statistical physics of ecological networks: from patterns to principles

From Patterns to Principles

Statistical Physics of Ecological Networks

@SamirSuweis

Outline•Statistical Physics & Ecology !

•Architecture of the species interaction network !

•Stability of ecological networks !

•From patterns to principles: explaining nested architectures in mutualistic ecological community

Stochastic approachesfor microbial mobility.Interacting particle models

Neutral Theory & Ecological PatternsPopulation dynamics & metapopulation modelsEcological Networks & Optimization Stability in Ecological CommunitiesStatistical Inference (Biodiversity) Everything :-)

VisitingPh.D. Master

studentPh.D. Post-Doc

Post-Doc

Masterstudent

Ph.D.

Not mixing our expertise, but summing them up

Emergent Pattern in Ecology: RSA

0 2 4 6 8 100

5

10

15

Num

ber o

f spe

cies

Coral Reefs

Abundance classes [log scale]

Tropical Forests

1 2 3 4 5 6 70

10

20

30

40

Num

ber o

f spe

cies

0Abundance classes [log scale]

Complex Patterns from Simple Rules

All species are equivalent Single trophic level

Basic (random) ecological processes

Birth & death Master Equation

dPn(t)

dt= bn�1Pn�1(t) + dn+1Pn+1(t)� (bn + dn)Pn(t)

Parameters: bn/dn and m = b0

Functional form of bn • Density dependent effects

Volkov et al., Nature 2007

0 2 4 6 8 100

5

10

15

Num

ber o

f spe

cies

Coral Reefs

Abundance classes [log scale]

Tropical Forests

1 2 3 4 5 6 70

10

20

30

40

Num

ber o

f spe

cies

0Abundance classes [log scale]

[…] and to reflect that these elaborately constructed forms, so different from each other in so complex a manner, have been all produced by laws acting around us. (Darwin, Origin of Species)

Darwin’s entangled bank

Our approach:

A. Einstein

“Make everything as simple as possible, but not simpler.”

“You don’t really understand something unless you can explain it to your grandmother.”

The architecture of mutualistic species interactions network

From patterns to principles

10/14/2014 Web of Life: ecological networks database

http://www.web-of-life.es/map.php 1/1

web of life ascompte labbCredits

Networks All Data All Species >0 & <10000 Interactions >0 & <10000 Reset Results Download(89) Help

Ecological Networks

Find the pattern :-)

A closer look to the nested structure

Plant Pollinator web in Chile Arroyo, et al.

Random same S,C

Random same S,C

Avian fruit web in Puerto Rico Carlo, 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

25

1 10 20 30 36

NODF=0.072

1 10 20 321

5

10

15

20 NODF=0.133

Bascompte et al., PNAS 2003

The number of common partners the i-th and the j-th plant have

NODF measureAlmeida et al., Oikos 2008

Quantitative measures of nestedness :-(

Overlap

Network data vs Randomization 1Null model 1: we keep fixed S and C,

and place at random the edges

# 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

0.1 0.2 0.3 0.4 0.5 0.6 0.7

0.1

0.2

0.3

0.4

0.5

0.6

0.7

NODF DATA

NO

DF

Nul

l Mod

el

Network data vs Randomization 2Null model 2: we keep fix p(k) while randomizing the edges

Why this recurrent topological structure?

Does the nested structure give more stability to these ecological communities?

Ecological implications

Are nested networks more stable?

Many ways to quantify stability (>13 definitions !) + no analytical results

Persistence

dPi

dt= ↵Pi � IPiP

2i +

NaX

j=1

�ijAjPi

h�1ij +

Pk,hji>0 Ak

dAi

dt= ↵Ai � IAiA

2i +

NpX

j=1

�jiAiPj

h�1ji +

Pk,hji>0 Pk

.Model In

divi

dual

surv

ival

Pers

isten

ce

Pers

isten

ce

0 10 200

0.5

1

r2 = 0.60r2 = 0.35

Partners

Strong mutualism

0 0.2 0.4 0.60

0.5

1

r2 = 0.87r2 = 0.77

Connectance

0

0.5

1

r2 = 0.77

Network magnitude102 104

ab c

architecture canminimize local stability (9), havea negative effect on community persistence (10),and have a low resilience to perturbations (12).Not surprisingly, the majority of these studieshave been based on either local stability or nu-merical simulations with arbitrary parameter-izations [but, see (6)].

