Supplementary information
Analysis of the ecological dynamics (no evolution)
Consider a plant P and an herbivore H whose dynamics follow the following ODE
system:
ì dP
æ æ aPö
ö
= P ç r ç1-b H÷
÷
ï
ï dt
K ø
è è
ø
í
ï dH
= H ( f b P - m)
ï
î dt
Equilibria can be obtained by solving dP/dt=0 and dH/dt=0. Local stability can be
determined using the eigenvalues of the related jacobian matrix, assessed at the
ecological equilibrium. Three different equilibria exist:
1) A trivial equilibrium at which the plant and the herbivore are extinct (P1*=0,
H1*=0). Assuming that the intrinsic growth rate of the plant population is positive
(r>0), this equilibrium is always unstable.
2) An equilibrium that allows the existence of the plant population while the
herbivore population is extinct. (P2*=K/α, H2*=0). This equilibrium is stable when
the plant carrying capacity is small K a < m f b . Herbivores then have too little
energy available to bear their intrinsic mortality rate m.
3) A coexistence equilibrium ( P3* = m f b , H 3* = r 1- a P3* K b ) that is feasible and
(
)
stable provided K a > m f b , that is, when the plant carrying capacity is
sufficient to sustain the herbivore population.
By differentiating the plant and herbivore biomass at equilibrium with parameter K, it
may easily be seen that the increasing the carrying capacity has a positive effect on the
herbivore population, while the plant population remains constant. This is classical in
such Lotka-Volterra models (Oksanen et al. 1981; Oksanen and Oksanen 2000; Loeuille
and Loreau 2004) in which lower trophic levels are top-down controlled.
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Evolution of quantitative defenses
We consider a trait y that embodies the amount of quantitative defenses produced by
the plant (resident population). Within this resident population, rare mutation occurs,
and the dynamics of such mutants, whose trait is noted ym, follows the dynamics:
æ æ a (y , y )P + a (y , y )P ö
ö
dPm
m
m
m
m
r
r
= Pm ç r ç1 b
(y
)
H
÷ with b (y) = b0 e -a y and
÷
m
dt
K(y m )
ø
è è
ø
-b y
K(y) = K 0 e (see the main text for the justification of costs and benefits of quantitative
defenses)
Assuming that the mutant is initially rare and that mutations are rare enough for the
resident population to be at the ecological equilibrium, the relative fitness of mutant
individuals can then be approximated by the per capita growth rate of the mutant
population (Metz et al. 1992; Dieckmann and Law 1996):
æ æ a (y , y )P* ö
ö
*
m
r
W (ym , yr ) = çç r ç1÷ - b (ym )H ÷÷
K(ym ) ø
è è
ø
We explore two scenarios regarding the effects of traits on the competition rate:
- Competition is independent of trait similarity: a (y i , y k ) =1
- Competition increases when traits are more similar. Normalized competition rate  is
then written as a (y i , y k ) =
a0
s 2p
e
- ( yi -yk ) 2
2s 2
We use adaptive dynamics methods to study variations in the trait (Dieckmann and Law
1996; Geritz et al. 1998).
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Scenario 1: Competition is independent on trait difference.
