Cooperation and Stability through Periodic Impulses
Supporting Information
Bo-Yu Zhang, Ross Cressman and Yi Tao
I.
PD Game
For the PD game, the evolution of the numbers of Defectors and Cooperators, denoted
respectively by n1 and n2 , is
dn1 n1
  Pn1  Tn2 
dt N
dn2 n2
  Sn1  Rn2 
dt
N
(S. 1)
for t  k ( k  1, 2,3,  ). When t  k ( k  1, 2,3,  ), the jumps in n1 and n2 at
t  k ( k  1, 2,3,  ) are given by
n1  k    1n1  k  
(S. 2)
n2  k    2 n2  k  

where ni  k  is defined as ni  k   ni k

  n  k  . With p  n

i
N , the replicator
1
equation for the time evolution of p is
dp
 p 1  p    P  S  p  T  R 1  p  
dt
(S. 3a)
for t  k . Similarly, the jump in p at t  k ( k  1, 2,3,  ) is given by
p  k   p  k    p  k  

n1  k  
n1  k

  n  k 

 p  k  
2
1   1  p  k  

 p  k  


1   1  p  k   1   2  1  p  k  

 p  k   1  p  k  
 1   p  k    1  p  k  .


2
1

1

2
That is, we have
p  p 1  p 
 2  1
1   1 p   2 1  p 
(S. 3b)
for t  k ( k  1, 2,3,  ). Combining Eqs. S.3a and S.3b, we get the replicator equation with
periodic impulses
dp
 p 1  p   a2  p  a1  a2  
dt
p  p 1  p U  p 
t  k
t  k
(S.4a)
for k  1,2,3, 
(S.4b)


where a1  S  P  0 , a2  T  R  0 and U  p    2   1  1   1 p   2 1  p   0
when
 2   1  1 . This system was analyzed for two-strategy symmetric games by Wang et al.
[1]. The following analysis uses a different approach that rests on translating the system to a
difference equation (Eq. S.5). The generalization of this difference equation method to the BS
 -periodic solution in this game (see
game is needed to prove the existence of an interior
Theorem 2 below).
Clearly, for Eq. S.4 both p  0 and p  1 are boundary equilibria. In order to determine
the existence of the interior
 -periodic solution of (S.4), let F  p   p  (k  1)    p  k   .
Then p  t  is an interior
 -periodic solution of (S. 4) if and only if F  p   0 . That is,

p  p  k    p  (k  1)   since
p  (k  1)    p  k    
( k 1)
k

where G  p   p 1  p  a2  p  a1  a2 

G  p  t   dt
,
and p  t  is the solution to (S. 4a) for
k  t  (k  1) with initial condition p  k    p  k    p  k  .
From the definition of F  p  , we have also
F  p   p  
( k 1)
k
G  p  dt .
The boundary equilibrium p  0 (or p  1 ), or an interior
only if it is a stable equilibrium of the difference equation
 -periodic solution, is stable if and
p  (k  1)    p  k    F  p 
,
(S. 5)
where F  p  is continuous for p  0,1 , and F  0  F 1  0 .
1. Case a1  a2  0
Notice that G  p   a2 p 1  p  . Thus, for k  t  (k  1) the solution of (S.4a), p  t  ,
can be expressed as p  t   Ce
a2t
1  Ce 
a2t
with C  p0 1  p0  where p0 denotes the
initial value. This implies that
F  p 

Cea2
p
1  Cea2
,

where C   p  p  1   p  p  . It is straightforward to show that F  p   0 for all


possible p in the interval 0  p  1 if a2  ln 1   1  1   2   0 . On the other hand,


if a2  ln 1   1  1   2   0 , then no solution of equation F  p   0 can exist. This


implies that under the condition a1  a2  0 , if a2  ln 1   1  1   2   0 , then for any
initial value in the interval 0  p0  1 , p  t , p0  will be a
 -periodic solution of (S.4), and it


is neutrally stable (see also [1]). Conversely, if a2  ln 1   1  1   2   0 under the
condition a1  a2  0 , then no interior
 -periodic solution can exist. In fact, since
Cea2 1  Cea2  is an increasing function of a2 , if a2  ln  1   1  1   2    0 , then
F  p   0 for all 0  p  1 , i.e., p  0 is globally stable. Similarly, if
a2  ln  1   1  1   2    0 , then F  p   0 for all 0  p  1 , i.e., p  1 is globally
stable.
2. Stability of boundary equilibria under a1  a2  0
Define   p  

