Emergence of Nested Architecture in Mutualistic Ecological Communities
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Samir Suweis, Filippo Simini, Jayanth Banavar and Amos Maritan Optimal Mutualistic Ecological Networks: Emergent Structural & Dynamical Properties [email protected] Post Doc Researcher, Physics and Astronomy Department, University of Padova
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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.
Transcript of Emergence of Nested Architecture in Mutualistic Ecological Communities
Samir Suweis, Filippo Simini, Jayanth Banavar and Amos Maritan
Optimal Mutualistic Ecological Networks: Emergent Structural & Dynamical Properties
[email protected] Post Doc Researcher, Physics and Astronomy
Department, University of Padova
Welcome to Amos Maritan Lab
Page 1 of 2http://www.pd.infn.it/~maritan/
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J-.% K%&%'$() L%-*1% L#?10('/0-+& F%'()0+2 6-11'?-$'/-$& "**-$/#+0/0%& 6-+/'(/&
Ecological Networks
Ladybug
Aphid
Mouse
BeetleCaterpillar
Towhee
Grasshopper
Louse
Owl
Sunflower
MUTALISTIC FOOD WEB
A =
aPPn1×n1
aPAn1×n2
aAPn2×n1
aAAn2×n2
W =
ΩPP
n1×n1ΓPAn1×n2
ΓAPn2×n1
ΩAAn2×n2
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 321510152025
1 10 20 30 36
NODF=0.424 NODF=0.192
15
10
15
20
251 10 20 30 36
NODF=0.0721 10 20 32
1
5
10
15
20 NODF=0.133
Architecture of MutualisLc Networks
Random same S,C
Random same S,C
nestedness
Pollinator
Pollinator
Pollinator
Pollinator
Plant
Plant
Plant
Plant
Pollina
tor
Plants
The number of common the i-th and the j-th plant have oPij ≡
k
aPAik aPA
jk
NODF =
i<j: i,j∈P TP
ij +
i<j: i,j∈A TAij
P (P−1)2
+A(A−1)
2
,
TXij = 0 if kXi = kXj
TXij =
oXijmin(kXi , kXj )
“Triangular” shape
hMp://www.nceas.ucsb.edu/interacLonweb/resources.html hMp://ieg.ebd.csic.es/JordiBascompte/
# Species [S]
Nes
tedn
ess [
NO
DF]
20 40 60 80 100 120 140 160 180 2000
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Random
Data
56 Networks
Network data
0.1 0.2 0.3 0.4 0.5 0.6 0.7
0.10.20.30.40.50.60.7
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
Are nested architecture more stable?
Real
Imag
inary
!7 !6 !5 !4 !3 !2 !1 0 1 2 3 4 5
A
B
!0.5
1.0
0.5
!0.5
1.0
0.5
x = Φx
4 3 2 1 0 1 24
2
0
2
4
(Allesina, Nature 2012)
Φij ∼ N (0,σ2) Φij ,Φji ∼ |N (0,σ2)|Random Structure MutualisLc (nested) Structure
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.
TheoreLcal Framework
• Abundances = x1,x2,...,xS
• σΩ , σΓ so that x* is stable • Community populaLon dynamics (HTI or HTII)
dxi
dt= xi
αi −S
j
Mijxj
≡ fi(x)
γij ∼ −|N (0,σΓ)|ωij ∼ |N (0,σΩ)|
ImplementaLon of the OpLmizaLon Principle
T T+1i j
l
k j
l
!"#$
Wil
Start with xi~ N(1,0.1) and random M (α, S, C fixed)
AdapLve EvoluLon
i
M → M
ifx∗
i > x∗i
We accept the swap
norm
aliz
ed m
utua
listic
stre
ngth
|!ij|/maxi,j=1..S|!ij|1
01 5 10 15 20 25
15
10
15
20
251 5 10 15 20 25
15
10
15
20
25
MAXIMIZATION OF SPECIES POPULATION ABUNDANCES
# Plants # Plants
# Po
llinat
or
# Po
llinat
or
Random Optimized
a
b
# optimization steps
Plan
ts/P
ollin
ator
s Po
pula
tion
0 T T+1
0.95
1.00
1.05
1.10
1.15
Steps
Popu
latio
n [x
i] !
