evolve to "xation for "nite c(even

in a saturated grid). For c"1, altruism will pre-

vail if b'4.78, corresponding to c'91. But if

c"10, then bneeds only be '43.9, correspond-

ing to c'33.

Figure 5 shows that the boundary between

weak and strong altruism is not as fundamental

as suggested by Taylor and WPD. In fact, we

argue in Section 4 below that the demarcation at

b"5cis an artifact of the geometry chosen by

WPD, 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 mvariable is required for this

model (remember that mwas introduced in order

to engineer a stable steady-state population

density less than unity). However, the &&"tness

of the void'' gwas an essential parameter of the

model, and gwas varied from 0.5 to 0.92. Ninety-

"ve percent of the occupied sites were randomly

voided every 50, 100 or 200 generations; in

488 J. MITTELDORF AND D. S. WILSON

FIG. 6. Dispersed culling results. 95% of sites are selected

at random and vacated every 100 generations. Patches of

altruists are able to regrow at a greater rate than non-

altruist patches. The parameter gcontrols the speed of

regrowth. The slower the regrowth, the more e!ective is the

disturbance for promoting the viability of altruism: g"0.50

( ); g"0.60 ( ); g"0.80 ( ); g"0.92 ( ).

FIG. 7. Dispersed culling results. A "xed proportion of

sites 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( ).

FIG. 8. Dispersed culling results. In this "gure, regrowth

rate and frequency of culling are held "xed while proportion

culled is varied from 75 to 98%. More severe culling creates

a more favorable environment for evolution of altruism.

separate trials, 50, 75, 90, 95, 96.5, and 98% were

culled every 100 generations.

In runs with deep population cullings, re-

growth takes place from small, isolated patches

that tend to be dominated by one variety or the

other (the founder e!ect). Much of the competi-

tion is not directly Aagainst Sat their common

border, but rather is a race for free expansion into

unpopulated regions. Higher values of gimply

slower growth rates for both varieties, but as

gapproaches 1, altruists are a!ected relatively

less; this is because at g"1, non-altruist popula-

tions do not grow at all, but altruist populations

may still grow for somewhat higher values (de-

pending on parameters band c). Therefore, high-

er gand deeper, more frequent culling are the

conditions more favorable to the emergence

of altruism, consistent with our model results

(Figs 6}8).

Three trends apparent in these charts are read-

ily explained. First, increasing gwhile holding

other parameters constant has the e!ect of in-

creasing the viability of altruism, i.e. decreasing

the stalemate c. This is because higher gmakes

the void a more formidable competitor, increas-

ing the importance of the altruists'competitive

growth advantage. Second, with gheld constant,

increasing the frequency of culling also makes

altruism more viable. For the entire range of

frequency values in the chart, there is ample time

to regrow and "ll the grid to saturation between

cullings; however, the runs with least frequent

culling (200 generations) spend more than 3/4 of

each cycle in a saturated, direct-competition

phase, while those with the most frequent cullings

(50 generations) spend almost the entire cycle in

a growth phase. In direct competition, altruists

are at a disadvantage, but during the growth

phase, regions with high concentrations of

POPULATION VISCOSITY EVOLUTION ALTRUISM 489

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. Note

that much deeper culling is required in the dispersed case in

order to have comparable e!ect; this is because regrowth

into scattered voids is much more rapid than into large,

vacant blocks. Conversely, culling in a compact block is

much 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 as

in Fig. 7 more frequent culling creates a more favorable

environment for evolution of altruism: 150 gen's( ); 100

gen's( ).

altruists advance and spread faster. Thirdly, and

for similar reasons, increasing the culling percent-

age has a salutary e!ect on the viability of altru-

ism. Thus, stalemate cdecreases monotonically

with decreasing culling factor (the percentage

that remain unculled) (see Figs 9 and 10).

A fourth trend presents more challenge: com-

paring stalemate cfor high and low c,we"nd

some upward trends with cand some downward.

(These are adjacent lines of data all through the

tables.) Where viability of altruism is high, it

tends to be higher when cis lower; however, when

viability of altruism is already low, it tends to be

lowest for low values of c. The reason for the

former trend is that increasing cand bin parallel

increases the growth rate for both A's and S's

(balways overpowers c); hence more of each cycle

is spent in the saturated phase. The latter trend is

just what we observed in Section 3.1: for direct

competition on a saturated grid, higher cand

bmakes altruism more viable. The reason for this

remains unclear.

3.3. POPULATION CULLING IN A COMPACT SWATH

In a variation on the population culling model

in 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 to

half the total grid, but was randomly placed each

time. (The swath was also permitted to straddle

the grid's periodic boundaries, leaving behind

a cross rather than a frame of occupied sites.)

