ARTICLE IN PRESS

Journal of Theoretical Biology 226 (2004) 421–428

Evolution of mutualism through spatial effects
Norio Yamamura*, Masahiko Higashi, Narayan Behera1, Joe Yuichiro Wakano2
Center for Ecological Research, Kyoto University, Otsu 520-2113, Japan
Received 15 April 2003; received in revised form 9 September 2003; accepted 26 September 2003

Abstract
Mutualism among species is ubiquitous in natural ecosystems but its evolution is not well understood. We provided a simple
lattice model to clarify the importance of spatial structure for the evolution of mutualism. We assumed reproductive rates of two
species are modiﬁed through interaction between species and examine conditions where mutualists of both species, that give somebeneﬁt to the other species with their own cost, invade non-mutualists populations. When dispersal of offspring is unlimited, we
veriﬁed the evolution of mutualism is impossible under any condition. On the other hand, when the dispersal is limited toneighboring lattice sites, mutualists can invade if the ratio of cost to beneﬁt is low and the intrinsic reproductive rate is low in casewhere the parameter values are symmetric between species. Under the same conditions, non-mutualists cannot invade mutualist
populations, that is, the latter are evolutionarily stable. In case of asymmetric parameters, mutualists tend to invade if the averagevalue of costs to two species is low or that of beneﬁts is high, and if the intrinsic reproductive rate is low for one of the two species. A
mechanistic explanation of why mutualists increase when the dispersal is limited is given by showing that mutualist pairs of the two
species at the same lattice site rapidly increase at the initial phase of the invasion.
r 2003 Elsevier Ltd. All rights reserved.
Keywords: Mutualism evolution; Mutual altruism; Spatial effects; Dual lattice

1. Introduction
Species interact constantly in diverse ways in ecosystems. In mutualistic interaction, one of the species
provides some kind of ‘‘service’’ that its partner species
cannot provide for itself and receives some kind of‘‘reward’’ in return. The two mutualistic species survive
and reproduce by helping each other. Such mutualism
plays a fundamental role in all ecosystems (Boucher,
1985; Smith and Douglas, 1987; Bronstein, 1994).
Evolution of mutualism has been studied as one of themost important problems in ecology (Doebeli andKnowlton, 1998; Herre et al., 1999; Hoeksema andBruna, 2000; de Mazancourt et al., 2001; van Baalen
and Jansen, 2001).
One important factor to promote the evolution ofmutualism has been suggested to be vertical transmis*Corresponding author. Tel.:/fax: +81-77-549-8257.
E-mail address: yamamura@ecology.kyoto-u.ac.jp(N. Yamamura).
1
Present address: Institute of Bioinformatics and Applied Biotechnology, ITPL, Whiteﬁeld Road, Bangalore-560 066, India.
2
Present address: School of Science, University of Tokyo, Hongo
7-3-1, Bunkyo-ku, Tokyo, Japan.
0022-5193/$ - see front matter r 2003 Elsevier Ltd. All rights reserved.
doi:10.1016/j.jtbi.2003.09.016

sion of symbionts through hosts, which assures a long
relationship between offspring of the interacting individuals of the two species over generations (MaynardSmith, 1991; Yamamura, 1993, 1996; Genkai-Kato andYamamura, 1999). If offspring of the symbiont interact
with offspring of the host, the symbiont could drive agreater advantage by helping rather than exploiting theirhosts. Even without vertical transmission, the interaction between individuals of two species will continue
over generations if dispersal of the two species is
spatially limited. Doebeli and Knowlton (1998) showed
the importance of spatial structure for mutualism toevolve in their Iterated Prisoner’s Dilemma (IPD) gamemodel.
The IPD was proposed (Axelrod and Hamilton, 1981)
and had been analysed (Nowak and May, 1992;
Nakamaru et al., 1997; Wakano and Yamamura 2001)
as a theoretical method to investigate the phenomena ofcooperation in a single species, mainly by simulating
higher animals’ or human’s cognitive processes. Here, inorder to clarify the effect of spatial structure, per se, on
the evolution of mutualism between different species, weapply a simpler spatial model of cost and beneﬁt, used
for the analysis of the evolution of altruism (Matsuda,

