Mathematical Biosciences 201 (2006) 72–89
www.elsevier.com/locate/mbs

A mathematical analysis on public goods gamesin the continuous space
Joe Yuichiro Wakano
Department of Biological Sciences, The University of Tokyo, Hongo 7-3-1, Bunkyo-ku, Tokyo 113-0033, Japan
Received 15 July 2004; received in revised form 6 December 2005; accepted 6 December 2005
Available online 8 February 2006

Abstract
We consider the population dynamics of two competing species sharing the same resource, which is modeled by the carrying capacity term of logistic equation. One species (farmer) increases the carrying capacity
in exchange for a decreased survival rate, while the other species (exploiter) does not. As the carrying capacity is shared by both species, farmer is altruistic. The eﬀect of continuous spatial structure on the performance of such strategies is studied using the reaction diﬀusion equations. Mathematical analysis on thetraveling wave solution of the system revealed; (1) Farmers can never expel exploiters in any traveling wave
solution. (2) The expanding velocity of the exploiter population invading the farmer population can be analytically determined and it depends only on a cost of altruism and the diﬀusion coeﬃcients while it isindependent of the beneﬁt of altruism. (3) When the eﬀect of altruism is small, the dynamics of the invasionof exploiters obeys the Fisher-KPP equation. Numerical calculations conﬁrm these results.
 2005 Elsevier Inc. All rights reserved.
Keywords: Public goods game; Reaction diﬀusion; Traveling wave solution; Altruism; Continuous space; Carrying
capacity

1. Introduction
Altruistic behavior among unrelated individuals has been widely studied since it cannot bedirectly explained by the standard kin-selection theory [1]. Spatial structure has been considered
E-mail address: joe@biol.s.u-tokyo.ac.jp0025-5564/$ - see front matter  2005 Elsevier Inc. All rights reserved.
doi:10.1016/j.mbs.2005.12.015

J.Y. Wakano / Mathematical Biosciences 201 (2006) 72–89

73

as a possible mechanism to explain the evolution of altruistic behavior. Most mathematical studies assume discrete spatial structure such as a lattice or randomly connected patches [2–6]. Discretely structured model is advantageous to deal analytically [7,8]. However, the discrete space
and the continuous space are essentially diﬀerent and many creatures in real biology inhabit continuous space. Reaction diﬀusion equation is one of the simplest models describing spatiotemporal dynamics of continuous entity in the continuous space. Individuals are discrete (countable)
entities but their density might be considered as continuous entity if the number of individuals
is large enough.
There are a few studies on the evolution of altruistic behavior in the continuous space [9]. Mostof them study the reaction diﬀusion equations modeling the competition among players adopting
Tit-For-Tat or All-D strategy in iterated prisoner’s dilemma game [10,11]. They showed that,when the corresponding non-spatial model has bistable solutions, the spatial model might havestable coexistence solutions (spatially homogeneous or inhomogeneous). As Tit-For-Tat is a conditional strategy with a memory, it is implicitly assumed that a player can recognize the opponent’s last behavior. Such a strategy might evolve if the interaction is direct and observable. In
general, however, altruism is not necessarily performed through a direct interaction. In this study
we assume that altruism is performed through the investment to the common resource that isshared by all players. The situation is referred to as ‘tragedy of commons’ or ‘public goods game’.
Unlike the game played by Tit-For-Tat and All-D, altruistic strategy in public goods games cannever win in the non-spatial model, regardless of the initial frequency. Recently, Brandt et al. [12]have studied public goods games in the discrete space but there has been no study in the continuous space. We introduce spatial structure so that each individual moves randomly and that thedensity distribution obeys a reaction diﬀusion equation. And we study the performance of thealtruistic strategy.
From the mathematical viewpoint, there are a few studies on game models in the continuous
space [13–15]. Some useful theorems are derived, especially on the traveling wave solutions. In
these models, the local carrying capacity is assumed to be constant and the relative payoﬀ determines the growth rate of each strategy. In our model, the local carrying capacity itself is a functionof the density of a strategy, which is diﬀerent from the preceding works that motivated the author
to begin this study.
In this study, we assume two types of strategies (altruistic and non-altruistic) competing in thecontinuous space. Population dynamics is modeled by the reaction diﬀusion equations and the
behavior of the system is studied. Especially, the existence and the velocity of the traveling wave
solutions in which one strategy expels the other are analyzed analytically. The results are conﬁrmed by numerical calculations.