This mechanism corresponds in nature to dam-

age from an extreme weather event, the outbreak

of a parasite, or any other catastrophe which may

devastate a compact geographic region, leaving

outlying areas una!ected.

This scheme was found to be an e$cient seg-

regating force, creating within a few culling cycles

very clearly de"ned regions "lled densely with

homogeneous populations of type Aor S(Fig. 3).

Competition takes place both at boundaries be-

tween Aand Sregions, and also in the rate at

which 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 "nd

a much lower stalemate c(greater hospitability to

strong altruism) in runs with 50% compact cull-

ing than with 50% distributed culling. In fact,

experiments with a compact culling factor of

50% are even more hospitable to altruism than

are uniform culling models in which only 5% are

left standing. Growth into a vacated region from

490 J. MITTELDORF AND D. S. WILSON

FIG. 11. In Section 3.4, evolution of altruism is made

possible by the greater population density in patches of

altruists. 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 of

non-altruists. This "gure demonstrates that the population

elasticity is an important determinant of the hospitability of

any parameter set to the evolution of altruism. The y-axis,

on a log scale, is &&stalemate c'', the ratio of communal bene"t

to individual cost. Low values of cindicate that altruism

may emerge more easily. The x-axis is grid vacancy rate at

stalemate, which represents the potential for population

elasticity.

the edges takes much longer than "lling in an

equivalent area of smaller, distributed voids, and

hence is a more sensitive test of a population's

ability to expand.

Two trends visible in Section 3.2 above may be

detected here as well, and the same reasoning

applies: "rst, increasing gwith everything else

held constant creates an environment more hos-

pitable to altruism because the growth rate of

non-altruist patches is reduced while the growth

rate for altruist patches is less a!ected. Second,

increasing the frequency of culling also enhances

the viability of altruism, by increasing the

fraction of the time that growth is taking place

unimpeded into void regimes.

Another clear trend noted in our results is that

the outcome of individual runs is less predictable

in this section (with culling across a swath). In

other words, the range of cvalues for which Aor

Smay randomly prevail in a given run is largest

with this paradigm. We speculate that the cor-

relation between maximum e!ectiveness of group

selection and minimum predictability is a broad

trend. The problem of group selection may be

stated: how can individual selection, which is

much quicker and more e$cient, be forestalled

long enough for intergroup di!erences to show

their e!ect? This formulation suggests that any

stochastic e!ect, making the short-term outcome

less certain, is likely to decrease the importance of

individual selection relative to group selection.

3.4. ELASTIC POPULATION DENSITIES

IN STEADY STATE

A trait with positive selective value may either

increase the rate of free population expansion, or

else it may augment the steady-state density at

which a population may be supported in a given

environment. Models in this section allow for

variable population densities, with patches of

altruists supported at a higher density than

patches of non-altruists, but the population is not

continually 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. Altruists

will always lose in head-to-head competition

with their sel"sh fellows but it is easy to arrange

for altruists to have a greater steady-state "lling

factor, so that regions dominated by altruism are

more densely populated than regions in which

non-altruists prevail (see Fig. 11).

In Table 2, the left three columns are input

parameters: gand mcontrol the &&"tness of voids''.

cis the gross cost of altruism, before the altruist's

bene"t to himself, b/5, is deducted. The critical

btabulated in the next column is the result of

many computer runs searching for a bvalue

which 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 columns

hold 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 next

column, cis the bene"t : cost ratio corresponding

to cand the critical b, as computed in eqn (6). The

last column is the observed grid "lling factor at

stalemate, the ratio of "lled sites to all grid sites.

Two trends are clearly visible: as grises, voids

are relatively more competitive, and the advant-

age of altruism becomes more important. This

can be seen by comparing each group of "ve lines

(in which gis held constant) with the other two

POPULATION VISCOSITY EVOLUTION ALTRUISM 491

TABLE 2

Results from the variable density model.¸ower cindicates greater viability for

altruists.Higher mincreases the dynamic range of grid vacancy,leading to lower

cvalues.=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

gmcbAltr oQQ Self-oQQ co

0.80 0.20 0.01 0.0395 0.821 0.80 15.0 0.73

0.80 0.20 0.03 0.1225 0.856 0.80 17.8 0.76

0.80 0.20 0.10 0.428 0.920 0.80 24 0.82

0.80 0.20 0.30 1.35 0.967 0.80 36 0.97

0.80 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 0.35

0.90 0.20 0.03 0.0880 0.709 0.60 5.7 0.38

0.90 0.20 0.10 0.3465 0.871 0.60 9.0 0.60

0.90 0.20 0.30 1.21 0.959 0.60 16.7 0.68

0.90 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 0.31

0.95 0.10 0.03 0.084 0.775 0.60 5.1 0.40

0.95 0.10 0.10 0.389 0.937 0.60 14.0 0.75

0.95 0.10 0.30 1.36 0.982 0.60 39 0.89

0.95 0.10 1.00 4.51 0.994 0.60 37 0.88

groups. Altruism can evolve at a lower cwhen gis

higher, because the voids are a bigger factor in

the competition.

