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Evolutionary Stability in the Asymmetric Volunteer’s Dilemma
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Supporting Information
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Jun-Zhou He, Rui-Wu Wang and Yao-Tang Li
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Appendix S1: The equilibrium points and Local stability analysis
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From the definition of equilibrium point of nonlinear systems, we know that the
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equilibrium points of the equation (2.3) in text are the solutions of the following
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equations
 S ( 1  S )K( S U S N 1W ) 0
.

N 2

(
1


)
K
(

U



)
0
W
W
W
W
S
W

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(1)
Now, we solve the equation (1). Solving  S (1   S )( K S  U S W N 1 )  0 , we get
 S  0,  S  1, W   K S U S 
1 ( N 1)
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.
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Solving W (1  W )( KW  UW  S W N  2 )  0 , we get W  0, W  1 and
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KW  UW  S W N  2  0 .
(2)
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Obviously, points (  S , W )  (0,1), (0, 0), (1, 0), (1,1) are the solutions of the
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equations (1). We denote them as A(0,1) , B (0, 0) , C (1, 0) , D (1,1) , respectively,
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where  S and W are respectively the frequencies of defectors of “strong” and
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“weak” players.
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Substituting  S  1 in Equation (2) yields:
W   KW UW 
1 ( N  2)
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  ,    1,  K
UW 
1 ( N 2)

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that is,
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denote it as E (1, W* ) ，here W*   KW UW 
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S
W
W
，
is also a solution of the equations (1), and we
1 ( N  2)
Substituting W   K S U S 
1 ( N 1)
.
in Equation (2), we obtain
1
 S   KW UW  K S U S 
  KW UW  K S U S 
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( N  2) ( N 1)
( N 11) ( N 1)
  KW UW  K S U S  K S U S 
1 ( N 1)
 U S KW K SUW  K S U S 
1 ( N 1)


, W   U S KW K SUW  K S U S 
.
,
 KS
denote
it
1 ( N 1)
US 
1 ( N 1)

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So
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the
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 S**  U S KW K SUW  K S U S 
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From above discussion, we obtain six equilibrium points of the nonlinear system (2.3):
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A(0,1) , B (0, 0) , C (1, 0) , D (1,1) , E (1, W* ) , F (  S** , W** ) ,
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S
equations
(1),
we
and
1 ( N 1)
where W*   KW UW 
1 ( N  2)
F (  S** , W** )
as
and W**   K S U S 
1 ( N 1)
,  S**  U S KW K SUW  K S U S 
1 ( N 1)
is also a solution of
，
here
.
, W**   K S U S 
1 ( N 1)
.
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The following is the local stability analysis of the system of volunteer dilemma
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game. In text, the linearization of the replicator dynamics (2.3) at an equilibrium point
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(  S , W ) is
 f (  S , W )
 g (  S , W )
  
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S
f ( S ,
g ( S ,
)  W
  (S
W)
S
.
W
,
W
)  ( S , W
(3)
)
For convenience, let
 f (  S , W )  S
J (  S , W )  
 g (  S , W )  S
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S
f (  S , W ) W 
，

g (  S , W )  S  (  ,  ) (  ,  )
S W
S W
(4)
where
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f (  S , W )  S  (1  2 S )( K S  U S W N 1 ) ,
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f (  S , W ) W  ( N  1)U S W N  2  S (1   S ) ,
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g (  S , W )  S  UW W N 1 (1  W ) ,
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g (  S , W )  S  (1  2W ) KW  [ N W  ( N  1)]UW  S W N  2 ,
2
(  S , W ) represents the equilibrium points of the nonlinear system (2.3),  S and
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and
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W are respectively the frequencies of defection of “strong” and “weak” players.
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From (4), we obtain the Jacobi matrices of the linear system (3) of the nonlinear
system (2.3) at these six equilibrium points:
0 
,
KW 
 K US
J ( 0 , 1) S
 0
0 
,
 KW 
 K
J ( 1 , 0) S
 0
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K
J (0, 0)   S
 0
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 [ K S  U S ( W* ) N 1 ]

0
0
U S  K S

*
J (1,1)  
 , J (1, W )  
* 
UW  KW 
0
( N  2) KW (1  W ) 
 0

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
0
( N  1)U S  S** (1   S** )( W** ) N 2 
J (  S** , W** )  
.
**
** N 1
( N  2)UW  S** (1  W** )( W** ) N 2 
 UW (1  W )( W )
0 
,
KW 
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The local stability of the nonlinear system (2.3) at equilibrium points is
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determined by the eigenvalues of the Jacobi matrices. Now, we analyze character of
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the eigenvalues of above six matrices.
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Since 0  K S  KW  UW  U S and
J (0, 0) , J (1,1) , J (0,1) and
J (1, 0) are
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diagonal matrices, we can easily get that the eigenvalues of J (0, 0) and J (1,1) are
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all positive, and the eigenvalues of J (0,1) are negative. On thus condition, one of
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the eigenvalues of J (1, 0) is positive and another is negative.
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For J (1, W* ) , we obtain easily that its eigenvalues are ( N  2) KW (1  W* ) and
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[ K S  U S ( W* ) N 1 ] . Obviously,
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0  K S U S  KW UW  1 , we know that yield
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W*   KW UW 
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negative if and only if K S U S   KW UW 
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J (1, W* ) are all negative under the special condition
1 N 2
( N  2) KW (1  W* )  0 .
) demand K S U S   KW UW 
[ K S  U S ( W* ) N 1 ]  0
N 1 N  2
N 1 N  2
In addition, from
(here
. So another eigenvalue is
, that is, the eigenvalues of
 KS
US 
N 2
  KW UW 
N 1
.
3
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As to J (  S** , W** ) , its characteristic equation is
 2  a22   a12 a21  0 .
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Solving Equation (8), we get

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2
a22  a22
 4a12 a21
，
2
where
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a12  ( N  1)U S  S** (1   S** )( W** ) N 2 ，
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a21  UW (1  W** )( W** ) N 1 ，
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a22  ( N  2)UW  S** (1  W** )( W** ) N  2 .
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(8)
Obviously, a12  0, a21  0
and
a22  0 ,
then
it
is
easy
to
obtain
2
a22  a22
 4a12 a21 . Hence, one of the eigenvalues of J (  S** , W** ) is a positive real
number and another is a negative.
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From above analysis of the eigenvalues and the stability theory of differential
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equations (Hofbauer and Sigmund 1998; Hirsch et al. 2004), we obtain that A(0,1)
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and E (1, W* ) are sinks (stable); and that B (0, 0) and D (1,1) are sources (unstable);
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C (1, 0) and
F (  S** , W** ) are saddles (unstable).
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References
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Hirsch, M. W., Smale, S., Devaney, R. L., 2004. Differential Equations, Dynamical
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Systems, and an Introduction to Chaos. Calif. Acad. Press.
Hofbauer, J., Sigmund, K., 1998. Evolutionary Games and Population Dynamics.
Cambridge Uni. Press.
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