2. ModelWe start from the standard logistic equation
dx
x
¼ rx 1 
;
dt
K

ð1Þ

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J.Y. Wakano / Mathematical Biosciences 201 (2006) 72–89

where K denotes the carrying capacity. We assume two strategies, namely farmer and exploiter.
Farmer increases the carrying capacity in exchange for a decreased survival rate. We assume that
all farmers and exploiters share the increased carrying capacity. Therefore, a farmer strategy canbe regarded as an altruism strategy. The model is denoted by


dx
xþy
¼ rx 1 
;
dt
KðyÞ


dy
xþy
ð2Þ
¼ ry 1 
 cy;
dt
KðyÞ
KðyÞ ¼ k 0 þ ky;
where x and y represent the density of exploiters and farmers, respectively. k measures the beneﬁtof altruism given to both strategies while c measures the cost of altruism. We consider the onedimensional continuous spatial structure by introducing the diﬀusion of both players.


ox
o2
xþy
¼ Dx 2 x þ rx 1 
;
ot
KðyÞ
oX
ð3Þ


oy
o2
xþy
¼ Dy 2 y þ ry 1 
 cy;
ot
KðyÞ
oX
X denotes the coordinates and Dx and Dy denote the diﬀusion coeﬃcients of exploiters and farmers, respectively. By the following transformation:rﬃﬃﬃﬃﬃﬃr
1
0
X ; t0 ¼ t;
X ¼
Dx
r
Dy
c
ð4Þ
d¼
; c0 ¼ ;rDxxykx0 ¼ ; y 0 ¼ ; k 0 ¼ ;k0
k0
k0
the non-dimensionalized system is denoted by


ox
o
xþy
xþx 1
¼;
ot oX 2
1 þ kyoyoxþy¼d 2yþy 1 cy;ot
1 þ kyoXð5Þwhere only d, c and k are independent parameters. Primes are omitted for convenience. Eqs. (5)have two spatially homogeneous equilibriums: (x, y) = (1, 0) and (x, y) = (0, y*) where
y ¼

1c
.
1  k þ ck

ð6Þ

We presuppose d > 0, 0 < k < 1 and 0 < c < 1 in order that y* is always positive and that the model
is biologically realistic. Neglecting the eﬀect on exploiters, the farmer strategy is considered adaptive if y* > 1 or k  c  ck > 0. However, without spatial structure, farmers always go to extinction because the per capita growth rate of exploiters is always larger than that of farmers andJ.Y. Wakano / Mathematical Biosciences 201 (2006) 72–89

75

because there exists the upper bound of the total population (Fig. 1). In order to study the eﬀect of
spatial structure, we consider the case in which exploiters initially inhabit the left half of the spaceand farmers the right half (Fig. 2). The populations are in the equilibrium states in both ends,
while competition occurs at the boundary of two populations (hereafter, we call this ‘the front’).
The direction of the motion of ‘the front’ determines which strategy will win the competition. As
long as the reaction term is considered, the per capita growth rate of exploiters is always larger
than that of farmers at any point in the space. However, the equilibrium density of farmers, y*,
y
1
1– k

y* =

1– c
1 – k + ckx
1– cx* = 1

Fig. 1. Phase plane analysis of the dynamics of non-spatial model. Non-dimensionalization denoted by Eq. (4) isapplied. Two lines indicate isoclines of x and y. Filled and open circles indicate stable and unstable equilibriums,
respectively. As the beneﬁt of altruism is increased (larger k), the monomorphism of farmers can maintain larger
population (larger y*). However, this monomorphism is always unstable against invasion of exploiters.

yy*diffusionx
1

local competition

space

Fig. 2. The schematic illustration of the farmer-exploiter boundary. Solid and bold lines indicate the density of farmers
and exploiters, respectively. In general, the equilibrium density of farmers is larger than that of exploiters and thus the
disadvantage of farmers in local competition at the boundary area might be compensated by the diﬀusion from thehigh-density area (the right half).

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J.Y. Wakano / Mathematical Biosciences 201 (2006) 72–89

can be much greater than that of exploiters when c is very small and k nearly equals one. Under
such a condition, the disadvantage of farmers in local competition might be compensated by thediﬀusion.

