1
Supporting Information
2
Figure S1: Parameter dependence of the dynamic regime in the community, assuming weak
3
interspecific competition (  < 1 (  = 0.3 and  = 1.3)). (a) Weak mutualistic interaction
4
(M = 0.1). (b) Strong mutualistic competition (M = 2). Other information is the same as in
5
Fig. 2.
6
Figure S2: Parameter dependence of the dynamic regime in the community, assuming strong
7
interspecific competition (  > 1 (  = 0.8 and  = 1.8)). (a) Weak mutualistic interaction
8
(M = 0.1). (b) Strong mutualistic competition (M = 2). In phase IIIa in (b), only population
9
cycles (PC) occur (Fig. 1c). In phase IIIa in (a), both population cycles and the trait cycle of
10
the mutualist (TC (b)) occur. In phase IIIb in (a), only population cycles occur. In phase IIIc,
11
both population cycles and trait cycles (TC) occur. In phase IIIa, only population cycles
12
occur. In phase IIIb and IIId, both population cycles and trait cycles occur. In phase IIIc in
13
(b), the equilibrium is stable. Other information is the same as in Fig. 2.
14
Figure S3: Parameter dependence of dynamic regime in the community. We assume weak
15
interspecific competition (  < 1 (  = 0.3 and  = 1.3)). Other information is the same as
16
in Fig. 2.
17
Figure S4: Effects of mutualism on the abundance of other species. Weak interspecific
18
competition (  > 1 (  = 0.8 and  = 1.8)). Phases shown in the upper side of the panels
1
1
correspond to those in Fig. S3. Other information is the same as in Fig. 4.
2
Figure S5: Parameter dependence of the dynamic regime in the community. Horizontal axes
3
are the strengths of the mutualistic interaction relative to the antagonistic interaction, S’. We
4
assume weak interspecific competition (  < 1 (  = 0.3 and  = 1.3)) and set A = 0.05 and
5
GY = 0.05, to prevent the subsystem of exploiter-two resources (inferior competitor does not
6
persist). In this setting, the subsystem of mutualist-two resources also does not persist
7
(inferior competitor does not persist). (a) Faster adaptation of mutualist (GZ = 0.01). (b)
8
Slower adaptation of mutualist (GZ = 0.001). Other information is the same as in Fig. 2.
9
Figure S6: Coexistence region in the absence of adaptation. (a) S = 0.1, (b) S = 0.5, (c) S =
10
1. Horizontal axes are the interaction effort of mutualist to resource species 1 and vertical
11
axes are the foraging effort of exploiter to resource species 1. In grey region, the all species
12
coexist. In white region, the coexistence of four species is impossible. These regions are
13
determined by mean values of population dynamics after the dynamics approach to the
14
asymptotic behaviors. Parameter values are same as in Fig. 2.
15
16
17
18
19
20
21
22
23
2
1
Figure S1
2
3
4
5
6
7
Weak interspecific competition
Mean abundance (log10)
Weak mutualistic interaction (M = 0.1)
1
a
IIa
I
IIIa
Strong mutualistic interaction(M = 2)
IIIc
IIIb
(α β < 1)
2
b
I
IIa IIb
IIIa
IIIb
1
0
0
-1
-1
-2
-2
-3
-3
Mean effort to
superior competitor
1
0
8
9
-2
-1
0
1
2
-3.3
-2.3
-1.3
Relative strength of antagonistic interaction S (log10)
10
11
12
13
14
15
16
17
18
3
-0.3
0.7
1
Figure S2
2
3
4
5
6
7
Strong interspecific competition
Mean abundance (log10)
Weak mutualistic interaction (M = 0.1)
1
a
IIa
I
IIIa IIIb
(α β > 1)
Strong mutualistic interaction(M = 2)
IIIc
2
b
I
IIa
IIb
IIIa
IIIb
PC
PC&TC
IIIc
IIId
1
0
0
-1
-1
-2
-2
-3
-3
Mean effort to
superior competitor
1
PC
PC & TC
TC
(b)
0
8
PC&TC
PC
-2
-1
0
1
2
-3.3
-2.3
-1.3
Relative strength of antagonistic interaction S (log10)
9
10
11
12
13
14
15
16
17
18
4
-0.3
0.7
1
Figure S3
2
3
4
5
6
Weak interspecific competition
IIaIIb
Mean abundance (log10)
I
IIIa
(α β < 1)
IIIc
IIIb
2
Y
Z
0
X1
X2
-2
Mean effort to
superior competitor
1
a
b
0
-3
7
-2
-1
0
1
Relative strength of antagonistic interaction S (log10)
8
9
10
11
5
1
Figure S4
2
3
4
5
IIIa
IIIc
IIIb
Effect of mutualism
10
0
6
-1
0
1
Relative strength of antagonistic interaction
S (log10)
7
8
9
10
11
12
6
1
Figure S5
2
3
4
Weak interspecific competition
Mean abundance (log10)
Faster adaptation of mutalist (Gz = 0.01)
3
a
Slower adaptation of mutualist (Gz = 0.001)
b
II
I
(α β < 1)
II
I
2
1
0
Mean effort to
superior competitor
1
0
5
-1.7
-0.7
0.3
1.3
2.3
-1.7
-0.7
0.3
Relative strength of mutualistic interaction S’ (log10)
6
7
8
9
10
7
1.3
2.3
1
Figure S6
2
3
4
5
a
S = 0.1
b
S = 0.5
c
S = 1.0
1
Coexistence
a1
No coexistence
0
6
1
b1
0
1
b1
7
8
9
10
11
12
13
14
15
8
0
1
b1
1
SI Appendix
2
Competitive exclusion caused by the mutualist
3
In this section, we mathematically show that competitive exclusion in the lower trophic level
4
is inevitable when the exploiter does not exist. In the absence of the exploiter, it is clear that
5
the mutualist always chooses the most abundant resource species. Given this assumption, the
6
equilibrium abundances of the two resource species are given by:
7


