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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