p
p p
ds G  s  , i.e.,   p  is the amount of time for the change of the
Defector frequency from p  p to p . Notice that
2
 1

p
a1  a2 

1
1
  p  
ds  




p p G s
p p a s
a1 1  s  a1a2  a2  s  a1  a2   
 
 2
p

a2  p  a1  a2 
a a
1
p
1
1 p
ln
 ln
 1 2 ln
a2 p  p a1 1   p  p 
a1a2
a2   p  p  a1  a2 

a2  p  a1  a2 
a a
1
1
1
1
ln
 ln
 1 2 ln
,
a2 1  1  p  U  p  a1 1  pU  p 
a1a2
a2   p  p  a1  a2 


and lim   p   1 a2  ln 1   2  1   1  . Thus, the boundary p  0 is stable if
p 0
1 a2  ln 1   2  1  1   


( a2  ln 1   1  1   2   0 ), and it is unstable if
a2  ln  1   1  1   2    0 . For the situation with a2  ln  1   2  1   1    0 (i.e.,
lim   p    ), notice that
p  0
d  p
dp

 a1  a2 U  p 
 a2  p  a1  a2    a2   p  p  a1  a2 
and lim d   p  dp    a1  a2  2   1 
p 0
 a 1    . Thus, when
2
2
2
p is sufficiently
close to 0 ,   p    if a1  a2  0 , i.e., p  0 is stable; and   p    if a1  a2  0 ,
i.e., p  0 is unstable.
For convenience, define
1
  ln  1   2  1   1   .

The stability of boundary equilibria is given by the following theorem and summarized in Table
S1. The above analysis of Cases 1 and 2 provides the proofs for (a), (b) and (e) in Theorem 1. The
proofs of (c) and (d) are similar.
Theorem 1. For the stability of the boundary p  0 , (a) if a1  a2  0 , then p  0 is stable
if and only if a2   and unstable if and only if a2   ; (b) if a1  a2  0 , then p  0 is
stable if and only if a2   and unstable if and only if a2   . Similarly, for the stability of the
boundary p  1 , (c) if a1  a2  0 , then p  1 is stable if and only if a1    0 and
unstable if and only if a1    0 ; (d) if a1  a2  0 , then p  1 is stable if and only if
a1    0 and unstable if and only if a1    0 . (e) As a special case with a1  a2  0 , if
a2   (i.e. a1    0 ), then for any initial value in the interval 0  p0  1 , p  t , p0  will
be a
 -periodic solution; and if a2   (i.e. a1    0 ), then no interior periodic solution
can exist. In this situation, if a2   (i.e. a1    0 ), p  0 is globally stable; and if
a2   (i.e. a1    0 ), p  1 is globally stable.
Table S1
Boundaries
p0
p 1
Stable
Unstable
Neutral
a1  a2  0
a2  
a2  
——
a1  a2  0
a2  
a2  
a2  
a1  a2  0
a2  
a2  
——
a1  a2  0
a1    0
a1    0
——
a1  a2  0
a1    0
a1    0
a1    0
a1  a2  0
a1    0
a1    0
——
3. Existence and stability of interior  -periodic solution under a1  a2  0
We first show that there exists at most one interior
 -periodic solution of Eq. S.4. Suppose that
p  t  is an interior  -periodic solution of (S.4), i.e. p  p  k    p  (k  1)   and
F  p    0 ). At p  p we have that
dF  p  d p d


dp
dp dt

( k 1)
k
G  p  dt
 dpdt
1  2 p  p   p   G  p   G  p  p 

p 1  p 
G p 
 a  p  a  a    1  2 p  p   p    G  p   G  p  p 

G p 

 2







2

1
 2



2


p  a1  a2 
G  p 
 p 1  p   p 1  2 p




 p 

since
G  p   G  p  p   p 1  p   a2  p  a1  a2  
  p  p 1  p  p   a2   p  p  a1  a2  
 p  a1  a2   p 1  p   p 1  2 p  p  
  a2  p  a1  a2   p 1  2 p  p  .
 0