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N. Yamamura et al. / Journal of Theoretical Biology 226 (2004) 421–428

1987; Taylor, 1992; Wilson et al., 1992), without any
complex interaction between partners, as assumed in the
IPD game. It is a new challenge that we apply themutual altruistic behavior for interaction of different
species, although evolution of altruism within a single
species has intensively been studied (for recent studies,see Irwin and Taylor, 2001; Le Galliard et al., 2003).
Our model includes several independent parameters:
beneﬁt to one species from another species, cost to eachspecies, and intrinsic reproductive rates that are theprimary determinants of population densities of both
species. First, we show analytically that the evolution ofmutualism is impossible in this simple model whendispersal is unlimited; that is, when there is no spatial
structure. Next, we investigate numerically the conditions of parameter values under which mutualism canevolve when dispersal is limited, where both symmetricand asymmetric parameter values between species are
examined.

rates are not. When a lattice site is occupied by amutualist of species A and the corresponding site in the‘‘dual’’ lattice also has a mutualist of species B, theirreproductive rates are rA þ bA  cA and rB þ bB  cB : If
the site in the dual site is occupied by a non-mutualist of
species B, their reproductive rates are rA  cA and rB þ
bB : Those for the contrary case are rA þ bA and rB  cB :That is, a mutualist pays a cost without discrimination
whether the opponent is a mutualist or non-mutualist. If
a mutualist is associated with an empty site, thereproductive rate of the mutualist is rA  cA for species
A and rB  cB for species B when mutualists pay costs
before assuring whether there is an opponent at the dual
lattice site or not (Case I). They are rA and rB whenmutualists do not pay any costs when the dual lattice site
is empty (Case II). The reproductive rates of nonmutualists remain rA and rB when they are associatedwith empty sites or non-mutualists of the other species.
The reproduction is successful only when offspring canenter empty sites for each species.

2. The model
3. Unlimited dispersal
We assume a two-dimensional lattice space where
species A and B inhabit; only one individual can occupy
one lattice site for each species. Individuals of different
species can coexist at the same lattice site and theninteract with each other. The individuals of two species
are divided into two classes each. AM and ANM are
frequencies of mutualists and non-mutualists of species
A, respectively. Similarly, BM and BNM are frequencies
of mutualists and non-mutualists of species B, respectively. AE and BE are frequencies of empty sites for
species A and B, respectively. One can imagine a lattice
habitat populated by species A and another lattice
parallel to it, called as its ‘‘dual’’, which is inhabited by
species B. Doebeli and Knowlton (1998) proposed thisdual-lattice description, but did not assume empty sites
in their model. We have the following identities for thetwo lattices:AM þ ANM þ AE ¼ 1

ð1aÞ

and
BM þ BNM þ BE ¼ 1:

ð1bÞIn our model, mutualists of A and B species
reproduce by helping each other. A cost to a reference
species (measured in terms of the reduction of reproductive rate) is always associated with the beneﬁt it
provides to the partner species. Let rA ; dA ; bA and cA
denote intrinsic reproductive rate, death rate, beneﬁt ofmutualism and cost of mutualism for species A,
respectively. Let rB ; dB ; bB and cB denote the corresponding parameters for species B, respectively. We
assume, for simplicity, that the reproductive rates are
modiﬁed through species interaction, but that death

When dispersal of reproduced offspring is unlimited
and they randomly reach any site, the changes infrequencies of AM ; ANM ; BM and BNM for Case I can be
written as differential equations:dAM¼ ðrA  cA ÞAE AM ðBE þ BNM Þ
dtþ ðrA  cA þ bA ÞAE AM BM  dA AM ;

ð2aÞ

dANM
¼ rA AE ANM ðBE þ BNM Þ
dtþ ðrA þ bA ÞAE ANM BM  dA ANM ;

