Mutualistic networks are formed when the interactions between two classes of species are mutually beneficial. They are important examples of cooperation shaped by evolution. Mutualism between animals and plants has a key role in the organization of ecological communities. Such networks in ecology have generally evolved a nested architecture independent of species composition and latitude; specialist species, with only few mutualistic links, tend to interact with a proper subset of the many mutualistic partners of any of the generalist species.Despite sustained efforts to explain observed network structure on the basis of community-level stability or persistence, such correlative studies have reached minimal consensus. Here we show that nested interaction networks could emerge as a consequence of an optimization principle aimed at maximizing the species abundance in mutualistic communities. Using analytical and numerical approaches, we show that because of the mutualistic interactions, an increase in abundance of a given species results in a corresponding increase in the total number of individuals in the community, and also an increase in the nestedness of the interaction matrix. Indeed, the species abundances and the nestedness of the interaction matrix are correlated by a factor that depends on the strength of the mutualistic interactions. Nestedness and the observed spontaneous emergence of generalist and specialist species occur for several dynamical implementations of the variational principle under stationary conditions. Optimized networks, although remaining stable, tend to be less resilient than their counterparts with randomly assigned interactions. In particular, we show analytically that the abundance of the rarest species is linked directly to the resilience of the community. Our work provides a unifying framework for studying the emergent structural and dynamical properties of ecological mutualistic networks.

1. Optimal Mutualistic Ecological Networks:
Emergent Structural & Dynamical Properties
Samir
Suweis,
Filippo
Simini,
Jayanth
Banavar
and
Amos
Maritan
suweis@pd.infn.it
Post
Doc
Researcher,
Physics
and
Astronomy
Department,
University
of
Padova
Welcome to Amos Maritan Lab
!!!!!!!!!!!!!!!!!!!!!
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<+!/)%!&*0$0/!-,!/)%!.-//-!=0+/%$70&(0*10+'$0/4!0&!70'1-2=!/)%!'0.!-,!/)%!>'?!0&!/-
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A-/!.0@0+2!-#$!%@*%$/0&%&B!?#/!&#..0+2!/)%.!#*5!

2. Ecological
Networks
Aphid
Ladybug
Mouse
Caterpillar Beetle
Grasshopper
Towhee
Louse
Owl
Sunflower
MUTALISTIC
FOOD
WEB
A =
aPP
n1×n1 aPA
n1×n2
aAP
n2×n1 aAA
n2×n2
W=
ΩPP
n1×n1 ΓPA
n1×n2
ΓAP
n2×n1 ΩAA
n2×n2

3. Architecture
of
MutualisLc
Networks
Avian
fruit
web
in
Puerto
Rico
Carlo,
et
al.
Plant
Pollinator
web
in
Chile
Arroyo,
et
al.
1
5
10
15
20
1 10 20 32
1
5
10
15
20
25
1 10 20 30 36
NODF=0.424 NODF=0.192
1
5
10
15
20
25
NODF=0.072
1 10 20 30 36
1 10 20 32
1
5
10
15
20 NODF=0.133
Random
same
S,C
Random
same
S,C

4. nestedness
Pollinator
Pollinator
Pollinator
Pollinator
Plant
Plant
Plant
Plant
Pollinator
Plants
The number of common the i-th
and the j-th plant have oP
ij ≡
k
aPA
ik aPA
jk
NODF =
ij: i,j∈P TP
ij +
ij: i,j∈A TA
ij
P(P−1)
2
+
A(A−1)
2
,
TX
ij = 0 if kX
i = kX
j
oX
TX
=
ij
ij min(kX
i , kX
j )
“Triangular” shape

5. # Species [S]
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
hMp://www.nceas.ucsb.edu/interacLonweb/resources.html
hMp://ieg.ebd.csic.es/JordiBascompte/
Nestedness [NODF]
20 40 60 80 100 120 140 160 180 200
0
Random
Data
56
Networks
Network
data
0.1 0.2 0.3 0.4 0.5 0.6 0.7
0.7
0.6
0.5
0.4
0.3
0.2
0.1
NODF Data
NODF CM
Null
model
1
We
keep
fixed
S
and
C
and
k1, k2,…,kS
Null
model
0
We
keep
fixed
S
and
C,
and
place
at
random
the
edges

6. Are
nested
architecture
more
stable?
Random
Structure
MutualisLc
(nested)
Structure
Φij ∼ N(0,σ2) Φij ,Φji ∼ |N(0,σ2)|
Real
Imaginary
x˙ = Φx
A
B
−7 −6 −5 −4 −3 −2 −1 0 1 2 3 4 5
0.5
1.0
−0.5
0.5
1.0
−0.5
4
2
0
2
4
4 3 2 1 0 1 2
(Allesina,
Nature
2012)

7. Unifying
framework
to
explain
the
emergent
structural
and
dynamical
properLes
of
mutualisLc
ecological
networks.
RelaLonship
between
species
abundances
in
the
community,
nestedness
of
the
interacLon
network
and
stability
of
the
system.