Notice that p  0 , G p

and
 p 1  p   p 1  2 p





 p   p 1  p 


1   1 1   2 
1   1 p   2 1  p 

 p 1  p  1  1  p  U  p  1  pU  p   0 .
Thus dF  p  dp
p  p
is positive if a1  a2  0 and negative if a1  a2  0 . In particular,
dF  p  dp p  p is non zero. Since dF  p  dp p  p has the same sign for all possible p  in


the interval 0  p  1 , there exists at most one p . That is, if an interior
 -periodic solution
exists, then it is unique.
Clearly, from the boundary stability, no interior
 -periodic solution can exist unless both
p  0 and p  1 are stable or both are unstable. If p  0 is stable (unstable), then
F  p   0 ( F  p   0 ) for p near 0 ; and if p  1 is stable (unstable), then F  p   0
( F  p   0 ) for p near 1 . From the continuity of F  p  ,
there is at least one p in the

interval 0  p  1 such that F  p   0 . Thus, only one p can exist under the condition
a1  a2  0 . That is, if both p  0 and p  1 are stable or unstable, then (S.4) has a unique
interior
 -periodic solution.

Notice that p (k  1)

  p  k   F  p 

(see Eq. S.5). Thus, p
asymptotically stable equilibrium of (S.5) if and only if 1  dF  p  dp
is a stable interior
p  p

is an
 0 , or p  t 
 -periodic solution of (S.4) if 1  dF  p  dp p  p  0 . It is easy to see

that if p  is locally asymptotically stable, then it must also be globally asymptotically stable.
For stability of the interior  -periodic solution we have that: (i) If both boundaries p  0


and p  1 are stable, or unstable, then there must exist a unique p with 0  p  1 such
   0 ; (ii) If one boundary is stable but the other unstable, then no interior
that F p

 -periodic solution can exist; (iii) If both boundaries are stable, then there is an
interior  -periodic solution but it is unstable since dF  p  dp
p  p
 0 ; and (iv) if both
boundaries are unstable, then there is an interior  -periodic solution and it is globally
asymptotically stable since 1  dF  p  dp
II.
p  p
0.
BS Game
For the BS game, let n1 and n2 be the numbers of philandering and faithful individuals in
the male population, respectively, and m1 and m2 be the numbers of coy and fast individuals in
the female population, respectively. Then we have
dn1
 n1G (1  q ),
dt
dn2
C
C
 n2 ((G   E )q  (G  )(1  q )),
dt
2
2
dm1
C
 m1 (G   E )(1  p ),
dt
2
dm2
C
 m2 ((G  C ) p  (G  )(1  p )),
dt
2
where p  n1 (n1  n2 ) is the frequency of philanders in the male population and
(S.6)
q  m1 (m1  m2 ) is the frequency of coy females in their population. At time t  k
( k  1, 2,  ), the jumps in ni and mi ( i  1, 2 ) are ni   i ni and mi   i mi ,
respectively.
The bimatrix replicator equation is
dp
 p 1  p   C 2  q  G  E   ,
dt
dq
 q 1  q    p  G  E  C   E 
dt
(S.7a)
for t  k ( k  1, 2,  ). Similar to the PD game model, the jumps in p and q at time
t  k ( k  1, 2,  ) are given by
 2  1
1   1 p   2 (1  p)
 2  1
q  q 1  q 
.
1  1q  2 (1  q )
p  p 1  p 
(S.7b)
Combining Eqs. S.7a and S.7b together, we have that
dp
 p 1  p   a2  q  a1  a2  
dt
dq
 q 1  q   b2  p  b1  b2  
dt
for t  k
for t  k
(S. 8a)
(S. 8b)
p  p 1  p U  p 
for t  k , k  1, 2,3, 
(S. 8c)
q  q 1  q V  q 
for t  k , k  1, 2,3, 
(S. 8d)
where (i) a1  G  C 2  E  0 , a2  C 2  0 , b1  G  C  0 and b2   E  0 ; (ii)
p  k   p  k    p  k   and q  k   q  k    q  k   denotes the jump in p and
q at moment t  k , respectively; and (iii) U ( p ) and V (q ) are given by
U  p 
V q 
 2  1
,
1   1 p   2 1  p 
 2  1
.
1  1q  2 1  q 
Clearly, Eq. S.8 can be equivalently expressed as the difference equations
p  (k  1)    p  k    p  k   
( k 1)
k
q  (k  1)    q  k    q  k   
( k 1)
k

where P  p, q   p 1  p  a2  q  a1  a2 

P  p, q  dt ,
(S. 9a)
Q  p, q  dt ,
(S. 9b)


and Q  p, q   q 1  q  b2  p  b1  b2  .
1. Boundary stability
First, consider the stability of the boundary (0,0) . If this vertex is to be stable, then it must
be stable on the two edges p  0 and q  0 of the unit square. On the edge q  0 , we have a
simplified PD game among males. From the analysis of these games, we know that p  0 is


stable if a2  ln 1   1  1   2   0 and unstable if this inequality is reversed. By a similar
analysis of the simplified PD game among females on the edge p  0 , we find q  0 is stable


if b2  ln 1  1  1  2   0 and unstable if this inequality is reversed. Thus, stability of
 0, 0 




requires that a2  ln 1   1  1   2   0 and b2  ln 1  1  1  2   0 .