ð2bÞ

dBM
¼ ðrB  cB ÞBE BM ðAE þ ANM Þdtþ ðrB  cB þ bB ÞBE BM AM  dB BM ;

ð2cÞ

dBNM
¼ rB BE BNM ðAE þ ANM Þdtþ ðrB þ bB ÞBE BNM AM  dB BNM :ð2dÞ

These equations are derived from the assumptions of
the model. For example, the ﬁrst term of the right-handside of (2a) represents the condition that mutualists ofspecies A (the frequency is AM ) reproduce rA  cA
offspring when they are associated with empty sites or
non-mutualists of species B (the frequency is BE þ BNM )and the offspring survives only when they reach empty
sites (the frequency is AE ). We omit the explanation of
the other terms and the other equations because they arestraightforward.

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From Eqs. (2a) and (2b), we have
1 dAM
1 dANM¼  cA AE ðBE þ BNM þ BM Þ
AM dt
ANM dt¼  cA AE o  eo0;
ð3Þ
where e is a positive value, because empty sites arealways produced due to the death of mutualists andnon-mutualists. That is, the speciﬁc increasing rate of
AM is always smaller than that of ANM : This equation
can be rewritten as


d
AM
log
o  e:
ð4Þ
dt
ANM
Integrating this in time interval from 0 to t; we haveAM ðtÞ
AM ð0ÞoexpðetÞ:ANM ðtÞ ANM ð0Þ

ð5Þ

As ANM ðtÞ is bounded under 1, AM ðtÞ must approachzero as t goes to inﬁnity. Similarly, since
1 dBM
1 dBNM¼  cB BE ðAE þ ANM þ AM Þ
BM dt
BNM dt¼  cB BE o  eo0;
ð6ÞBM must approach zero. Therefore, mutualists cannot
increase for any parameter values under the conditionsof Case I.
The changes in the frequencies of AM ; ANM ; BM andBNM for Case II can be written as the following
differential equations:dAM¼ rA AE AM BE þ ðrA  cA ÞAE AM BNM
dtþ ðrA  cA þ bA ÞAE AM BM  dA AM ;

ð7aÞ

dANM
¼ rA AE ANM ðBE þ BNM Þ
dtþ ðrA þ bA ÞAE ANM BM  dA ANM ;

ð7bÞ

dBM
¼ rB BE BM AE þ ðrB  cB ÞBE BM ANM
dtþ ðrB  cB þ bB ÞBE BM AM  dB BM ;

ð7cÞ

dBNM
¼ rB BE BNM ðAE þ ANM Þdtþ ðrB þ bB ÞBE BNM AM  dB BNM :ð7dÞ

These equations are derived from the assumptions,noting that mutualists do not pay costs when they areassociated with empty sites.
In this case,
1 dAM
1 dANM¼ cA AE ðBNM þ BM Þo0; ð8ÞAM dt
ANM dt423

that is, the speciﬁc increasing rate of AM is always
smaller than that of ANM : By the same procedure as inCase I, we can show that AM must approach zero.
Similarly, BM must approach zero.
We conclude that mutualists cannot survive under
existence of non-mutualists for any case when dispersal
is unlimited. It is reasonable that mutualists, which pay
some costs, are less advantageous than non-mutualists,
which pay no costs, if association between mutualists ofdifferent species is random due to unlimited dispersal.