The second trend is that within each group,

critical cincreases when both cand bare raised.

The reason for this is less transparent, but we

propose that it can also be understood in terms of

the relative competitive position of the void. Lar-

ger cleads to larger balancing b, which increases

the competitive advantage of both Aand Swith

respect to V. The result is that the stalemated

battle which de"nes critical cis fought in a grid

with a higher "lling factor.

Our results show generally that the addition to

the model of elastic population density creates

a hospitable environment for the evolution of

altruism, even without major disturbance events.

We "nd that altruism can prevail with cratios as

low as 4.7 (for comparison, the WPD adaptation

of Hamilton's rule in this situation would call for

c"2, while the WPD result for a "xed-density

grid corresponds to in"nite c). The prospects of

the altruists improve with decreasing steady-state

occupancy of non-altruists. Figure 11 charts

stalemate cas a function of the measured grid

vacancy rate. The roughly exponential decline of

cwith increasing vacancy rate suggests that this

variable, which is really a surrogate for popula-

tion elasticity, is the primary factor responsible

for the variance of stalemate c.

The original reason for including voids in the

grid was to allow the modeling of elastic popula-

tion densities. The population elasticity has

proven to be a critical factor; indeed Fig. 5 sug-

gests that population elasticity explains most of

the variation in the viability of altruism within

our model. Nevertheless, we have only begun to

explore the population elasticity variable, with

a dynamic range of only a factor 3: the present

model allows each site to be occupied by at most

one individual, so that maximum population

density is 100%; minimum population density is

about 30% because when parameters are ad-

justed for a density lower than about 30% the

population is too fragmented to survive, as it

is vulnerable to random extinction events. The

minimum c"4.7 that we observe may well be

surpassed in model variations that allow for

a population variable at each grid site. This sug-

gests a line of investigation for future research.

492 J. MITTELDORF AND D. S. WILSON

4. Discussion of Results

We have sought to construct models for altru-

ism in which population structure emerges purely

from localized interactions. We began with "xed-

density models after WPD, where the "nding was

that weak but not strong altruism could be sup-

ported, so long as band cwere both small. It is

tempting to seek general meaning in this "nding,

and to seek some correspondence to the well-

known rule that weak altruism but not strong may

evolve in a model where the altruist'sbene"tis

scattered randomly on a large population.

It is a lesson of MLS theory that weak and

strong altruism, as parts of a continuum, can be

understood with a single theory. The question of

whether either one can evolve depends upon the

balance between within-group and between-

group selection (Sober and Wilson, 1998). Our

analysis reveals that the coincidence between the

weak/strong altruism boundary and the min-

imum b/cratio for the emergence of altruism does

not signal any more general relationship, and

that only the narrowest signi"cance should be

attached to this result. Not only do vacancies

allow the evolution of strong altruism, but so also

does increasing the absolute value of band

c(Table 1). Furthermore, if the rules of the grid

are altered such that corners are included in each

neighborhood, then the demarcation becomes

approximately b"13c, though the boundary be-

tween weak and strong altruism is now at b"9c,

for neighborhoods of size 9 (results not tabulated

here). Thus, the signi"cance of the weak/strong

boundary is not even independent of geometry.

We have generalized the WPD model to allow

for variable population density, and found this

indeed to be a crucial factor for the viability of

altruism. We began with models in which groups

of altruists, though unable to compete head-to-

head against non-altruists, were nevertheless able

to grow into a vacant habitat at a faster rate. Not

surprisingly, models in which the populations

were periodically culled, creating empty space to

be repopulated, constituted a friendly environ-

ment for the emergence of altruism. We moved

from there to model variations in which dispersed

vacancies were built into the population, in such

a way that clusters of altruists could exist in

steady state at densities up to 3 times the

corresponding population density for clusters of

non-altruists. We saw that altruists were able to

succeed in this model through a mechanism that

was somewhat less transparent than that of free

expansion: "rst, viscosity supports partition of

the lattice into patches dominated by one or the

other variety. The "tness advantage which altru-

ists confer upon their neighborhood permits

those patches dominated by altruists to establish

denser populations. In the competition that takes

place at patch boundaries, the greater density of

altruists permits them to counteract the "tness

advantage 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 great

generality. We have seen that altruism may be

supported in these models quite generally, and

that the sorts of strong altruism that can be

supported are such that the altruistic trait must

contribute to a higher population density in

patches of altruists than exists in patches of non-

altruists. It is not di$cult to think of examples of

traits that have this property, and to contrast

them with other kinds of altruistic traits that do

not. A forest canopy is completely closed, and

new trees can only grow up as old ones die out.

