Ecosystem stability in space: α, β, and γ variability
Shaopeng Wang and Michel Loreau
Centre for Biodiversity Theory and Modelling, Station d’Ecologie Expérimentale du CNRS,
09200 Moulis, France
Email: shaopeng.wang@ecoex-moulis.cnrs.fr, michel.loreau@ecoex-moulis.cnrs.fr
Overview:
Appendix S1. Derivation and partitioning of spatial variability ( CV
S
2
)
Appendix S2. Effect of unevenness on the multi-scale variability
1

Appendix S1. Derivation and partitioning of spatial variability (CV
S
2 )
Following the definition in the main text, the spatial variance at time t is (see Table 1 for definition of notations):
V
S
( t )

 i
( N i
( t )

N ( t ))
2 m
(S1) where the sum of the squared differences ( SSD ) between local community biomass and their spatial average at time t can be expressed as:
SSD ( t )




 i i
[ N i
[( N i
 i
( t )
( t )

( N i

 i
( t )

 i
)
2
 i


N

(
 i

( t ))
2




)
2

2 ( N i
( t )
  i
)(
 
N ( t ))

N ( t )]
2

(

2 (
 i


N ( t ))
2

)(
 

2 ( N i
( t )

N ( t ))]
 i
)(
 i
 
)

(S2)
To obtain the temporal expectation of SSD ( t ), we calculate the expectation of the six terms in the right-hand side of Eq. (2). Recall that N ( t ) = ( N
1
( t ), N
2
( t ), ..., N m
( t )) is a stationary random vector with mean
μ
= (
μ
1
,
μ
2
, ...,
μ m
) and covariance matrix W = ( w ij
). Therefore we have:
E

 i
( N i
( t )
  i
)
2

  i w ii
  i
( w ii
 w )
2  m w
2
E

 i
(
 i
 
)
2

  i
(
 i
 
)
2
E

 i
(
 
N ( t ))
2

 m

E

(
 
N ( t ))
2

 m

E

(

M
  i
N i
( t ) )
2 m
2


1 m
 i , j w ij
E

 i
[ 2 ( N i
( t )
  i
)(
 i
 
)]


0
E

 i
[ 2 ( N i
( t )
  i
)(
 
N ( t ))]

 
2 m

E

(
 
N ( t ))
2

 
2 m
 i , j w ij
E

 i
[ 2 (
 i
 
)(
 
N ( t ))]


0
We obtain the expectation of spatial variance as follows:
2

v
S

E

V
S
( t )


1 m
E
  i
( N i
( t )

N ( t ))
2



1 m
[
 i
( w
2

1 m 2 w ii
 w )
2  m w
2
 i , j w ij

1 m
 i
[(
  i
(
 i w ii

 
)
2 
1 m
 i , j w ij
] w )
2 
(
 i
 
)
2
]
We then define the spatial variability as follows:
CV
S
2  v

S
2


1

2
[ w
2 w
2

2
[ 1


1 m
2
 i , j w ij
]

1 m

2
 i , j m
2 w ij
2
]
 w
1 m

2
 i
[(
 i
[( w ii
 w ii
 w )
2 w )
2 
(
 i

(
 i
 
)
2
]
 
)
2
]
(S3)
Spatial variability arises from two factors: biomass unevenness among patches, and asynchronous dynamics among patches. By biomass unevenness, we mean any spatial or interspecific variation in the mean and/or variance of biomass. When the dynamics of all patches are perfectly correlated, i.e. the correlation coefficient between any two patches equals one (or equivalently: w ij
 w ii w jj
), the first term in eqn (S3) equals zero: w
2

2
[ 1

 i , j m
2 w ij w
2
]

1 m
2

2
[ m
2 w
2
  i , j w ij
]

1 m
2

2
[

 i w ii
2

  i , j w ij
]

0 .
So the spatial variability is caused solely by spatial biomass unevenness:
CV
S
2
.
uneven

 i
[(
 i
 
) 2 
( m
 2 w ii
 w ) 2 ]
. When the dynamics are at least partially
3

asynchronous, this asynchrony can further increase the spatial variability through the additional term:
CV 2
S .
asyn