Plants alone at equilibrium (equilibrium (2))
Equilibrium biomass
P* = K0 e-by and H * = 0
Conditions of feasibility
and stability
K < m f b (y)
Fitness
W (y m , y r ) = r 1 - e b ( ym -yr )
Fitness gradient
¶W (ym , yr )
¶ ym
y
y > y feas with y feas =
(
ln(b0 fK 0 m)
a+b
)
= -br
m ®yr
Trait variation
Evolutionary analysis
dy
1
= -r b K 0 e-by mw 2
dt
2
No singularity, the trait ever decreases. As a consequence,
equilibrium biomass P* increases continuously
Coexistence equilibrium (equilibrium (3))
Equilibrium biomass
m e ay
r e ay æ m e(a+b) y ö
*
*
P =
and H =
ç1÷
b0 f
b 0 è b0 f K 0 ø
Conditions of feasibility
and stability
Fitness
Fitness gradient
Trait variation
ln(b0 fK 0 m)
m (a+b) y
Û K0 >
e
a+b
b0 f
æ
ö
m
2a+b y
a+b y +ay
W (ym , yr ) = r e-aym ç( eaym - eayr ) +
e ( ) r - e( ) m r ÷
b0 f K 0
è
ø
æ m(a + b) (a+b) y ö
¶W (ym, yr )
r
=r ç a e
÷
¶ ym
b
f
K
è
ø
0
0
ym ®yr
H * > 0 Û y < y feas and y feas =
(
dy
¶W (ym , yr )
= kmw 2 P*
dt
¶ ym
ym ®yr
)
is >0 if y<y* and <0 if y>y*
Evolutionary singularity
y = y feas + ln( a (a + b)) (a + b)
Invasibility and
Convergence
¶ 2W (ym, yr )
¶ 2 ym
y
*
=- abr
m =yr =y
¶ W (ym, yr )
¶ 2 yr
y
2
m =yr =y
<0
*
=ar (2a + b) >
*
¶ 2W (ym, yr )
¶ 2 ym
y
m =yr =y
*
y* is a CSS (convergent and not invasible, Eshel 1983).
Quantitative defenses evolve to reach y*.
Pairwise invisibility plot
(positive fitness is in
grey)
Effects of enrichment
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CSS
¶P* (y* )
¶H * (y* )
¶y*
> 0,
> 0 and
>0
¶K 0
¶K 0
¶K 0
,
3
biomass and traits increase
Scenario 2: Competition increases with trait similarity
Plants alone at equilibrium (equilibrium (2))
Equilibrium biomass
K s 2p -by
P* = 0
e and H * = 0
a0
Conditions of feasibility
and stability
Fitness
æ b f K s 2p ö
0
y > y feas and y feas = l n ç 0
÷ (a + b)
m
a0 ø
è
( y -y ) ( 2bs 2 -ym +yr) 2s 2 ö
æ
÷
W (y m , y r ) = r ç1 - e m r
è
ø
Fitness gradient
Trait variation
Evolutionary analysis
dy
s p
= -r b K 0 e-by
mw 2
dt
a0 2
No singularity, the trait ever decreases. As a consequence,
equilibrium biomass P* increases continuously
Coexistence equilibrium (equilibrium (3))
Equilibrium biomass
m e ay
r e ay æ
a 0 me(a+b) y ö
P* =
and H * =
ç1÷
b0 f
b 0 è s 2p b 0 f K 0 ø
æ b f K s 2p ö
Conditions of feasibility
*
0
H
>
0
Û
y
<
y
and
y
=
ln
ç 0
÷ (a + b)
feas
feas
and stability
m
a0 ø
è
Fitness gradient
æ
¶W (y , y )
a m(a + b) (a+b)y ö
m
¶ ym
Trait variation
Evolutionary singularity
Invasibility and
Convergence
r
ym ®yr
0
=r ç a e
s
2
p b0 f K 0
è
r
÷
ø
dy
¶W (ym , yr )
= kmw 2 P*
dt
¶ ym
ym ®yr
y * = y feas + ln( a (a + b)) (a + b)
æ
ö
¶ 2W (ym, yr )
1
=ar
-b
+
> or < 0
ç
2÷
¶ 2 ym
(a
+
b)
s
*
è
ø
y =y =y
m
r
æ
ö ¶ 2W (ym , yr )
1
=ar ç 2a + b +
>
2÷
(a
+
b)
s
¶ 2 ym
*
è
ø
=y
=y
ym =yr =y*
m r
¶ W (ym, yr )
¶ 2 yr
y
2
y* is a CSS (convergent and not invasible) if and only if
1
s>
ab + b 2
1
When s <
, it becomes an evolutionary branching
ab + b 2
(convergent but invasible), so that disruptive selection leads
to the coexistence of strongly and lowly defended plants.