8
X 1*  rX  X 2* / 
[1a]
9
X 2*  rX  rX  C   / 1   ,
[1b]
10




11
where
12
than the intraspecific competition (i.e., 1 –  > 0), X2* is always negative because
13
rX  rX  C   0. This fact implies the competitive exclusion of the inferior competitor.
14
When the interspecific competition is greater than intraspecific competition (i.e., 1 –  < 0),
15
X1* < 0, implying the competitive exclusion of the superior competitor. These analyses
16
suggest that the competitive exclusion of one species at the lower trophic level is inevitable in
17
the absence of an exploiter. Although the trade-off between some traits could prevent
18
competitive exclusion, this condition is not within the scope of this study.
C  u1Z1* / 1  Z1*  u2 Z 2* / 1  Z 2*  0.
When the interspecific competition is weaker
19
9
1
Mathematical analysis of phase II
2
The following equations offer a simple form of the model for phase II (see Fig. 2):
3

4
X 1  rX  X 1  AY  u1Z / h  Z X 1 ,
5
Y   AgX 1  d Y ,
6
Z  rZ   Z Z  v1 X 1 / h  X 1 Z ,
[2a]

[2b]

[2c]
7
8
From these equations, we can obtain the nontrivial equilibrium:
9
10
X 1* 
d
,
Ag
11
Y* 
1
rX  X 1*  D ,
A
[3b]
12
Z* 
1 
v X* 
 rZ  1 1 * ,
Z 
h  X1 
[3c]

[3a]

13
14
where D = u1/[1+{HεZ(h+X1*)/{rZ(h+X1*)+ v1X1*}]. In this study, we assume S = A/M (M =
15
1). The equilibrium response to the increasing A (or S) in this system is given by:
16
17
dX 1*
  X 1* / A  0,
dA
[4a]
18
dY *
 rX  2 X 1*  E ,
dA
[4b]
10
1
dZ *
Hv1 X 1*

dA
A h  X 1*


2
 0,
[4c]
2
3
*
*
*
where E  u1 Ah Z dZ / dA  Z 2h Z  Z . The equilibrium responses of the superior
h Z  Z * 
4
competitor and the mutualist are always negative, but that of the exploiter depends on the
5
parameters. We assume A  0 , because A is very small in phase IIa. Thus, E  D. The
6
equilibrium response of the exploiter is approximately given by:
7
8
dY *
 rX  2 X 1*  D,
dA
[5]
9
10
If the exploiter is not feasible in the absence of the mutualist (D = 0 and rX < X1*), then
11
dY*/dA is always positive. Otherwise (D = 0 and rX > X1*), dY*/dA can be negative.
12
13
Mathematical analysis of phase IIIa
14
The simplest model of phase IIIa (Fig. 2) is given by the following equations:
15

16
X 1  rX  X 1  X 2  AY  u1Z / h  Z X 1 ,
17
X 2  rX  X 2  X 1 X 2 ,
18
Y   AgX1  d Y ,
19
Z  rZ  ZZ  v1 X 1 / h  X 1 Z ,
[6a]

[6b]

[6c]

[6d]
11
1
2
Thus, the nontrivial equilibrium is as follows:
3
4
X 1* 
d
,
Ag
[7a]
5
X 2*  rX  X 1* ,
[7b]
6
Y* 

1
u1 hrZ  X 1* rZ  v1 
*




r
1



X
1



,
X
1

*
A
H rZ   Z h   X 1 rZ   Z h  v1 
[7c]
7
Z* 
rZ h  X 1* rZ  v1 
,
 Z h  X 1*
[7d]




8
9
We evaluate the response of the equilibrium to the increase in the strength of the antagonistic
10
interaction A (or S), shown in Fig. 2. The equilibrium responses, except for that of the
11
exploiter, are given by:
12
13
dX 1*
dX 2*
dZ *
 0,
 0,
 0.
dA
dA
dA
[8]
14
15
The response of the exploiter depends on the magnitude of the interspecific competition
16
relative to the intraspecific competition. When   1 ,
17
18
dY *
 0,
dA
[9]
19
12
1
but when   1 , it might be positive or negative depending on the parameters.
2
13