CV
S
2 
CV
S
2
.
uneven
 w
2
  i ,

2 j w ij m
2
.
We call this term the asynchrony-related spatial variability ( CV
S.asyn
2 ). In homogeneous metacommunities where all patches have identical mean and variance of biomass, CV
S.asyn
2 results solely from spatially asynchronous dynamics; otherwise, it is also influenced by spatial biomass unevenness (see Appendix S2 for details).
To link spatial variability and local variability explicitly, we substitute
CV
L

 i

M w ii  m m
 w

 w
and
 
 i ,
 i w ij
 j w ii
2
 
 i , j w ij m
2 w
2
into eqn (S3), which yields:
CV
S
2 
CV
L
2
( 1
 
)

CV
S
2
.
uneven
(S4)
Finally, note that another way to define spatial variability is to first define a timedependent spatial variability, i.e. CV
S
2
( t )

V
S
( t )
N
2
( t )
, and then derive its temporal expectation.
Under this definition, however, more information about the probability distributions of N ( t ) (in addition to its mean and covariance matrix) is required to derive the temporal expectation of
CV
S
2
( t ) .
4

Appendix S2. Effect of unevenness on the multi-scale variability
Gamma variability ( γ cv
 cv
  cv
/

1
= CV
M
2 )
Alpha variability ( α cv
CV
L

 i

 i
M

= CV
L
2 )
CV i
Beta variability ( β
1
  
 i
 i
, j w ij w ii

2
= 1/ φ)
Spatial unevenness Variability of community i (CV i
2 )
CV i
2 
CV 2 species _ i
  species _ i
Spatial unevenness
Averaged species variability (CV species_i
2 )
CV species _ i
  j
 j
 i
_ i 
CV j _ i
Species asynchrony ( φ species_i
)
 species _ i
 

 k k
, l w kl _ i w kk _ i
2

Species unevenness in patch i
Variability of the j-th species in patch i (CV j_i
2 )
Species unevenness in patch i
Figure S1. A hierarchical framework of ecosystem variability in heterogeneous metacommunities, where biomass varies among species within local communities and among local communities. Notations are defined in Table 1, except for μ j_i and w kl_i
which represent the mean biomass of species j and the covariance between species k and l , respectively, within patch i . For perfect spatial evenness (
μ
1
=
μ
2
= ... =
μ m
and w
11
= w
22
= ... = w
SS
), alpha variability equals the variability of any specific local community. For perfect species evenness within local
5

patches (for instance in patch i ,
μ
1_i
=
μ
2_i
= ... = μ
S_i
and w
11_i
= w
22_i
= ... = w
SS_i
), species-level variability equals the variability of any specific species. Figure 2 represents a simplified scenario where patches are perfectly even and species are perfectly even within patches.
In Figure 2, we present a hierarchical framework of variability corresponding to a metacommunity with perfect evenness among species and among local communities. In reality, however, biomass is generally unevenly distributed, both among species within a local community and among local communities. Such unevenness can affect variability at multiple scales. Figure S1 shows an extended framework that takes into account the effects of biomass unevenness among species and among local communities. Specifically, species unevenness affects the weighted averages of species-level variability and species synchrony (Thibaut &
Connolly 2013). Similarly, spatial unevenness can affect the weighted averages of local-scale variability and spatial synchrony. The effects of species unevenness have been well documented in the recent paper by Thibaut & Connolly (2013). Here we develop a single-species metapopulation model to investigate the effect of spatial unevenness on variability at different scales.
We consider a metapopulation composed of m local populations. For each local population ( i ), we assume a power relationship between the mean (
μ i
) and variance ( w ii
) of population biomass: w ii
= cμ i z
(Taylor 1961). Consequently, spatial unevenness in the mean will also generate unevenness in the variance. The temporal CV of population i is given by:
CV i
 w ii
 i
 c
 i z

2
. We also assume a constant correlation (
ρ
) of biomass dynamics between
6

local populations, so that the covariance between any two patches i and j is: w ij
  w ii w jj
. We then derive the local-scale temporal CV :
CV
L
  i


M i

CV i


 i i
 w ii i

 i
 c
 i
 i i z and the spatial synchrony index:
 
 i ,
 i w ij
 j w ii
2
 
( 1
 
)
 i  w ii
 i
  w ii


2
 i w ii
2


( 1
 
)

 i
 i
 i z

 i z
2
  
(S5)
(S6)
With eqns (S5) and (S6), we obtain the variability at alpha, beta, and gamma scales:

CV

CV
L
2  c




 i i

 i i z

2
2

(S7)

1

1


( 1
 
)
 
1
 i
 i

 i i z b z
  

2
  cv

( 1
 
)
 c

( 1
 
)


 i 
 i z

 i
2
 i


2
 i
 i z

CV
  cv
   c

( 1
 
)
  i
 i z
 i

 i

 
2


 i
 i z
2

(S8)
(S9)
(S10)
We also derive the total spatial variability by substituting
μ i
and w ii
into eqn 3 from the main text:
CV
S
2  w
2

1 m
2
 i , j w ij

1 m
 i
[(
 i

2
 
)
2 
( w ii
 w )
2
]
.
7

We now investigate the effects of spatial unevenness using equations (S7-S10). When z <
2, a large patch has lower variability than a small one (this is easily seen from
CV i
 c
 i z

2 ).
Because alpha variability (
α cv
) is calculated as a size-weighted average (where large patches have larger weights), a higher unevenness reduces alpha variability when z < 2 (Fig. S2-a). But when z
= 2, unevenness has no effect on alpha variability because the local CV is scale-dependent.
Similar results were obtained for species unevenness in Thibaut & Connolly (2013).
A higher unevenness also reduces spatial asynchrony (multiplicative beta variability β
1
), except in the special case when patches are perfectly correlated (
ρ
= 1) (Fig. S2-b). This is because as spatial unevenness increases, the dynamics of the metacommunity becomes more dominated by one or a few large patches, thereby reducing the overall level of asynchrony between patches. A higher unevenness also decreases the asynchrony-related spatial variability
( CV
S.asyn
2 , i.e. β
2
) (Fig. S2-c), which is positively related to both alpha variability and multiplicative beta variability (

2
  cv

( 1

1

1
) ). Lastly, a higher unevenness increases the spatial variability that is solely due to biomass unevenness ( CV
S.uneven
2
). This effect turns out to be much stronger than that on CV
S.asyn
2
, so that total spatial variability ( CV
S
2
= CV
S.uneven
2
+
CV
S.asyn
2
) increases with spatial unevenness (Fig. S2-c). Note, however, that the effect on
CV
S.uneven
2 does not influence alpha, beta, and gamma variability directly.
Finally, while spatial unevenness decreases both alpha and beta variability, its net effect on gamma variability ( γ cv
) can be either stabilizing or destabilizing (Fig. S2-d). Under large z and/or small
ρ
, spatial unevenness tends to increase gamma variability through strong synchronizing effects. Under small z and/or large
ρ
, it tends to decrease gamma variability through strong stabilizing effects on alpha variability.
8

(a) (b)
0
0.5
1 b=1 b=1.5
b=1 b=1.5
b=2
0
0.5
1
(c)
0.0 0.5
1.0 1.5
2.0
2.5 3.0
Spatial unevenness
(d)
0.0 0.5
1.0 1.5
2.0
2.5 3.0
Spatial unevenness
50 20 10 5 2 1
Number of patches
50 20 10 5 2 1
Number of patches
50 20 10 5 2 1
Number of patches
50 20 10 5 2 1
Number of patches
50 20 10 5 2 1
Number of patches
50 20 10 5 2 1
Number of patches
50 20 10 5 2 1
Number of patches
50 20 10 5 2 1
Number of patches
0.0 0.5
1.0 1.5
2.0
2.5 3.0
Spatial unevenness
0.0 0.5
1.0 1.5
2.0
2.5 3.0
Spatial unevenness
Figure S2. Effects of spatial unevenness on variability at the various scales. Parameter values: m
0.0
1.0
2.0
= 10, c = 1, and average population biomass
μ Unevenness local population biomass (
μ
1
,
μ
2
, ... ,
μ
10
) are examined, and spatial unevenness is calculated as
0.0
1.0
2.0
3.0
0.0
3.0
0.0
1.0
2.0
Unevenness total spatial variability ( CV
S
2
). For clarity, we only show the CV
S
2
under
ρ
= 0, but the same increasing trends hold for ρ = 0.5 or 1.
1.0
2.0
Unevenness
3.0
0.0
3.0
0.0
1.0
2.0
3.0
Unevenness
1.0
2.0
3.0
Unevenness
0.0
1.0
0.0
1.0
2.0
Unevenness
3.0
2.0
3.0
Unevenness
9