3. Analytic result
A traveling wave solution (TWS) is the solution satisfying
xðX ; tÞ ¼ xðzÞ;yðX ; tÞ ¼ yðzÞ;ð7Þ

z ¼ X  vt.
Analysis of a TWS is almost only one analytic tool that is available to study reaction diﬀusion
system. However, the analysis is diﬃcult in multi-species system. One of the simplest models of
biological competition in the continuous space is a two-species Lotka–Volterra system with diﬀusion terms. This is equivalent to the present model with a constant carrying capacity (k = 0). Evenin this simplest example, the general existence proof of a TWS (as well as a solution itself) is
unknown. Only in some special cases, the existence and some characteristics of a TWS are proved[16–19]. Here, we explore the possibility of the existence of a TWS in the present model that is
another special case with density dependent carrying capacity.
If there exists a TWS in Eqs. (5), it satisﬁes


xþy
;
 v_x ¼ €x þ x 1 1 þ kyxþy
 cy
 vy_ ¼ d€y þ y 1 
1 þ kyor
x_ ¼ p;xþy;
p_ ¼ vp  x 1 1 þ kyy_ ¼ q;

ð8Þ



v
y
xþy
c
1
þ y;
q_ ¼  q dd1 þ kydwhere a dot symbol represents the derivative with respect to z. Eq. (8) has two equilibriums:
Px = (1, 0, 0, 0) and Py = (0, 0, y*, 0). Jacobian matrix around the point (x*, 0, y*, 0) is
0
1
0 1 0
0
B A v B
0 C
B
C
B
C;
@0 0 01 AC

0D v=dJ.Y. Wakano / Mathematical Biosciences 201 (2006) 72–89

where
A ¼ 1 þ
B¼
C¼

2x
y
þ
;
1 þ ky  1 þ ky x ð1  kx Þ
ð1 þ ky  Þ

2

;

y
;
dð1 þ ky  Þ1x þ y  y  ð1  kx Þ
c1þ
þ
D¼
d
1 þ ky  ð1 þ ky  Þ2!

and the characteristic equation is
k4 þ ðv þ v=dÞk3 þ ðv2 =d  A  DÞk2  ðA=d þ DÞvk þ AD  BC ¼ 0.
When BC = 0, the equation can be factorized asðk2 þ vk  AÞðk2 þ ðv=dÞk  DÞ ¼ 0.
In the neighborhood of Px,
A ¼ 1;
B ¼ 1  k;
C ¼ 0;cD¼
d
and the eigenvalues are
pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
v  v2 þ 4 v  v2 þ 4cd
k¼
;
;
2
2d
which means Px is a saddle point. In the neighborhood of Py,
A ¼ c;B ¼ 0;
1c
;
C¼
d
1c
D¼
dð1 þ ky  Þand the eigenvalues are
pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
v  v2  4c

k1 ¼;
p2ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
v  v2 þ 4Dd
k
.
2 ¼
2d

77

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J.Y. Wakano / Mathematical Biosciences 201 (2006) 72–89

As c, d, D > 0, Py has a three-dimensional stable manifold and a one-dimensional unstablemanifold.
To proceed further, we introduce the following lemma and deﬁnition.
Lemma 1. Let us consider the four-dimensional dynamical system with a hyperbolic fixed point P(none of the eigenvalues is on the imaginary axis). Assume the hyperplane I = {(x1, x2, x3, x4)jx1 =x2 = 0} is a two-dimensional invariant manifold of P (if x1 = x2 = 0 holds initially, it holds forever).
Assume the only one of the four eigenvalues of P has a positive real part and the corresponding
eigenvector is tangent to I. Let Wu(P) be the one-dimensional unstable manifold of P. Then, Wu(P) isa curve on the invariant hyperplane I.
Proof. See Appendix A. h
Deﬁnition. A realistic TWS is deﬁned as a heteroclinic orbit connecting Px and Py in the regionx > 0 and y > 0.
Now we can prove the following theorem.
Theorem 1. Eqs. (8) do not have any realistic TWS starting from Py.
Proof. Consider Wu(Py), the one-dimensional unstable manifold of Py. Any orbit which converges to Py as z ! 1 must be on Wu(Py). Wu(Py) is tangent to the eigenvector for kþ
2,
0