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Pairwise invisibility plot
(positive fitness is in
grey)
Effects of enrichment
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If s <
1
ab + b
2 :
If s >
1
ab + b 2
:
Branching point
CSS
*
*
*
*
¶P (y )
¶H (y )
¶y*
> 0,
> 0 and
>0
¶K 0
¶K 0
¶K 0
,
biomass and traits increase
5
Evolution of qualitative defenses
We now consider the evolution of qualitative defenses (trait x). The dynamics of the
mutant population, whose trait is xm, can be described by the following equation:
- ( p-x ) 2
æ æ a (x m , x m )Pm + a (x m , x r )Pr ö
ö
dPm
b
2
= Pm ç r ç1 ÷ - b (x m ) H ÷ with b (x) = 0 e 2g .
ø
dt
K
è è
ø
g 2p
The relative fitness of mutant is then:
æ æ a (x , x )P* ö
ö
*
m
r
W (xm , xr ) = ç r ç1÷ - b (xm ) H ÷
K
ø
è è
ø
As done for trait y, we analyse two scenarios:
- Competition is not dependent on trait similarity a (x i , x k ) =1
- Direct competition increases when plants are more similar in traits:
a (x i , x k ) =
a0
s 2p
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e
- ( xi -xk ) 2
2s 2
6
Scenario 1: Competition is not dependent of trait difference
Plants alone at equilibrium (equilibrium (2))
Equilibrium biomass
P* = K and H * = 0
Conditions of
feasibility and stability
Fitness
Fitness gradient
Trait variation
Evolutionary analysis
ì
b fK
ïï
A= 0
>1
mg 2p
í
ï
ïî x £ p - g 2 ln(A) et x ³ p + g 2 ln(A)
W (x m , x r ) = 0
ü
( p-x )2
ïï
m g 2p 2g2
e
ýÛK £
b0 f
ï
ïþ
dx
=0
dt
No evolutionary singularity. The trait is always neutral and its
dynamics only depends on genetic drift.
Coexistence equilibrium (equilibrium (3))
( p -x )2
( p-x)2 æ
( p-x )2 ö
Equilibrium biomass
m g 2p 2g 2
r g 2 p 2 g2 ç
2p m g 2 g2 ÷
*
*
and H =
e
1e
P =
e
ç
÷
b0 f
b0
b0 f K
è
ø
ì
Conditions of
ü
b fK
( p-x )2
ïï
A > 1 with A = 0
ïï
feasibility and stability
2
m
g
2
p
*
mg 2p
H > 0Ûí
e 2g
ýÛK >
b0 f
ï
ï
p
g
2
ln(A)
<
x
<
p
+
g
2
ln(A)
ï
ïî
þ
Fitness gradient
Trait variation
dx
¶W (xm, xr )
= kmw 2 P*
dt
¶ xm
xm ®xr
Evolutionary
singularity
Invasibility and
Convergence
x* = p
¶ 2W (x m , x r )
¶ 2 xm
x
m =x r
=x *
r æ 1 ö ¶ 2W (x m , x r )
= 2 ç1 - ÷ = g è Aø
¶ 2 xr
x
m =x r
=x *
x* is a repeller as it is not convergent and is invasible. Depending on
the relative position of the initial trait compared to p, the trait
increases or decreases, to evolve away from p. Eventually the
herbivore disappears and subsequent evolution is driven by drift.