8. TheoreLcal
Framework
• Abundances
=
{x1,x2,...,xS}
• σΩ , σΓ so
that
x*
γij ∼ −|N(0,σΓ)|
ωij ∼ |N(0,σΩ)|
is
stable
• Community
populaLon
dynamics
(HTI
or
HTII)
dxi
dt
= xi

αi −
S
j
Mijxj

 ≡ fi(x)

9. ImplementaLon
of
the
OpLmizaLon
Principle
T T+1
i j
l
k j
l
!#$
bWil
Start
with
xi~
N(1,0.1)
and
random M (α,
S,
C
fixed)
AdapLve
EvoluLon
i
M → M
if x∗
i x∗i
We accept the swap

10. |!ij|/max{i,j=1..S}|!ij|
normalized mutualistic strength
1
MAXIMIZATION OF SPECIES POPULATION ABUNDANCES
Random Optimized
1 5 10 15 20 25 0
1
5
10
15
20
25
1 5 10 15 20 25
1
5
10
15
20
25
# Plants # Plants
# Pollinator
# Pollinator
a
b
# optimization steps
Plants/Pollinators Population

11. Result
1:
OpLmizaLon
of
single
species
abundance
leads
to
an
average
increase
of
the
total
number
community
abundance
Steps Population [xi]
0 T T+1
1.15
1.10
1.05
1.00
0.95
T
!
#
!
!
#
a!#

12. a!#
T+1
a!#
!
#
a!#

13. !
$%'
: :
0.803522
1.08178
1.05803
1.05014
0.977939
1.01422
0.958128
1.13397
1.04078
1.0356
0.9664
1.02013
1.00682
0.67361
1.10131
1.07571
1.10289
0.959658
0.996913
0.918892
1.15298
1.03813
1.0223
1.01314
0.958794
1.00217
x* = x* =
δx∗tot =
m
δx∗m = δx∗k
T T+1
i j
l
k j
l
!#$
bWil

14. Result
2:
OpLmized
Networks
are
nested
null model 0 Optimization Single Species
null model 1
HTI HTII
0.2 0.3 0.4 0.5 0.6 0.7 0.8
Nestedness [NODF]
1 2 3 4 5 6 7
Null Model 0
Optimiz Total Pop HTI
Null Model 1
Optimiz Total Pop HTI
Null Model 0
Optimiz Total Pop HTII
Null Model 1
Optimiz Total Pop HTII
0.3 0.4 0.5 0.6 0.7 0.8
12
10
8
6
4
2
Nestedness [NODF]
12
10
0.2 0.3 0.4 0.5 0.6
20
15
10
12
10
8
6
4
2
0.2 0.3 0.4 0.5 0.6
8
6
4
2
Nestedness [NODF]
Nestedness [NODF]
PDF
8
6
4
PDF
PDF
PDF
0.2 0.3 0.4 0.5
5
PDF
Nestedness [NODF]
0.2 0.3 0.4 0.5 0.6 0.7 0.8
2
PDF
Nestedness [NODF]

15. AnalyLcal
relaLon
between
Nestedness
and
Species
populaLon
xtot = ¯α
InteracLon
Strength
dependence
i,j
W−1
ij W = W0 + V =
I+Ω O
O I+Ω
+
O Γ
ΓT O
250
W−1 = W−1
0 (I +VW−1
0 )−1 = W−1
0 −W−1
0 VW−1
0 +W−1
0 VW−1
0 VW−1
0 + ...
200
150
o =
1
γ2
ij
k
ΓkiΓkj +ΓikΓjk
0.06 0.07 0.08 0.09 0.10
100
50
a
!!
ta
ta
t

16. a
o ∝ K + C−1(ω, γ) · xtot

17. Stability
- 0.8 - 0.7 - 0.6 - 0.5 - 0.4
25
20
15
10
mK
ï ï
Max[Re( h)]
-1.5 -1.0 0.5 0.0
Min !
0.2
0.1
0.0
-0.1
-0.2
!#h$
%#h$
Random
Optimized
S=50
4321
0 5 10 15 20 25
5
4
3
2
1
0
number of connections [k]
species abundance ‹x›
si=|!jij|
a
‹x›
0 1 2
5
0
pdf
Max[Re(!)]
5
λi = −x∗i
+ o(σ2)

18. Conclusions
1) Optimization of single species abundance
increases the total population abundance
2) Population abundance is positively correlated to the
nestedness of the network.
3) Population size of the rarest species in the community
is related to community resilience.
4) Optimized Networks are less stable with respect to
their random counterparts.

19. Thanks
for
your
aMenLon!

20. Robustness
of
the
results
a b
HTII individual species i
HTII total
Random
HTI individual species
HTI total
ii
iii
iv
Nestedness [NODF] Relative Nestedness [NODF*]
i
ii
iii
iv
v
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.0 0.5 1.0 1.5
1 10 20 30 40
1
10
20
30
40
1
2
3
4
5
6
7
8
9
10
1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10
1
2
3
4
5
6
7
8
9
10
1
5
10
15
20
1 5 10 15 20
1
5
10
15
20
Random Optimization + Assembling Fully Optimized

21. Architecture
of
Ecological
Networks
A =
0 aPA
aPA 0
Fontaine
et
al.,
Eco.
LeM,
2011

22. 0 200 400 600 800 1000 1200 1400 1600 1800 2000
14.5
14
13.5
13
12.5
12
11.5
11
10.5
10
9.5
Attempted Swaps [T]
Population [!]
#$$%'(#)*+,-.%-+
$'(*+,-.%-+

23. σc ∼
1
√SC
σc ∼
1
SC

25. 50 100 200 500
0.50
0.20
0.10
0.05
0.02
C~1/S
# species
C
ConnecLvity

26. Holling
Type
II