4. Computer simulation for limited dispersal
We now investigate the evolution of mutualism, i.e.,
invasibility of mutant mutualists against non-mutualists
in the lattice model by means of computer simulation.
The evolution of mutualism is more difﬁcult for Case I,
where mutualists pay a cost even when the corresponding site of species B is empty, than for Case II. Thus, we
examine Case I in order to clarify whether or not theevolution of mutualism can be realized through spatial
effects. A lattice of size 25  25 is modeled and theperiodic boundary conditions of a torus are applied. The
lattice is dual, composed of lattice A and lattice B. Eachlattice site can be in one of the three possible states:
empty, non-mutualist or mutualist.
First, the model parameters are assumed to besymmetric between species, i.e., rA ¼ rB ¼ r; cA ¼ cB ¼ c;
bA ¼ bB ¼ b and dA ¼ dB ¼ d: In order to study theinvasion condition, the initial frequencies of mutualists ofboth species are chosen with low frequencies. Our aim is tosee if mutualists can invade the population of nonmutualists.
The following loop is repeated 625 times in one
generation; this number is the number of sites in thelattice. (i) A lattice site X is chosen at random. (ii) If
there is an individual at site X on lattice A, the average
number of offspring b is calculated, depending on thestate of the corresponding site of lattice B. For example,
b ¼ r  c þ b when both the site X on lattice A and thesite X on lattice B are inhabited by mutualists (see Case I
in the model section). (iii) At each neighboring site on
lattice A (four neighbor assumption adopted), theoffspring is produced at probability b=4 when the site
is empty. (iv) The same reproductive process is repeatedaround the site X on lattice B. (v) Individual at site X on
lattice A is killed at probability d. (vi) Individual at site
X on lattice B is killed at probability d.
In order to study invasibility, we put ‘‘wild-type’’
individuals on all sites on lattice A and B, and ran
simulations for 1000 generations to obtain the equilibrium state. Then we introduced mutants by placing
mutant individuals at random sites on lattice A and B at
frequency 0.04 (25 individuals). After 1000 generations,the average mutant frequencies on lattice A and B wereARTICLE IN PRESS
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N. Yamamura et al. / Journal of Theoretical Biology 226 (2004) 421–428

calculated. The invasion was deﬁned to be successfulwhen both of the average frequencies are higher than
0.04 (the initial frequencies). We also calculated
invasibility of non-mutualists in wild-type populationsof mutualists by the similar procedure.
In order to examine effects of the lattice size on theinvasibility of mutualists and non-mutualists, we practiced similar computer similation on 50  50 and
100  100 lattices.
Time series of the numbers of mutualist pairs on thesame site of lattice A and B, and neighboring mutualist
pairs on the same lattice were also calculated andplotted in order to illustrate how mutualist mutants
increase their frequencies. Finally, we investigatedinvasibility of mutualists in case of asymmetric parameter values between two species.

5. Results of the simulation
Fig. 1 shows invasibility of mutualists in theparameter space of cost (c) and beneﬁt (b) in case ofsymmetric parameter values between two species. Deathrate and intrinsic reproductive rate were ﬁxed (d ¼ 0:25
and r ¼ 0:4 in Fig. 1a). Mutualist mutants could
successfully invade wild-type populations of non-mutualists when c was low and b was high (represented byhorizontal segments ‘‘’’). Generally, only mutualists
remained after 1000 generations when the invasion was
successful and only non-mutualists remained whenunsuccessful, although mutualists and non-mutualists
continued to coexist in some cases owing to stochastic
effects. The invasion of non-mutualists in wild-type
populations of mutualists was successful when c washigh and b was low (represented by vertical segments‘‘|’’), while unsuccessful in the opposite case, implying
the mutualist population was evolutionarily stable. The
region occupied mainly by ‘‘’’ signs and that by ‘|’
signs were almost separated by a straight line (nearlyc ¼ 0:13b in this case). The results near the boundaryline were not constant, due to stochasticity of the model,
especially depending on the initial spatial distribution ofmutants. Generally speaking, a necessary condition formutualists to invade or for non-mutualists not to invadeis that the ratio of cost of mutualists to beneﬁt to theindividual of other species is lower than a constantvalue. As the intrinsic reproductive rate is increased(r ¼ 0:5 in Fig. 1b), the critical ratio of cost to beneﬁt
becomes lower (nearly c ¼ 0:075b). In other words, theevolution of mutualism becomes more difﬁcult withincreasing reproductive rate.
Fig. 2 shows effects of the lattice size on theinvasibility of mutualists and non-mutualists. The
parameter values are the same as in Fig. 1a (d ¼ 0:25
and r ¼ 0:4), where the initial frequencies of mutants arealso the same (0.04). The critical boundaries ofFig. 1. Invasibility of mutants in the space of beneﬁt ðbÞ and cost ðcÞ:One run was conducted at each point of combination of b
ð0; 0:025; 0:05; y; 1:0Þ and c ð0; 0:0025; 0:005; y; 0:1Þ: A horizontal
bar means that mutualists invaded non-mutualist populations. A
vertical bar means that non-mutualists invaded mutualist populations.
We set death rate as d ¼ 0:25; and intrinsic reproductive rate as (a)r ¼ 0:4 and (b) r ¼ 0:5: The region for successful invasion of mutualists
is smaller for higher reproductive rate.