A new variety of tree that produces more seeds

than the old will take over such a forest via

individual selection, assuming that each seed has

the same chance of "nding a vacant site as any

other. However, an altruistic trait that enables all

trees in a neighborhood to produce more seeds

cannot prevail. But consider a trait that enables

a tree to make more e$cient use of available

light, such that the same number of seeds is

generated by a tree covering a smaller patch of

sky. A single tree of this sort has no advantage

over other trees in its neighborhood, since it

produces no more seeds than they do. It cannot

emerge in a grove via individual selection; how-

ever, a grove of constant area will support more

such trees, and our viscous model predicts that

this variety will come to dominate a forest

through a group selection process.

Our model avoids imposing assumptions about

genetic relatedness, population density and popula-

tion structure; rather, these properties are allowed

to emerge spontaneously from local interactions

POPULATION VISCOSITY EVOLUTION ALTRUISM 493

and geometry. We introduce as a measure of the

viability of strong altruism the parameter c, equal

to the ratio of total communal bene"t to net

individual cost of altruism. The lower the value of

cat which altruism may sustain itself, the more

hospitable we say a model is to the emergence of

altruism. cis bounded from below at unity, since

it is impossible under very general assumptions

for altruism to prevail when its average impact

on "tness of self and neighbors is negative. The

smallest values of cto emerge from our model are

between 3 and 4. These values are greater than

predicted by Hamilton's rule (which ignores the

variation in local competition) but may be highly

signi"cant for the evolution of group-level ad-

aptations, which do not always require extreme

self-sacri"ce (Sober and Wilson, 1998).

Evolutionary theory during the 1960s and 1970s

was informed largely by the great body of popula-

tion genetics literature created and inspired by

Fisher (1930) in the middle part of this century. The

approach which Fisher pioneered treats population

growth di!erentially, using continuous functions to

approximate their discrete analogs. Large popula-

tion size is implicit to this framework, and random

mating is frequently invoked. In the ensuing dec-

ades, computers have become ubiquitous and

large-scale modeling has become practical and gen-

erally available. The words &&chaos'' and &&complex-

ity'' haveemergedintocommonparlance,andthe

conventional wisdom has gained respect for the

disparity that frequently arises between systems of

discrete, random events and the continuous models

used to approximate such systems analytically

(Wilensky & Reisman, 1998). One of our "ndings

(Section 3.3) is that stochasticity itself is a factor

favoring the selection of altruism. If computer sys-

tems to support models such as the present one had

been conveniently available in 1966, it is possible

that they might have played a helpful role in the

groupselectiondebate,whichotherwisewasprone

to abstraction. Perhaps the present availability of

computer models treating group selection is su$-

cient reason to re-open that debate and re-examine

its essential conclusions.

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APPENDIX A

Consider a grid fully occupied by N

altruists

and N1non-altruists, for a total of N

#N1"N

occupants.

Let the number of A}Aneighbor pairs be de-

noted N

, and similarly for N

1 and N

11 . The

total number of neighbor pairs is N

#N11#

N

1"2N, divided as follows:

N

#

N

1"2N

,

N

11#

N

1"2N1.

Now de"ne o"N

/Nas the proportion of A's

in the grid, and let k"

N

/N

be the probability

494 J. MITTELDORF AND D. S. WILSON

of "nding an Ain a random site adjacent to

another A.

Then the correlation coe$cient can be de"ned

in the usual way,

R"1xy2!1x21y2

1xx2!1x2,

where xis any two-valued function, say x"1

at a site holding an altruist and x"0 for a

non-altruist. yis then the same function evalu-

ated at an adjacent site, so that 1y2has the

same meaning as 1x2, and 1xy2is the product of

the function with itself, averaged over all adjacent

pairs.