0

1

B C
B 0 C
C
C
B 1 C
@ A

B
~
eþ
2 ¼ B

kþ
2
at Py. On the other hand, the hyperplane I = {(x, p, y, q)jx = p = 0} is a two-dimensional invariantumanifold of Eqs. (8). I is also tangent to ~eþ
2 at Py. From Lemma 1, W (Py) is a curve on the hyperplane I. As Px is not on I, Wu(Py) does not contain Px. This completes the proof. h
According to this theorem, farmers cannot expel exploiters in any TWS, regardless of parametervalues. By the next theorem, we obtain the minimum velocity of the TWS. We empirically knowthat the TWS of the minimum velocity is most often chosen as a result of a Cauchy problem witha regular initial condition.
Theorem 2. If Eqs. (8) have a realistic TWS entering Py, the velocity satisfies
pﬃﬃﬃ
v P 2 c.
pﬃﬃﬃ
Proof. We argue by contradiction. Assume there exists a realistic TWS with v < 2 c. Py has threeeigenvalues whose real parts are negative. k
2 is a negative real and the corresponding eigenvector isJ.Y. Wakano / Mathematical Biosciences 201 (2006) 72–89

79

1
0
B 0 C
B C;
~
e
2 ¼ @
1 A
k
2
0


which is tangent to I. k
1 are complex conjugates ðImðk1 Þ 6¼ 0Þ with negative real parts and thecorresponding eigenvectors are
1
0
1

C
B
k1
C
B
C
C
B

C
B
~
e1 ¼ B
2 C;Dðv=dÞk

k
C
B
A
@
kC
D  ðv=dÞk  k2

which have non-zero components in both x- and p-direction. A realistic TWS must be W s2 ðP y Þ,e
the one-dimensional stable manifold of Py that is tangent to ~2 at Py. Otherwise, the orbit has
e2 is tangent to I; W s2 ðP y Þ is a curve
to pass the region x < 0 as it approaches Py. However, as ~on I (this might seem almost trivial from Lemma 1 but see Appendix B for proof). Therefore,
W s2 ðP y Þ does not contain Px. The contradiction completes the proof. hIn addition to these analytic results, we can derive the following conjecture.
Conjecture 1. Assume d = 1. For sufficiently small c and k, the traveling wave solution exists in Eqs.
(5) and it approaches to the solution ofy ¼ y  ð1  xÞ;oxoxþ cxð1  xÞ¼
ot oX 2
as c ! 0 and k ! 0.

ð9Þ

Derivation. See Appendix C.
As the second equation is the Fisher-KPP equation,pﬃﬃﬃ a TWS exists in Eqs. (9) and the system
will converge to the TWS with the velocity v ¼ 2 c for most regular initial conditions [18].
The conjecture claims that Eqs. (5) is reduced into Eqs. (9).
According to these results, the beneﬁt of altruism (k) has no impact on the ﬁnal state of thesystem. However eﬃcient the altruism is, farmers can never expel exploiters. When k ! 1 andc ! 0, y* ! 1 and the diﬀusion of farmers from the right half of the space to ‘the front’ isexpected to diverge to the inﬁnity, while the per capita growth rates in local competition diﬀer
only by c. Despite these facts, our result predicts that exploiters will always win. The striking
result derived analytically has some limitations. Theorem 1 does not deny the possibility that
farmers expel exploiters in any form except TWS. Theorem 2 deals only with the minimum velocity. Only when c and k are small and d = 1, we have a reasonable conjecture that the PDE system(Eqs. (5)) develops into a TWS. However, the general PDE problem is diﬃcult to attack. Next, we
perform numerical calculations to check the validity of the analysis.

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J.Y. Wakano / Mathematical Biosciences 201 (2006) 72–89

4. Numerical result
We performed the numerical calculation of Eqs. (5) in the region 0 6 X 6 L with a zero-ﬂux
boundary condition. L denotes the size of the space and we ﬁx L = 4000. Based on preliminary
calculations as well as the analytical result, the initial condition is set as8
ð1; 0Þ; 0 6 X 6 L=10;>><
ðx; yÞ ¼ ð0; 0Þ; L=10 < X < L=2;>>:
ð0; y  Þ; L=2 6 X 6 L.
For all parameters we have tested, a traveling wave is formed in which exploiters expel farmers(Fig. 3). The dependence of the velocity of the TWS on c and k is studied. When d = 1, the velocity
does not depend on k, as is predicted analytically (Table 1). The dependence on c is quantitatively

5
4
3
2
1
0
200

300

400

500

Space

(a)
5
4
3
2
1
0
500

(b)

600

700

800

900

Space

Fig. 3. The proﬁles of the waves (k = 0.9, c = 0.1, d = 0.25). (a) The propagating waves of farmers (solid) and
exploiters (bold) approach each other (t = 100) and then collide (t = 160) and form the stable boundary (t = 240)
moving right slowly. (b) After some transient time, the stable traveling wave with the constant velocity is formed(t = 640, 800, 960).