Suppose that a2  ln 1   1  1   2   0 and b2  ln 1  1  1  2   0 .


Since a2  ln 1   1  1   2   0 , on the edge q  0 , dp dt  p 1  p  a2  0 and
p  (k  1)    p  k   when 0  p  1 (since p is sufficiently negative to more than
reverse the increase in p during the season). For q  0 , dp dt  p 1  p  a2 and  p is

the same as when q  0 . Thus, the difference between p ( k  1)

when q  0 than it is when q  0 . In particular, p (k  1)




and p k
  p  k 



is greater
for all 0  q  1 .
Thus, along any trajectory (that does not start on the edge p  1 ) we have p  0 . Once p is

sufficiently close to p  0 , we know that q (k  1)

  q  k 

b2  ln  1 1  1  2    0 . Thus,  0, 0  is globally stable.
For convenience, define
since
1
 m  ln  1   2  1   1  

1
 f  ln  1  2  1  1   .

The above analysis yields the stability conditions in the first row of Table S2. The stability
conditions in the other three rows follow from a similar argument showing that stability of the
simplified PD games on the adjacent edges to a vertex implies global stability in the unit square.
In particular, if one vertex is stable, it is globally stable.
Table S2.
Boundary Equilibrium
Stability Conditions
 0, 0 
a2  m and b2   f
1,0
a2  m and b1   f  0
 0,1
a1  m  0 and b2   f
1,1
a1   m  0 and b1   f  0
2. Existence of interior  -periodic solutions
In the rest of this section, suppose that no vertex is stable. For the existence of
 -periodic solution, we have following theorem.
Theorem 2.
If no vertex is stable in equation Eq. S.8, then there is at least one
 -periodic
solution.
Proof. Obviously, Eq. S.8 has a
 -periodic solution if and only if Eq. S.9 has an interior
equilibrium, which is the solution of equation
F  p, q   p  
( k 1)
G  p, q   q  
( k 1)
k
k
P  p, q  dt  0 ,
(S. 10a)
Q  p, q  dt  0 .
(S. 10b)


Without loss of generality, we assume that a2  ln 1   1  1   2   0 . This implies
a1  ln  1   2  1   1    0 , b1  ln  1  2  1  1    0 and
b2  ln  1 1  1  2    0 since no vertex is stable (see Table 2).
Then F  p,0   0 , F  p,1  0 , G ( p, 0)  0 and G ( p,1)  0 for 0  p  1 .
Also G  0, q   0 , G 1, q   0 , F (0, q)  0 and F (1, q )  0 for 0  q  1 . Consider the
continuous-time dynamical system dp dt  F ( p, q) , dq dt  G( p, q) with continuous
vector field given by F ( p, q) and G ( p, q) . The edges of the unit square form a
counterclockwise heteroclinic cycle for this dynamics. Since no interior trajectories converge to a
point on one of these edges, the Poincare-Bendixson theorem [2] can be used to show that there is
at least one interior equilibrium. Each such equilibrium corresponds to a
 -periodic solution of
Eq. S.8.
Unlike the corresponding result for the PD game with periodic impulses, the uniqueness of
 -periodic solutions in the BS game with periodic impulses cannot be guaranteed in general.
Numerical simulations suggest that there is only one interior
and that the interior
 -periodic solution if  is small,
 -periodic solution is not unique if  is large. Generally, the number of
 -periodic solutions depends on the relationship between  and the periodic orbits of Eq.
S.8ab.
Let Tmin denote the shortest period of periodic orbit of Eq. S.8ab. In general, it is difficult
to get the exact expression of Tmin , but simulation shows that it can be approximated by the
*
*
period of interior periodic cycles near the interior equilibrium ( p , q ) . When the system state is
near the interior equilibrium, Eq. S.8ab can be approximated by
dp
 a  q  q 
dt
dq
 b  p  p 
dt


where a  p 1  p

a  a 
1
2


and b  q 1  q
(S.11a)
(S.11b)

  b  b  . By setting
1
2
y  q  q and
x  p  p , Eq. S.11 becomes
 dx 
 dt   0 a  x 
 
  .
 dy   b 0  y 
 
 dt 
(S.12)
 0 a 
 are 1 , 2  i ab . Clearly, the solution of Eq.
 b 0 
The eigenvalues of the matrix 
S.12 is given by
 x(t ) 

  l exp i abt  m exp i abt
 y (t ) 