Model of mutualism

To study the structural stability and explain theapparently contradictory results found in studies

ofmutualistic networks, we first need to introducean appropriate model describing the dynamicsbetween and within plants and animals. We usethe same set of differential equations as in (6).We chose these dynamics because they are sim-ple enough to provide analytical insights and yetcomplex enough to incorporate key elements—such as saturating, functional responses (37, 38)and interspecific competitionwithin a guild (6)—recently adduced as necessary ingredients for areasonable theoretical exploration of mutualistic

interactions. Specifically, the dynamical modelhas the following form

dPi

dt¼ Pi aðPÞi − ∑ jb

ðPÞij Pj þ

∑ jgðPÞij Aj

1þ h∑ jgðPÞij Aj

!

dAi

dt¼ Ai aðAÞi − ∑ jb

ðAÞij Aj þ

∑ jgðAÞij Pj

1þ h∑ jgðAÞij Pj

!

8>>>>><

>>>>>:

ð2Þ

where the variables Pi and Ai denote the abun-dance of plant and animal species i, respectively.

SCIENCE sciencemag.org 25 JULY 2014 • VOL 345 ISSUE 6195 1253497-3

Fig. 2. Numerical analysis of species persistence as a function of modelparameterization. This figure shows the simulated dynamics of speciesabundance and the fraction of surviving species (positive abundance at theend of the simulation) using the mutualistic model of (6). Simulations areperformed by using an empirical network located in Hickling, Norfolk, UK(table S1), a randomized version of this network using the probabilistic model

of (32), and the network without mutualism (only competition). Each rowcorresponds to a different set of growth rate values. It is always possible tochoose the intrinsic growth rates so that all species are persistent in each ofthe three scenarios, and at the same time, the community persistencedefined as the fraction of surviving species is lower in the alternativescenarios.

RESEARCH | RESEARCH ARTICLE

James et al., Nature 2012

Rohr et al., Science 2014

Bastolla et al., Nature 2009

Eigenvalues of Random Matrixdx

dt= �x

�ij ⇠ N (0,�)

�c =1pSC

-20 -10 0 10 20-20

-10

0

10

20

0.6 0.8 1.0 1.2 1.4

0

0.2

0.4

0.6

0.8

1.0

σ SC

P(stability)

Random

Re�h

Im h

R. May Random Structure

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

Stability [Resilience] Max[Re(�)]

Mutualistic &

Nested Structure

LETTERdoi:10.1038/nature10832

Stability criteria for complex ecosystemsStefano Allesina1,2 & Si Tang1

Forty years ago, May proved1,2 that sufficiently large or complexecological networks have a probability of persisting that is close tozero, contrary to previous expectations3–5. May analysed largenetworks in which species interact at random1,2,6. However, innatural systems pairs of species have well-defined interactions(for example predator–prey, mutualistic or competitive). Herewe extend May’s results to these relationships and find remarkabledifferences between predator–prey interactions, which are stabil-izing, and mutualistic and competitive interactions, which aredestabilizing. We provide analytic stability criteria for all cases.We use the criteria to prove that, counterintuitively, the probabilityof stability for predator–prey networks decreases when a realisticfood web structure is imposed7,8 or if there is a large preponderanceof weak interactions9,10. Similarly, stability is negatively affected bynestedness11–14 in bipartite mutualistic networks. These results arefound by separating the contribution of network structure andinteraction strengths to stability. Stable predator–prey networkscan be arbitrarily large and complex, provided that predator–preypairs are tightly coupled. The stability criteria are widely applicable,because they hold for any system of differential equations.

May’s theorem deals with community matrices1,2,6 M, of size S 3 S,where S is the number of species. Mij describes the effect that species jhas on i around a feasible equilibrium point (that is, species havepositive densities) of an unspecified dynamical system describing thespecies’ densities through time.