successful mutualists invasion in cases of (a) 50  50
lattice and (b) 100  100 lattice were almost the same asin the 25  25 lattice, but the boundaries became clearer
as the lattice size increased. We could conﬁrm the resultson the critical cost–beneﬁt ratio are robust even in arather small lattice such as 25  25: In Fig. 2b with lessstochasticity, we can see small region, at an upper partof the boundary, where both mutualists and nonmutualists invade and another small region, at lower
part, where neither mutualists nor non-mutualists
invades. Therefore, there may exist the region ofcoexistence between mutualists and non-mutualists at

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0.1

c

0
0

1.0
b

(a)

0.1

c

0
0
(b)

1.0
b

Fig. 2. Effects of the lattice size on the invisibility of mutualists andnon-mutualists. The meaning of horizontal and vertical bars are thesame as in Fig. 1, and the parameter values are the same as in Fig. 1a(d ¼ 0:25; r ¼ 0:4). As the lattice size increases ((a) 50  50 and (b)100  100), the critical boundary for successful invasion of mutualists
becomes clearer.

the former part, and the region of bistability at the latterpart. If so, such regions are small and most of theparameter space will be ﬁlled with the two regions wheremutualists can invade and are stable, and where nonmutualists can invade and are stable.
Fig. 3a shows an example of time series of thenumbers of mutualists on lattice A and lattice B, wheremutualist mutants invaded the populations dominatedby non-mutualists (r ¼ 0:4; b ¼ 0:7; c ¼ 0:01 and d ¼
0:25; see also Fig. 1a). The mutualists of the two species
seem to increase in an approximately corresponding
manner. Time series of the numbers of mutualist pairson the same site of lattice A and B and neighboring

425

mutualist pairs on the lattice A and B were also
calculated. Fig. 3b shows the ratios of the numbers totheir expected values if the mutualists were randomly
distributed on the two lattices. At the initial phase when
the numbers of mutualists were low, the relative
numbers of neighboring mutualist pairs took considerably high values, implying the mutualists increased tomake clusters by reproducing in neighboring latticesites. The high value of the relative number of pairs
between different species during the same phase implies
that the mutualists could increase by pairing with eachother at a high probability. Fig. 3c shows an example oftime series of the numbers of non-mutualists on lattice A
and lattice B, where non-mutualist mutants invaded the
populations dominated by mutualists (r ¼ 0:5; b ¼ 0:2;
c ¼ 0:05 and d ¼ 0:25; see Fig. 1b). The non-mutualists
seem to increase independently. As shown in Fig. 3d,
during the initial phase when the numbers of nonmutualists were low, the relative numbers of neighboring non-mutualist pairs also took high values, but not ashigh as in case of the mutualists invasion, implying thedegree of clustering of non-mutualists was weaker. The
relative number of pairs between different species during
the same phase was remarkably low and became lessthan one, implying that the non-mutualists of different
species kept away from each other. This may be because
non-mutualist mutants could increase by exploiting
mutualists of the other species.
Fig. 4 shows invasibility of mutualists in case ofasymmetric parameter values between two species where
the invasibility is represented as frequency isoclines ofinvading cases among 100 trials. Although the isoclines
are unsteady due to highly stochastic nature of themodel, the following tendency is clearly recognized.
When reproductive rates of species A ðrA Þ and species
B ðrB Þ were different but cost and beneﬁt were the same
(Fig. 4a), the invading probability was higher if the
reproductive rate of either species was higher, corresponding to the result of symmetric parameter cases(compare Fig. 1a and b, having different r values). An
interesting fact is that the isoclines tend to be convex
downwards, which implies that mutualism may evolve
even when the reproductive rate of only one species islow. Fig. 4b shows the case of asymmetric costs between
species while reproductive rates and beneﬁts are thesame. The invasion probability was higher if the cost ofeither species was higher, corresponding to the result of
symmetric parameter cases (Fig. 1). The frequency
isoclines seem to be weakly convex, but close to astraight line. Fig. 3c illustrates the case of asymmetricbeneﬁts between species, while reproductive rates andcosts are the same. The frequency isoclines also seem tobe near a straight line. We can say that in the case ofasymmetric values of cost or beneﬁt, the probability ofmutualism evolution is determined by the average value
between different species.