,
(S.13)
where l and m are eigenvectors corresponding to eigenvalues 1 and 2 , and its period is
2
ab which is the period of interior periodic cycles near the interior equilibrium ( p* , q* ) .
Thus, Tmin can be approximated as 2
 p 1  p   a1  a2  q* (1  q* )(b1  b2 ) .
From the numerical simulations, there appears to be a simple law for the maximum number
of
 -periodic solutions, denoted by N , which is given by N  2  Tmin   1. Thus, if  is
less than Tmin , there is a unique interior  -periodic solution which is given by Theorem 2.
Assume that
 is greater than Tmin from now on and denote by  T  the periodic orbit of
Eq. S.8ab with period T . Numerical simulations indicate that there are at most two
 -periodic
orbits between    and the boundary of the unit square (region III in Figure S1a); at most two
 -periodic orbits between    M 1  and   M  for M  2,3, ,  Tmin  (region
II in Figure S1a); and one in the region inside   M  for M   Tmin  (region I in Figure
S1a). Figure S1 illustrates that this maximum number is achieved for specific impulsive
parameters when payoff parameters are given by a1  3, a2  2, b1  2, b2  3 and

Tmin   2 (i.e. there are exactly five  -periodic orbits).
Numerical simulations also indicate that the actual number of interior
 -periodic solutions
decreases as the impulsive effects increase (Figure S2). The payoffs for Figures S1 and S2 (i.e.
a1  3, a2  2, b1  2, b2  3 ) are different than the payoffs used by Dawkins (see Figure 3
of the main text) since the three regions corresponding to Figure S1a for these latter payoffs are
more difficult to discern. Nevertheless, our simulations using Dawkins’ payoffs show that there
are also five interior
 -periodic solutions for small impulsive effects and that this number
decreases as the impulsive effects increase (Figure S2).
References:
1. Wang SC, Zhang BY, Li ZQ, Cressman R Tao Y (2008) Evolutionary game dynamics with
impulsive effects. J Theor Biol 254: 384–389.
2. Hofbauer J, Sigmund K (1998) Evolutionary Games and Population Dynamics. Cambridge
University Press, Cambridge.
Figure Captions
Fig. S1. Five interior  -periodic solutions of the bimatrix replicator equation with periodic
impulses of fixed intermediate strength for the BS game. The equations F  p, q   0 and
G  p, q   0 are represented by red and green curves, respectively, and the yellow curves
represent   2 (the interior periodic cycles of the bimatrix replicator dynamic with period
  2 ) (small cycle) and    (big cycle), where the parameters are taken as
a1  3, a2  2, b1  2, b2  3 ,  1   1  0 ,  2   2  0.1, Tmin  5.3 ,   15 and

Tmin   2 . Fig. S1a shows the three regions in the unit square formed by   2 and
   . From Figs. S1b-d, we can see that the solutions of F  p, q   0 and G  p, q   0
are isolated in three parts by   2 and    . Fig. S1d shows that there are five
 -periodic solutions (indicated by ), where two are in region III, two are in region II, and one is
in region I.
Fig. S2. Interior  -periodic solutions of the bimatrix replicator equation with periodic
impulses of varying intermediate strength for the BS game. The equations F  p, q   0 and
G  p, q   0 are represented by red and green curves, respectively, where the parameters are
taken as a1  3, a2  2, b1  2, b2  3 ,

 1   1  0 , Tmin  5.3 ,   15 and
Tmin   2 . In Fig.S2a-c,  2 and  2 are 0.5, 0.9 and 0.99, respectively, and the average
impulsive effects,  m and  f , are -0.034, -0.11 and -0.23, respectively.
can see that the number of
From Figs.S2a-c, we
 -periodic solutions decreases with the increase of impulsive effects.
Figures
Fig.S1
Fig.S1a
Fig. S1b
Fig. S1c
Fig. S1d
Fig.S2
Fig. S2a
Fig. S2b
Fig. S2c