In May’s work1,2, the diagonal coefficients are 21, and the off-diagonal coefficients are drawn from a distribution with mean 0 andvariance s2 with probability C and are 0 otherwise. For these matrices,the probability of stability is close to 0 whenever the ‘complexity’sffiffiffiffiffiffiSCp

w1. Local stability measures the tendency of the system toreturn to equilibrium after perturbations. In unstable systems, eveninfinitesimal perturbations cause the system to move away fromequilibrium, potentially leading to the loss of species. Thus, it shouldbe extremely improbable to observe rich (large S) or highly connected(large C) persistent ecosystems1,2. Mathematically, an equilibriumpoint is stable if all the eigenvalues of the community matrix havenegative real parts1,2,6.

Local stability can only describe the behaviour of the system aroundan equilibrium point, whereas natural systems are believed to operatefar from a steady state5,15. However, methods based on local stability arewell suited to the study of large systems1,16,17, whose empirical para-meterization would be unfeasible. Moreover, the methods are general,so that they can be applied to any system of differential equations.

May’s matrices have random structure: each pair of species interactswith the same probability. However, this randomness translates, forlarge S, into fixed interaction frequencies, so that these matrices followa precise mixture of interaction types. For example, in May’s matricespredator–prey interactions are twice as frequent as mutualistic ones(Supplementary Table 1). Here we extend May’s work to differenttypes of interaction, starting from the random case.

Suppose that two species j and i interact with probability C, and thatthe interaction strength is drawn from a distribution: Mij takes thevalue of a random variable X with mean E Xð Þ~0 and variance

Var(X) 5 s2. The diagonal elements of the community matrix, repre-senting self-regulation, are set to 2d. For large systems, the eigenvaluesare contained in a circle18 in the complex plane (Fig. 1 and Supplemen-tary Information). The circle is centred at (2d, 0) and the radius issffiffiffiffiffiffiSCp

. In stable systems, the whole circle is contained in the left half-plane (that is, all eigenvalues have negative real parts). Thus, thesystem is stable when the radius is smaller than d:

ffiffiffiffiffiffiSCp

vh~d=s.In predator–prey networks, interactions come in pairs with opposite

signs: whenever Mij . 0, then Mji , 0. With probability C, we sample oneinteraction strength from the distribution of jXj and the other from 2jXj,whereas with probability (1 2 C) both are zero. The eigenvalues of largepredator–prey matrices are contained in a vertically stretched ellipse19,centred at (2d, 0), with horizontal radius s


1{E2 Xj jð Þ"

s2# $and

thus the stability criterion isffiffiffiffiffiffiSCp

vh"

1{E2 Xj jð Þ=s2# $(Fig. 1 and

Supplementary Information).When we constrain Mij and Mji to have the same sign, and thus

impose a mixture of competition and mutualism with equal probability,the eigenvalues are enclosed in a horizontally stretched ellipse19 andthe criterion becomes


vh"

1zE2 Xj jð Þ=s2# $

(Fig. 1 and Sup-plementary Information).

Take C 5 0.1, X , N(0, 1/4) (that is, X follows a normal distributionwith mean 0 and variance 1/4), and d 5 1. The criterion becomesffiffiffiffiffiffi

SCp

v2 for random matrices, and is violated whenever S $ 41. Forpredator–prey we find


v2p= p{2ð Þ (violated for S $ 303) andfor the mixture of competition and mutualism


v2p= pz2ð Þ(violated for S $ 15). Since E Xj jð Þ=sw0 for any distribution of X,the stability criteria form a strict hierarchy in which the mixture matricesare the least likely to be stable, the random matrices are intermediate,and the predator–prey matrices are the most likely to be stable (Fig. 2and Table 1). Considerations based on qualitative stability2 andnumerical simulations16 are consistent with this hierarchy.

In the three cases above, the mean interaction strength is zero, andthe coefficients come from the same distribution. In fact we can shufflethe interaction strengths, thereby transforming a network of one typeinto another: the difference in stability is driven exclusively by thearrangement of the coefficients in pairs with random, opposite andsame signs, respectively. This feature allows us to further derive thestability criteria for all intermediate cases by using linear combinationsof the three cases above (Supplementary Information).