J.Y. Wakano / Mathematical Biosciences 201 (2006) 72–89

81

Table 1The velocity of the TWS in Eqs. (5) as a function of c and kkc (v*)
0.001 (0.0632)

0.01 (0.2000)

0.01
0.1
0.5
0.9
0.999

0.05830.1946
0.05830.1946
0.05830.1946
0.05830.1946
0.05790.1946pﬃﬃﬃ
d = 1.0, v ¼ 2 c. The bold values indicates y* > 1.

0.1 (0.6325)

0.5 (1.4142)

0.9 (1.8974)

0.62740.62740.62740.62740.6274

1.4136
1.4136
1.41371.4136
1.41371.90441.90441.90441.90441.9044

Table 2The velocity of the TWS in Eqs. (5) as a function of d and k
k
0.01
0.1
0.5
0.9
0.999

d
0.01

0.1

1

10

100

0.62490.62490.62500.62500.62510.62600.62600.62600.62610.62610.62740.62740.62740.62740.62740.63970.63950.63830.63530.63530.79330.78970.76830.72510.7026c = 0.1. v* = 0.6325.

similar to the predicted value. These results hold when d is decreased, while the discrepancy
appears when d is increased (Table 2). Even in the case when c = 0.001 and k = 0.999 (the corresponding y* = 500), the density of farmers reaches y* temporally, but exploiters invade the highly
eﬃcient population of farmers very slowly and ﬁnally expel them. The ﬁnal realization is always
the spatially homogeneous equilibrium state of exploiters, x = 1.
We also have a conjecture that the spatiotemporal dynamics of exploiters will converge to thatof the Fisher-KPP equation. Our conjecture is valid when c and k are small. In this case, a numerical calculation suggests that the conjecture is true (Fig. 4(a)). When k is not small (k = 0.9), thewave is sharper but still aﬃne congruent to the solution of the corresponding Fisher-KPP equation (Fig. 4(b)). If d is increased (d = 10), the dynamics behaves quite diﬀerently from the FisherKPP equation (Fig. 4(c)).
The reason farmers cannot win is also explained by the following verbal discussion. The (local)
net per capita growth rate is the sum of the reaction term and the second derivative of spatial distribution. In order to enjoy the diﬀusion eﬀect, the spatial distribution function must be concave
at a focal point. However, due to the nature of a reaction diﬀusion equation, exploiters instantly
spread into the whole space regardless of the initial distribution. Since the zero-ﬂux boundary condition is used, the second derivative cannot be concave for the whole space, which means there is apoint where exploiters locally expel farmers. Therefore, farmers cannot expel exploiters in theform of a TWS. The nature (inﬁnite spreading velocity) is a major defect of the reaction diﬀusion

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J.Y. Wakano / Mathematical Biosciences 201 (2006) 72–89
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1

(a) 0
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1

(b) 0
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1

(c)

0

Fig. 4. The comparison between the original model and the Fisher-KPP equation. The solid curve represents the wave
proﬁle of Fisher-KPP equation (c = 0.1, d = 1). The circles represent the wave proﬁle of the solution of Eqs. (5). The
two wave proﬁles are translated so that they intersects at x = 0.5. c = 0.1 for all ﬁgures. (a) Two proﬁles are almost
identical (d = 1, k = 0.1). (b) They are slightly diﬀerent but still aﬃne congruent (by rescaling space by a coeﬃcient 1.16)
when k is not small (d = 1, k = 0.9). (c) They are completely diﬀerent when d > 1 (d = 10, k = 0.9).

equation as a model for biological invasions. The defect might be eliminated by introducing theleast viable density e. In this ‘cutoﬀ’ model, when the density is below e, it decreases exponentially.
Formally,J.Y. Wakano / Mathematical Biosciences 201 (2006) 72–89

83

Table 3The velocity of the TWS in the model with cutoﬀ eﬀect
e0.00010.001
0.01

c (v*)
0.001 (0.0632)

0.01 (0.2000)

0.1 (0.6325)

0.5 (1.4142)

0.9 (1.8974)

0.05340.04370.01640.17780.15830.1094

0.58300.54590.45991.32161.25481.10741.77591.68551.4875d = 1.0 and k = 0.999.

8 <x 1  x þ yðx > eÞ;1 þ kyRx ðx; yÞ ¼:
x
ðx 6 eÞ;8 < y 1  x þ y  cy ðy > eÞ;1 þ kyRy ðx; yÞ ¼:
yðy 6 eÞ;where Rx and Ry are reaction terms of exploiters and farmers, respectively. The numerical resultof the model shows that the cutoﬀ eﬀect decrease the invading velocity of exploiters, however,farmers still cannot win for all parameters we have tested (Table 3).

