Localization in Mutualistic Ecological Networks
Physics and Astronomy Department,
University of Padova
Welcome to Amos Maritan Lab
Page 1 of 2http://www.pd.infn.it/~maritan/
!!!!!!!!!!!!!!!!!!!!!
Our! research! spans! from! statistical!mechanicsto!organization!of!ecosystems...
29#01#2013
Claudio!wrote!his!thesis.!Good!luck!with!it!
In!the!spirit!of!the!motto!"interdisciplinarity!is!dialog"!the!aim!of!the!Lab!is!toface!biological!and!ecological!problems!in!collaboration!with!experts!of!the!field.Not!mixing!our!expertises,!but!summing!them!up.!
!!!!!!!!!!!!!!!
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CONTACTS
# 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
59 networks Interac5on-‐web
database
Architecture of Mutualistic Networks
Null model 0 We keep fixed S and C, and place at random the edges
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
NESTEDNESS
Why do we found this ubiquitous structure?
Time
Pop
ulat
ion
0 T TA'
Nestedness correlates with species abundance
Suweis et al., Nature 2013
Are nested architecture more stable??
~̇x = �~x Max(Re[�(�)]) = �1 ) Resilience
Mutualistic Nested Networks are less resilient than their random counterpart!
!1
!
0
0.5
maxA B C
MF [� = 0] CE [� = 0.5] OPT [� = �0.5]
ASSIGN INTERACTION STRENGTHS
�ij = Bij�0k�i
Null Model Bran ! �ran
Beyond Resilience: Localization
A↵
rIPR =
* PSi=1 v1(i)
4
PSi=1 v
ran1 (i)4
+> 1 ) Localized
Eigenvecor components
0
1PERTURBATION
⇠⇠⇠all(i) / (1 + ki)N (1, ⇣)
Quantifying Stability
�1(�, C, S) ! resilience
St!1(C,�) ! persistence
vvv1(�, C, S)? ! spread of⇠⇠⇠
uuu1(�, C, S)? ! A1
Transient Stability
Eigenvecor components
0
1
Reactivity: H = (�+ �
T)/2 ! {�H ,wwwH}
Amplification Envelope: ⇢(tmax) = Maxt2[0,1)||�xxx(t)||
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0 200 400 600 8001.0
10.0
5.0
2.03.0
1.5
15.0
7.0
Size !S"
rIPRright
A B
Mutualistic Ecological Networks are Localized
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1.1
Size @SD
AêA ra
nxall
A↵
Localization attenuates perturbations
»v1\»u1\»wH\k\
kmax
0 20 40 600.0
0.2
0.4
0.6
0.8
1.0
1.2
Species
lmax=-0.0813779 lH=0.145052
Real Data Example
20 40 60 80050100150200250300
r@d=0D
r ran@d=
0D
0 100 200 300 400 5000
100
200
300
400
500
r@d=0.5D
r ran@d=
0.5D
20 40 60 80050100150200250300
r@d=0.5D
r ran@d=-0.5D
A
4 5 6 7 8 9
4
6
8
10
12
14
A1Exp@-l1D
A1ranExp@-l 1
ran D B
4 5 6 7 8 94
6
8
10
12
14
16
A1Exp@-l1D
A1ranExp@-l 1
ran D C
4 6 8 10 12 1446810121416
A1Exp@-l1D
A1ranExp@-l 1
ran D
Localization and AmpliRication Envelope
Stability: putting ingridients together (t=1)
0.0 0.2 0.4 0.6 0.8 1.020406080100120140
Time @t D
⁄ i»dx iHtL\
0.0 0.2 0.4 0.6 0.8 1.05
10
15
20
25
Time @t D
⁄ i»dx iHtL\
0.0 0.2 0.4 0.6 0.8 1.0
1015202530
Time @t D
⁄ i»dx iHtL\
0.0 0.2 0.4 0.6 0.8 1.091011121314
Time @t D
⁄ i»dx iHtL\
Stability: transient results
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0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.70
1
2
3
4
5
NODF
Localization
Nestedness is not the whole story…
Take Home Messages
�1 ! Resilience
vvv1 ! Spreading of the perturbation
uuu1 ! Attenuation of the perturbation
{�H ,wwwH} ! Reactivity
⇢ ! Amplification Envelope
Asymptotic Stability
Transient Stability
Architecture of mutualistic ecological networks = Nestedness +Localization!
Thanks for your attention! Questions?
Join the LIVING Satellite!Thursday 25, 2 pm @ IMT library (San Ponziano Church)
Robustness, Adaptability and Critical Transitions in Living Systems
SamirSuweis
Optimization: Nature, 500 (449); 2013 Localization: soon in Arxiv
0.6 0.7 0.8 0.9 1.01
2
3
4
5
6
7
8
Max@lDêMax@lranD
rIPR
g=0.00483092
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-0.9
-0.8
-0.7
-0.6
-0.5
-0.4
Size @SD
Max@lD
g0=constant
TRADE-‐OFF between resilience and localiza5on
~̇x = �~x
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
�4 �3 �2 �1 0 1 2�4
�2
0
2
4
(Allesina, Nature 2012)
�ij ⇠ N (0,�2) �ij ,�ji ⇠ |N (0,�2)|Random Structure Mutualis5c (nested) Structure
See also Staniczenko et al., Nat Comm.; Suweis et al. Oikos 2013
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. LeT, 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 [x
]
Pollinator*species
Plant*species
�c ⇠1pSC
�c ⇠1
SC
0. 2 0. 3 0. 4 0. 5 0. 6 0. 7 0. 850
54
58
62
66
Nestedness [NODF]
Mea
n Po
pula
tion
B
Correla5on between Popula5on and Nestedness
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: Op5mized Networks are nested
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>
50 100 200 500
0.02
0.05
0.10
0.20
0.50
C~1/S
# species
C
Connec5vity
Holling Type II
Degree @kD »v1\ »z\ »x\
0 20 40 60 80 100
0.0
0.5
1.0
1.5
2.0
Species i
RandomWeights
Degree @kD »v1\ »z\ »x\
0 20 40 60 80 100
0.5
1.0
1.5
Species i
RandomCase
Degree @kD »v1\ »z\ »x\
0 20 40 60 80 1000.0
0.5
1.0
1.5
2.0
Species i
OptimalCa
se
Degree @kD »v1\ »z\ »x\
0 20 40 60 80 1000.0
0.5
1.0
1.5
Species i
ConstantE
ffortCase
Summary