Pairwise invisibility
plot (positive fitness is
in grey)
0.60
0.55
0.50
0.45
0.40
0.40
Effects of enrichment
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0.45
0.50
0.55
0.60
¶P (x )
¶H (x * )
¶x *
,
and
=0
>0
=0
¶K
¶K
¶K
*
*
*
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Scenario 2: Competition increases with trait similarity
Plants alone at equilibrium (equilibrium (2))
Equilibrium biomass
Ks 2 p
*
P* =
and H = 0
a0
Conditions of feasibility
and stability
Fitness
ì
b f Ks
ïï
A= 0
>1
mga 0
í
ï
ïî p - g 2 ln(A) < x < p + g 2 ln(A)
(x -x )2 ö
æ
- m 2r
W (x m , x r ) = r çç1 - e 2s ÷÷
è
ø
ü
( p-x )2
ïï
m ga 0 2 g2
e
ýÛK £
b0 f s
ï
ïþ
Fitness gradient
Trait variation
Evolutionary analysis
dx
=0
dt
No evolutionary singularity. Neutral trait. Variations are expected
to happen due to genetic drift.
Coexistence equilibrium (equilibrium (3))
( p -x )2
( p-x)2 æ
( p-x)2 ö
Equilibrium biomass
2
m
g
2
p
r g 2 p 2 g2 ç
m ga0
*
*
2g 2
and H =
e
1e 2g ÷
P =
e
ç b0 f K s
÷
b0 f
b0
è
ø
ì
Conditions of feasibility
ü
b f Ks
( p-x )2
ïï
A= 0
>1
ïï
and stability
2
m
g
a
*
0
mga 0
H > 0Ûí
e 2g
ýÛK >
b0 f s
ï
ï
ïî p - g 2 ln(A) < x < p + g 2 ln(A) ïþ
Fitness gradient
Trait variation
dx
¶W (xm, xr )
= kmw 2 P*
dt
¶ xm
xm ®xr
Evolutionary singularity
x* = p
Invasibility and
Convergence
¶ 2W (x m , x r )
¶ 2 xm
x
¶ 2W (xm , xr )
¶ 2 xr
x
m =x r
2
2
r æ (s - g ) ö
ç
÷
= 2 ç1 g è
A s 2 ÷ø
always positive
æ
2
s +g ö
r
÷ < ¶ W (xm , xr )
= 2 ç -1+
g çè
A s 2 ÷ø
¶ 2 xr
xm =xr =x*
=x *
(
m =xr =x
*
2
2
)
(NB: A is positive, but inferior to 1)
Singularity x* is invasible and not convergent, so it is a repellor.
Interpretation is similar to scenario 1.
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Pairwise invisibility plot
(positive fitness is in grey)
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0.55
0.50
0.45
0.40
0.40
Effects of enrichment
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0.45
0.50
0.55
0.60
¶P* (x * )
¶H * (x * )
¶x *
= 0,
> 0 and
=0
¶K
¶K
¶K
Quantitative defenses do not vary with enrichment; herbivores
increase while the plants are top down controlled (similar to the
ecological scenario).
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Evolution of herbivore preference p
We now consider the evolution of herbivore trait p. Consider a mutant that has a trait pm
within a resident population having a trait p. Its dynamics can be described by the
following equation:
b
dH m
= H m ( f b (pm )P- m) with b ( p) = 0 e
g 2p
dt
- ( p-x ) 2
2g2
.