The choice of values 0 and 1 simpli"es the

computation, but the pair correlation so de"ned

is independent of this choice. 1x2"1y2"ofol-

lows immediately from the de"nition, and

1xy2"ko is derived from counting the A}A

pairs. Similarly, 1xx2"obecause it evaluates to

1 for each Aand 0 for each S. Hence,

R"k!o

1!o. (A.1)

Alternatively, Rcan be de"ned as the di!er-

ence in probability of "nding an Awhen looking

next to an Aand looking next to an S. The former

is kby de"nition. The latter is

P(A"S)"

N

1 /(N

11#

N

1)"N

1/(4N

1)

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

So this alternative de"nition of Ris seen to be

R"k!o(1!k)

(1!o)"k!o

1!o.

APPENDIX B

We modify the WPD model by randomizing

the grid locations of all individuals once in each

generation. This may be conceived as a zero-

viscosity version of the model, in which the

population is thoroughly mixed. Another way to

characterize the di!erence from the original

WPD 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 approached

quasi-analytically. We assume that the grid holds

a random mixture of half A's and half S's, and

ask, for a given c, what value of bwill lead to

a lottery result which replicates the equal popula-

tion proportions. With the random mixing of

each generation as speci"ed, this must lead to

a stalemate in the competition.

We note that there are just ten types of "ve-site

lotteries, and we analyse them exhaustively, then

combine them in statistical proportions. The ten

di!erent possibilities are:

SSAAA

SAS AAS AAS AAA AAA

SSSSA

SAAAA

SSS SSS ASS ASA ASA

SSSSA

1:5:10:10:5:1

The numbers under each pair of diagram are

their 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 are

further subdivided 1 : 4 or 4 : 6, re#ecting the

probability that an Aor an Swill appear at the

central location. A full accounting of frequencies

for the ten diagrams is

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

The sum of the proportion numbers is 32, so

we proceed using 32 as a denominator to seek

a weighted average of the results of the ten lotte-

ries. To determine the average "tness of the lot-

tery participants, we surround each diagram with

individuals of type X, 50% of whom are A's. X's

are reckoned as contributing b/2 to the "tness of

each neighboring individual. For example, con-

sider the diagram

X

XSX

XAASX

XSX

X

POPULATION VISCOSITY EVOLUTION ALTRUISM 495

The central Ahas itself and 1 other Afor

a neighbor, so its "tness is 1#2b!c.TheAin

the wing has 3Xneighbors, one Aand itself, for

a"tness of 1#7b/2!c. The 3 S's each have

"tness 1#5/b2 by the same reckoning. So the

result for this lottery is that the probability of

Abeing victorious is

(1#2b!c)#(1#7b/2!c)

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

"2#11b/2!2c

5#13b!2c. (B.1)

The above contribution is weighted by 4/32

and combined with nine other expressions sim-

ilarly derived to create an expression for the

proportion of A's in the daughter generation. The

sum of nine fractions (one is zero) with di!erent

denominators 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) for

bfor 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 band c, the sum of fractions can be

linearized, neglecting quadratic and higher terms,

to produce a simple expression in band c, which

can 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: The

bthat is exported by each altruist is partitioned

as 2/5 that bene"ts direct competitors within that

altruist's own lotteries and 3/5 which is scattered

to other lotteries. (This comes from the fact that

1/5 of the altruists will be at the center of their

groups, and all of their bgo to direct competitors,

while 4/5 are located in the wings, where only 1/4

of their bsupport direct competitors.) In the

average "vesome, there are 2.5 A's. Each receives

1bfrom itself and 2b/5 from each of the 1.2 other

A's, for a total "tness of 1#8b/5!c. The 2.5 S's

in the "vesome each receive an average 2b/5 from

2.5 A's in the group, for an average "tness of

1#b. To specify that the lottery results in an

Avictory just half the time, we equate the averge

"tness of A's and S's in the group: 1#b"

1#8b/5!creduces to b"(5/3)c.

APPENDIX C

Consider a grid "lled with S's and V's (holes),

thoroughly homogenized. The proportion of S's

is o, and the proportion of V'sis1-o. The average

lottery will include 5oA's, each with "tness 1, and

5(1!o)V's, each with "tness g. Additional lot-

tery tickets numbering mare assigned to the V's. If

the grid is in steady state, the lottery will result in

an S a fraction oof the time. Equating the lottery

odds to o, we have

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

When this equation is solved for o, the result is

eqn (4), oQQ"1!m/(5(1!g)).

For a binary population of A's and V's,

the same logic applies, but with shared bene"t

adding an extra ripple. The average A, sur-

rounded by 4 cells, receives an altruistic contribu-

tion to its "tness of 4obin addition to the bwhich

it confers upon itself. As above, there are 5(1!o)

V's, each with "tness g, plus a bonus of m. The

steady-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 into

a quadratic equation in o, whose solution is

eqn (5) for o.

496 J. MITTELDORF AND D. S. WILSON