5. Discussion
We show that the emergence of altruism is impossible in the present model framework. We analytically show that farmers can never expel exploiters in any form of a TWS. Numerical simulationof the reaction diﬀusion equations shows that exploiters always win. This result is in marked contrast to many studies showing that spatial structure promotes the evolution of cooperative behavior. The discrepancy comes from the essential diﬀerence of model, i.e. discrete and continuous
spatial structure. Discrete space model is easier to deal with and might be more appropriate forsome biological phenomena than continuous space model. However, space is essentially continuous and continuous space model has more applications to the real biological world. If continuous
space does not promote the evolution of altruism at all, we must reconsider the eﬀect of spatial
structure on the evolution of altruism.
What diﬀerence between continuous and discrete space brought the present result? There aremany studies on the evolution of altruistic behavior in the population with local interaction
and local dispersal (viscous population). Among them, the studies on lattice population using
inclusive ﬁtness theory [4,7] may be a clue to the question. Based on the Hamilton’s rule, relatedness must be positive for altruism to evolve. In this case, relatedness represents the fact that altruistic individual more likely interacts with altruistic individuals than population average in viscous
population. In the lattice models, relatedness is generally positive because the size of local population (patch) is ﬁnite. On the other hand, in reaction diﬀusion models, individuals are represented
by density, that is continuous. Thus, relatedness of local population might be always zero,

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J.Y. Wakano / Mathematical Biosciences 201 (2006) 72–89

although we do not know any formal deﬁnition of relatedness in general reaction diﬀusion
models. This might be the reason why the emergence of altruistic behavior is not promoted in
the continuous space.
From the mathematical point of view, we provide the non-existence proof of a TWS in which
farmers win. We numerically expect the existence of a TWS in which exploiters win. When k = 0,
our model falls into a category of the diﬀusive Lotka–Volterra competition model that has been
intensively studied. For example, Hosono [18] shows that the velocity becomes an increasing function of Dy (or d in non-dimensionalized model) for suﬃciently large d. This agrees with ournumerical result (Table 2) although the present model is not a Lotka–Volterra type. In a realistic
situation d = O(1) the velocity does not depend on d, which motivated the analysis of the special
case d = 0 of the diﬀusive Lotka–Volterra models. The existence of a TWS in the corresponding
three-dimensional phase space is proved by Hosono [17]. We showed that our model might be
reduced to the two-dimensional when d = 1, but considering the case d = 0 of our case might
be another way to approach the existence proof.
This study does not positively contribute to the understanding of the evolution of altruistic
behavior. Our result suggests that any eﬀort with cost to increase the shared carrying capacity
is maladaptive, however eﬃcient it is. One possible explanation for the evolution of altruistic
behavior in the continuous space is that the common resource is not allocated evenly. If (for somebiological reasons) farmers have the priority to use the addition of the common resource by them,
they might be able to resist the invasion of exploiters. This situation might be denoted by the timedelay in carry capacity term of exploiters. The author is currently working on this type of model.
Although altruism does not spread in the present model, we hope that our analysis will contributeto the understanding of the evolution of altruism in public goods games in the continuous space.

Acknowledgments
The author would like to thank Y. Hosono, T. Horita, G. Tanaka, Y. Otake, and T. Miki formany valuable comments. The author also thanks M. F. Boni for his comment on the Proof of
Lemma 1.

Appendix A
Proof of Lemma 1. We consider the four-dimensional dynamical system satisfying
x_ ¼ f ðxÞ;ðA1Þ

f ð0Þ ¼ 0;

ðA2Þ8x3 ; x4 ;f1 ð0; 0; x3 ; x4 Þ ¼ 0;
f2 ð0; 0; x3 ; x4 Þ ¼ 0;

ðA3Þ

J.Y. Wakano / Mathematical Biosciences 201 (2006) 72–89

85(A3) means the hyperplane I = {(x1, x2, x3, x4)jx1 = x2 = 0} is an invariant manifold of O. Jacobian matrix of (A1) around O is generally denoted by


A 0
J¼
;
C B
where A, B and C are 2 · 2 square matrices. With proper base transformations in a x1  x2 subspace and a x3  x4 subspace, A and B can be taken as Jordan canonical forms without loss ofgenerality. Thus, we assume0
1
k1 0 0 0
Ba k
0 0C
2
B
C
ðA4Þ
J ¼B
C;
@ c1 c3 k3 0 A
c2

c4

b

k4

where ki are eigenvalues. We also assume
Reðk4 Þ > 0

and Reðk1 Þ; Reðk2 Þ; Reðk3 Þ < 0.