"!
#
!
#
"
!"#
!"#
T
!
"!
#
!"#
!"#
T+1$%&'
0.8035221.081781.058031.050140.9779391.014220.9581281.133971.040781.03560.96641.020131.00682
0.673611.101311.075711.102890.9596580.9969130.9188921.152981.038131.02231.013140.9587941.00217
::
x* = x* =
δx∗tot =
m
δx∗m = δx∗
k
Result 1: OpLmizaLon of single species abundance leads to an average increase of the total number community abundance
T T+1i j
l
k j
l
!"#$
Wil
Nestedness [NODF]0.2 0.3 0.4 0.5 0.6 0.7 0.8
1234567
Null Model 1
Optimiz Total Pop HTI
Null Model 0
Optimiz Total Pop HTII
0.3 0.4 0.5 0.6 0.7 0.8
24681012
Nestedness [NODF]
Null Model 1
Optimiz Total Pop HTII
0.2 0.3 0.4 0.5 0.6
2468
1012
0.2 0.3 0.4 0.5 0.6
2
4
6
8Nestedness [NODF]
Nestedness [NODF]
Null Model 0
Optimiz Total Pop HTI
0.2 0.3 0.4 0.5
5
10
15
20
Nestedness [NODF]
null model 0 Optimization Single Speciesnull model 1
0.2 0.3 0.4 0.5 0.6 0.7 0.8
2468
1012
PD
F
Nestedness [NODF]
HTI HTII
Result 2: OpLmized Networks are nested
xtot = α
i,j
W−1ij W = W0 + V =
I+ Ω O
O I+ Ω
+
O ΓΓT
O
W−1 = W−10 (I+ VW−1
0 )−1 = W−10 −W−1
0 VW−10 +W−1
0 VW−10 VW−1
0 + ...
o =1
γ2
ij
k
ΓkiΓkj + ΓikΓjk
AnalyLcal relaLon between Nestedness and Species populaLon
0.06 0.07 0.08 0.09 0.10
50
100
150
200
250
!!"
InteracLon Strength dependence
o ∝ K + C−1(ω, γ) · xtot
Stability
Max[Re( )]
Min
!"
-1.5 -1.0 0.5 0.0
-0.2
-0.1
0.0
0.1
0.2
!"# $
%&#$
RandomOptimizedS=50
0 5 10 15 20 250
1
2
3
4
5
number of connections [k]
spec
ies a
bund
ance
‹x›
si=|!j"ij|
a
‹x›
0 1 2
5
0
4321
Max[Re(!)]-0.8 -0.7 -0.6 - 0.5 - 0.4
5
10
15
20
25
λi = −x∗i + o(σ2)
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.
Thanks for your aMenLon!
Robustness of the results
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
Random
HTI total
HTII total
HTI individual species
HTII individual speciesi
ii
iii
iv
a b
1 10 20 30 40
1
10
20
30
40
12345678910
1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10
12345678910
1
5
10
15
20
1 5 10 15 20
1
5
10
15
20
Optimization + AssemblingRandom Fully Optimized
Architecture of Ecological Networks
A =
0 aPA
aPA 0
Fontaine et al., Eco. LeM, 2011
0 200 400 600 800 1000 1200 1400 1600 1800 20009.5
10
10.5
11
11.5
12
12.5
13
13.5
14
14.5
Attempted Swaps [T]
Popu
latio
n [!
]
"#$$%&'(#)*+,-.%-+
"$'&(*+,-.%-+
σc ∼1√SC
σc ∼1
SC
50 100 200 500
0.02
0.05
0.10
0.20
0.50
C~1/S
# species
C
ConnecLvity
Holling Type II