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426

12

500

10

400

8
300

6
200

4
100

2
0
0

100

200

300

400

500

generation

(a)

0
0

100

200

300

400

500

400

500

generation

(b)
10

300

8
200

6

4
100

2

0

0
0

(c)

100

200

300

400

0

500

generation

(d)

100

200

300

generation

Fig. 3. When mutualists invade populations of non-mutualists ðd ¼ 0:25; r ¼ 0:4; b ¼ 0:7; c ¼ 0:01Þ; (a) the numbers of mutualists in lattice A (thickline) and in lattice B (thin line) increase rather correspondingly, and (b) the ratio of the number of pairs of neighboring mutants in lattice A to thenumber expected from random distribution (thick line), the same index in lattice B (thin line) and the ratio of the number of pairs of mutants acrossthe lattices to the expected number (gray line) took high values during the initial phase. When non-mutualists invade populations of mutualists
ðd ¼ 0:25; r ¼ 0:5; b ¼ 0:2; c ¼ 0:05Þ; (c) the numbers of non-mutualists in lattice A (thick line) and in lattice B (thin line) increase independently, and(d) the ratio of the number of pairs of neighboring mutants in lattice A to the expected number (thick line) and the same index in lattice B (solid line)
took high values but the ratio of the number of pairs of mutants across the lattices to the expected number (gray line) took low values during theinitial phase.

6. Discussion
We provided a simple lattice model to clarify theimportance of spatial structure for the evolution ofmutualism. When dispersal of offspring is unlimited, we
veriﬁed analytically that the evolution of mutualism
evolution is impossible under any condition. On theother hand, when the dispersal is limited to neighboring
lattice sites, mutualists can invade non-mutualists if theratio of cost to beneﬁt is low and the intrinsic
reproductive rate is also low. Although our model isvery simple, these results do suggest that the evolutionof mutualism between species is more likely in organisms that have low dispersal abilities.
We can regard the mechanism of mutualism evolution
in our model as mutual ‘‘altruism’’ between different

species. Altruism has been investigated as a behavior of
an individual that give beneﬁt to another individual,with its own cost, in the same species (Hamilton, 1964;
Yamamura and Higashi, 1992). Apparent altruism may
evolve when the ratio of cost to beneﬁt is low andgenetic relatedness between the interacting individuals ishigh. Yet even when individuals cannot discriminate
whether or not their opponents are relatives, altruism
can evolve if dispersal is limited, because there is a highprobability that the interacting individuals are relatives
to each other (Matsuda, 1987). In our model, theinteracting individuals are of different species and thus
are not clearly relatives. However, a mutualistic
individual which facilitates reproduction of a mutualistic individual of the other species may neverthelessobtain returns if the dispersal is limited for both species:ARTICLE IN PRESS
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427

0.1

0.7

rB 0.5cB 0.05

20

10

40
60

20

80
30

100
40
0.3
0.3
(a)

0.5
rA

50
0.0
0.0

0.7

0.05
cA

(b)

0.1

1.4

60
50
1.0

40
30
20

bB

10
0.5

0.0
0.0
(c)