ðA5ÞThis means that the unstable eigenvector is tangent to I. Here we prove the global unstablemanifold of O, Wu(O), is contained by I.
Let W uloc ðOÞ be the local unstable manifold of O. Let z  x4 then stable manifold theorem
ensures that W uloc ðOÞ is a curve (x1, x2, x3) = u(z) where Du(0) = (0, 0, 0). The following must be
the identity equation of z;
f4 ðu1 ðzÞ; u2 ðzÞ; u3 ðzÞ; zÞ  u0 ðzÞ ¼ fi ðu1 ðzÞ; u2 ðzÞ; u3 ðzÞ; zÞ;

ðA6Þ
j

j+1

where i = 1, 2, 3. Substituting a Taylor expansion ui(z) = aijz + O(z ) (j P 2) into a Taylor
expansion of f4 around O, the left hand side of (A6) becomes
"
#



3 
X
of4 
of4 
2
j1
j
j
jþ1
 uk ðzÞ þ ox  z þ Oðz Þ ðjaij z þ Oðz ÞÞ ¼ jaij k4 z þ OðZ Þ.
ox
k
4
O
O
k¼1
Similarly, for i = 1, the right hand side of (A6) becomes




3 
X
of1 
of1 
1 o2 f1  2jjþ1
 uk ðzÞ þ ox  z þ 2 ox2  z þ    ¼ k1 a1j z þ OðZ Þ
ox
k
4
4 O
O
O
k¼1


l 
of1 
because oxk  ¼ 0 for k = 2, 3, 4 form (A4) and because ooxfli  ¼ 0 for i = 1, 2 from (A3). ComparingO4

the coefﬁcients, we obtaink1 ¼ jk4

Oor a1j ¼ 0.

From (A5), the former condition is impossible and we obtaina1j ¼ 0.
For i = 2, (A6) becomes
ja2j k4 zj ¼ aa1j zj þ a2j k2 zj þ Oðz3 Þ

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J.Y. Wakano / Mathematical Biosciences 201 (2006) 72–89

and we obtain a2j = 0. Therefore, we obtainui ðzÞ ¼ 0
for i = 1, 2. Thus, W uloc ðOÞ is a curve on the hyperplane I. As Wu(O) is the extension of W uloc ðOÞ and
I is invariant, Wu(O) is a curve on I. This completes the proof. hAppendix B
Proof. We consider the same dynamical system denoted by (A1)–(A5). In addition, we assume
that k1 and k2 are complex (corresponding to k
1 in the text) and that k3 and k4 are realinthetext),
formally,(corresponding to k2Imðki Þ 6¼ 0 for i ¼ 1; 2;

ðB1ÞImðki Þ ¼ 0 for i ¼ 3; 4.

e
Let W s2 ðOÞ be the one-dimensional stable manifold of O that is tangent to ~2 at O. Here we proves
W 2 ðOÞ is contained by I.
2
2
Let z  x3 and W sloc
ðOÞ be W s2 ðOÞ in the neighborhood of O. Then W sloc
ðOÞ is a curve(x1, x2, x4) = u(z) where Du(0) = (0, 0, 0). The following must be the identity equation of z:
f3 ðu1 ðzÞ; u2 ðzÞ; z; u4 ðzÞÞ  u0i ðzÞ ¼ fi ðu1 ðzÞ; u2 ðzÞ; z; u4 ðzÞÞ;
j

ðB2Þ
j+1

where i = 1, 2, 4. Substituting a Taylor expansion ui(z) = aijz + O(z) (j P 2), we obtainja1j k3 ¼ a1j k1
and
ja2j k3 ¼ aa1j þ a2j k2 .
From (B1), neither k1 = jk3 nor k2 = jk3 is possible. Thus, we obtainui ðzÞ ¼ 0
2
2
for i = 1, 2 and W sloc
ðOÞ is a curve on the hyperplane I. As W s2 ðOÞ is the extension of W sloc
ðOÞ ands2
I is invariant, W ðOÞ is a curve on I. This completes the proof. hAppendix C
Derivation of Conjecture 1. Here we show that our model reduces to a Fisher-KPP equation if weneglect the second and higher order terms of c and k. The central idea is that the quantity
y
ðC1Þ
xþ 
y
will remain one at all points on the orbit when the diﬀerence between the property of farmers andthat of exploiters is very small, i.e. c, k  1. Based on the idea, we apply the following
transformation:

J.Y. Wakano / Mathematical Biosciences 201 (2006) 72–89

ym ¼ x þ   1;yqn¼ pþ ;yyu¼ x ;yq
w ¼ p .
y

87

ðC2Þ

We expect that m = n = 0 is an invariant manifold of both Px and Py. If this is true, the system canbe reduced into the two-dimensional dynamics on the manifold. Eqs. (8) can be rewritten by the
new four parameters using
1þmþu
;
2
1þmu 
y¼
y ;
2
ðC3Þ
nþw
p¼
;
2
nw 
q¼
y .
2
Substituting Eqs. (A3) into Eqs. (8) yields complicated equations. However, linearization aroundc = k = 0 becomes rather simple. To the ﬁrst order of c and k, we obtainx¼

u_ ¼ w;vv
cc
w_ ¼  Jw  Kn þ Ju2  J þ OðmÞ;2
2
4
4
m_ ¼ n;
v
v
c
c
ðC4Þ
n_ ¼  Jn  Kw þ Ku2  K þ OðmÞ;
2
2
4
4
1
J ¼1þ ;d1
k ¼1 ;dwhere O(m) represent the terms including m, which vanish when m = 0. Unfortunately, m = n = 0
is not invariant in general. However, in the special case K ¼ 0 () d ¼ 1, the equations become
simpleru_ ¼ w;cc
w_ ¼ vw þ u2  þ OðmÞ;2
2
m_ ¼ n;n_ ¼ vn þ OðmÞðC40 Þ

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J.Y. Wakano / Mathematical Biosciences 201 (2006) 72–89

and m = n = 0 becomes invariant. Hereafter, we assume d = 1. The reduced two-dimensional
system isu_ ¼ w;c
w_ ¼ vw þ ðu  1Þðu þ 1Þ.
2

ðC5Þ

The realistic TWS of Eqs. (8) corresponds to the orbit of Eqs. (A5) starting (u, w) = (1, 0) andentering (1, 0) in the region 1 < u < 1. Applying the transformation,
uþ1
;
2
w
e¼ ;
w
2
Eqs. (A5) become
e
u¼

e
e;
u_ ¼ we_ ¼ v w
e  cewu ð1  e
u Þ;

ðC6Þ

ðC7Þ

which is exactly the ODE governing a TWS in the Fisher-KPP equationoeu o2 eu¼
þ ceu ð1  eu Þ;ot oX 2

ðC8Þ

with the wave velocity v. We obtain Eq. (9) because eup¼ﬃﬃﬃx if m = 0. The existence of a realistic
TWS in Eq. (B2) had been already proved for v P 2 c [19]. Thus, a realistic TWS exists in
Eqs. (C4 0 ). The stability and the convergence to the TWS of the minimum velocity are also proved
for the Fisher-KPP equation. To check these in our case, we rewrite Eqs. (5) as
ou
ou
c
¼
 ðu  1Þðu þ 1Þ þ OðmÞ;
2
ot oX
2 

om
om
cþk
¼
ð1  uÞ þ Oðm2 Þ;
m 1
ot
2
oX 2

ðC9Þ

where we used d = 1 and neglected the second and higher order terms of c and k. Apparently,
m = 0 is invariant and we obtainououc ðu  1Þðu þ 1Þ;
¼
2
ot oX
2

ðC10Þ

which is essentially identicalp
toﬃﬃﬃ the Fisher-KPP equation. Therefore, the stability and the convergence to the TWS for v ¼ 2 c may also hold in our case.
The above argument proves that Eqs. (5) reduces to a Fisher-KPP equation and have the same
property (existence and stability of a TWS) if we neglect all the second and higher order terms of cand k. Actually, Eqs. (8) converges to Eqs. (C4 0 ) as c ! 0 and k ! 0. Unfortunately, however, theexistence of a TWS in Eqs. (C4 0 ) does not in general ensure the existence of a TWS in Eqs. (8). We
could only show that there is an orbit in Eqs. (8) connecting Px and P such that P ! Py. However,as we have seen, our system is closely related to Fisher-KPP equation although they look very
different.

J.Y. Wakano / Mathematical Biosciences 201 (2006) 72–89

89

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