Its relative fitness may then be assessed by its per capita intrinsic growth rate when
rare:
W(pm, pr ) = f b (pm )P* - m
Coexistence equilibrium (equilibrium (3))
( p -x )2
( p-x)2 æ
( p-x )2 ö
Equilibrium biomass
m g 2p 2g 2
r g 2 p 2 g2 ç
2p m g 2g2 ÷
*
*
and H =
e
1e
P =
e
ç
÷
b0 f
b0
b
f
K
0
è
ø
ü
Conditions of feasibility
ì
b fK
( p-x )2
ïï
ïï
A= 0
>1
and stability
2
m
g
a
0
mg 2p
H* > 0 Û í
e 2g
ýÛK >
b0 f s
ï
ï
x
g
2
ln(A)
<
p
<
x
+
g
2
ln(A)
ïî
ïþ
( p -p )( p +p -2 x)
Fitness
- m r m2 r
2g
W(pm , pr ) = m(-1+ e
)
Fitness gradient
Trait variation
dp
¶W (pm , pr )
= kmw 2 H *
dt
¶ pm
pm ®pr
Evolutionary singularity
p* = x
Invasibility and
Convergence
¶ 2W (x m , x r )
¶ 2 xm
x
Evolutionary outcome
Pairwise invisibility plot
(positive fitness is in
grey)
=m =x r
=x *
m
¶ 2W (x m , x r )
=
g2
¶ 2 xr
x
m =x r
=x *
Singular strategy p*=x is both convergent and non-invasible (CSS)
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0.55
0.50
0.45
0.40
0.40
Effects of enrichment
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0.45
0.50
0.55
0.60
¶P (p )
¶H (p* )
¶p*
= 0,
> 0 et
=0
¶K
¶K
¶K
The preference trait is not affected by evolution, so that the
effects of enrichment are similar to the outcome observed in the
ecological scenario: the herbivore biomass increases while plant
biomass does not vary (top down control)
*
*
*
10
Evolution of herbivore generalism
We now consider the evolutionary dynamics of herbivore generalism g. Consider a
mutant, whose trait is gm, within a resident population of trait g. The population
dynamics of this mutant population may be described by the following equation:
dH m
= H m ( f b (gm )P- m)
dt
with b (g) =
b0
g 2p
e
- ( p-x ) 2
2g2
.
Relative fitness of individual mutant, as assessed through this dynamical equation is:
W(gm, gr ) = f b (gm )P3* - m
Coexistence equilibrium (equilibrium (3))
( p -x )2
( p-x)2 æ
( p-x )2 ö
Equilibrium biomass
m
g
2
p
r g 2 p 2 g2 ç
2p m g 2g2 ÷
*
*
2g 2
and H =
e
1e
P =
e
ç
÷
b0 f
b0
b
f
K
0
è
ø
Conditions of feasibility
ì
b fK
ïï
B= 0
> p- x
and stability
m 2p e
H* > 0 Û í
ï
ïî g min< g < g max avec g min< p - x < g max
( p -x )2 ( p -x )2 ö
æ
Fitness
+
2
2
g
r
W (gm ,gr ) = mç -1+ e 2g m 2g r ÷
ç
÷
gm
è
ø
Fitness gradient
Trait variation
dg
¶W (gm , gr )
= kmw 2 H *
dt
¶ gm
gm ®gr
Evolutionary singularity
g* = p - x
¶ 2W (gm ,gr )
¶ 2 gm
g
Invasibility and
Convergence
Evolutionary outcome
Pairwise invisibility plot
(positive fitness is in grey)
m =g r
=g *
2m
¶ 2W (gm ,gr )
==( p - x) 2
¶ 2 gr
g
From the above line, singular strategy
convergent (CSS)
g*
m =g r
=g *
is not invasible and
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Effects of enrichment
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0.18
0.20
0.22
0.24
¶P* (p* )
¶H * (p* )
¶g*
= 0,
> 0 et
=0
¶K
¶K
¶K
The preference trait is not affected by evolution, so that the effects
of enrichment are similar to the outcome observed in the
ecological scenario: the herbivore biomass increases while plant
biomass does not vary (top down control)
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References
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Geritz SAH, Kisdi É, Meszéna G, Metz JAJ (1998) Evolutionary singular strategies and the
adaptive growth and branching of the evolutionary tree. Evol Ecol 12:35–57.
Loeuille N, Loreau M (2004) Nutrient enrichment and food chains: can evolution buffer
top-down control? Theor Popul Biol 65:285–298.
Metz JAJ, Nisbet RM, Geritz SAH (1992) How should we define fitness for general
ecological scenarios? Trends Ecol Evol 7:198–202.
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