0.5

1.0

1.4

bA

Fig. 4. Invasibility of mutualists are represented as isoclines of probabilities of the successful invasion when parameters are asymmetric between
species. Parameters are set as r ¼ 0:5; d ¼ 0:25; b ¼ 0:7; c ¼ 0:03 unless they are not changed. (a) Intrinsic reproductive rates ðrA ; rB Þ are changed
within (0.3, 0.3)(0.7, 0.7) at 0.02 intervals. (b) Costs of mutualism ðcA ; cB Þ are changed within (0, 0)(0.1, 0.1) at 0.01 intervals. (c) Beneﬁts ofmutualism ðbA ; bB Þ are changed within (0, 0)(1, 1) by at 0.1 intervals. At each data point, the invasion probability was calculated, based on 100
trials, from which the probability isoclines were interporated.

offspring of the former individual is likely to behelped by offspring of the latter individual. As shownin Fig. 2b, the high values of the number of pairs ofmutualists during the initial phase of mutualists invasionsuggests that the mechanism stated above really canoperate. The high probability of mutualist pairs ofdifferent species functions as the primary role for theevolution of mutualism and the high probability ofneighboring mutualist pairs of the same species is theresult of the process, while the latter directly contributes
to the altruism evolution in a single species.
Mutualism in our model may evolve under the same
condition as in the evolution of altruism, where the ratioof cost to beneﬁt is low. This condition would be morefeasible in the evolution of mutualism because different
species require different resources for reproduction andthus it is likely that a small cost for one species becomesa large beneﬁt for the other species (Yamamura, 1993).
Therefore, values of the critical cost–beneﬁt ratio suchas 0.13 or 0.075 in Fig. 1 do not seem to be unrealistic.
Mutualism in our model may evolve more easily when
intrinsic reproductive rates are small (compare Fig. 1aand b). This would be because equilibrium frequenciesof wild-type non-mutualists are low under the low
intrinsic reproductive rate and mutualists of different
species easily make clusters together using the highfrequencies of empty sites. A similar explanation wasgiven for the evolution of altruism when the equilibrium
frequency is low (Matsuda, 1987). On the other hand, ina spatial model of the evolution of altruism without
empty sites, such evolution does not occur, because
competition for reproductive sites cancels out any
helping effect that may be conferred by neighboring
altruists (Taylor, 1992; Wilson et al., 1992).

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N. Yamamura et al. / Journal of Theoretical Biology 226 (2004) 421–428

The evolution of mutualism does not hinge on theequivalent costs and beneﬁts between interacting species
because the average value of costs and that of beneﬁts
tend to determine the probability of the evolution even if
the parameters are asymmetric (Fig. 3b and c). Regarding the asymmetry of intrinsic reproductive rates, theprobability of the evolution is high when this parameter
is low for either one of the two species (Fig. 3a). We did
not examine the asymmetry of dispersal rates, withrelaxing the assumption of neighborhood dispersal.
However, in a special case of neighborhood dispersal
for one species and random dispersal for the otherspecies, we conﬁrmed that mutualism could not evolve.
This would be because mutualist pairs between different
species are frequently lost by the random dispersal of
one species.
Our study in case of limited dispersal is based oncomputer simulation. There are several analyticalmethods to investigate spatial dynamics such as pairapproximation (Nakamaru et al., 1997; Le Galliard,
2003). The results that we obtained here may be
reinforced by applying such analytical methods, but
our computer simulation could clearly have shown theimportance of spatial structure in evolution of mutualism.
Spatial effects are generally very important indynamics and evolution of interacting species (Tilman
and Kareiva, 1997; Thompson, 1999). The present study
elaborates on this conclusion, and provides a robust
example of the importance of spatial structure in theevolution of mutualism.

Acknowledgements
We thank members of mathematical ecology group in
Center for Ecological Research, Kyoto University forvaluable comments on the manuscript. This work waspartly supported by the MEXT Grant-in Aid for N.Y.
(No. 15657007) and for the 21st COE Century Programof Kyoto University (A2).

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