Two ecologically important cases, however, cannot produce a meaninteraction strength of zero. In mutualistic networks all interactionsare positive, whereas in competitive networks they are negative. Inthese cases, for large systems, all the eigenvalues except one (equal tothe row sum) are contained in an ellipse (Fig. 3 and SupplementaryFigs 1 and 2). In mutualistic networks in which all interaction pairs arepositive and drawn from the distribution of jXj independently withprobability C, the stability criterion becomes S{1ð ÞCE Xj jð Þ=svh(that is, row sum , 0; Supplementary Information). For competitivematrices, in which interaction pairs are drawn from the distribution of2jXj with probability C, the criterion isffiffiffiffiffiffiSCp

1z 1{2Cð ÞE2 Xj jð Þ=s2# $% ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1{CE2 Xj jð Þ=s2

qzCE Xj jð Þ=svh

1Department of Ecology and Evolution, University of Chicago, 1101 East 57th Street, Chicago, Illinois 60637, USA. 2Computation Institute, University of Chicago, 5735 South Ellis Avenue, Chicago, Illinois60637, USA.

0 0 M O N T H 2 0 1 2 | V O L 0 0 0 | N A T U R E | 1

Macmillan Publishers Limited. All rights reserved©2012

Nestedness reduces system resilience!

Localized Not Localized

0 50 100 150 2000.0

0.2

0.4

0.6

0.8

Component @SpeciesD

»v i\

Beyond Resilience: Localization

rIPR =

* PSi=1

��v1(i)|4PS

i=1 |vran1 (i)|4

+

�x

x

x(t) =SX

↵=1

⇠

⇠

⇠ · uuu↵

u

u

u↵ · vvv↵e

�↵tv

v

v↵

!1

!

0

0.5

max

ASSIGN INTERACTION STRENGTHS

�ij = aij�0k�i

Null Model ) aran

!

!!

!

!!!!!!

!!!!!

!!!!!!!

!!

!!

!

!

!

!!!

!!!

!

!

!

!

!!!!!!!

!

!!

!

!!!!!

!!

!

!

"

""

"

""""""

"

"

"

"

""""

""

""

""

""

"

"

"

"""

""

"

"

"

"

"

""""

"""

"

"" """

""

"

"

"

"

"#

#########

#

######

######

###

# # #######

##

#

#

########

##

#

######

#

#

#

0 200 400 600 800

12

51020

50

Size !S"

rIPRleft

!

!!

!

!!!!!!

!

!!

!

!

!!!!!

!!

!!

!

!

!

!

!

!!!

!!!

!

!

!

!

!!!!

!!!

!

!!

!

!!

!!!

!

!

!

!

""

""

""""""

"

""""""

""""" """

"

""

"

"" """""

"" """""""""

"""

"

"

"""

""

"

"

#

##

#

######

##

#

#

####

##

##

#

#

##

#

#

#

###

###

#

#

#

#

#####

##

#

##

#

####

#

#

#

#

#

0 200 400 600 8001.0

10.0

5.0

2.03.0

1.5

15.0

7.0

Size !S"

rIPRright

A B

Ecological networks are localized!

A1 = |⇠0|(X

j

v1,j |)2

Localization attenuates perturbations

Max|{v1}A1 = 1/pS

Min|{v1}A1 = �i,j⇤

�

��

��

�

�

��

�

�

��

�

��

��

�

�

��

�

�

�

�

�

��

�

��

�

�

�

�

��

�

�

�

�

�

��

�

�

��

�

�

�

�

�

�

��

��

�

�

��

�

�

��

�

��

��

�

�

��

�

�

�

�

�

��

�

� ��

�

�

�

�

��

�

�

�

�

�

��

�

�

�

� �

�

�

�

�

�

�

��

��

��

�

�

��

�

��

��

�

�

��

�

�

�

� ��

� �

�

�

�

�

�

�

�

��

�

�

�

�

��

�

�

�

��

�

�

�

�

� ��

��

��

��

��

��

��

�

�� [�]

�/�

��

Trade Off with May!

Suweis et al., 2014

»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

Localization occurs on the hubs

Suweis et al., 2014

Ê

ÊÊ

Ê

ÊÊÊ

Ê

Ê Ê

Ê

Ê

Ê

Ê

Ê

Ê ÊÊ

ÊÊ

ÊÊ

Ê

Ê

Ê

Ê

Ê

Ê

ÊÊÊ

ÊÊ

Ê

Ê

Ê

Ê

Ê

Ê ÊÊÊ

ÊÊ

Ê

Ê

ÊÊ

Ê

Ê Ê

ÊÊ

Ê

Ê

Ê

Ê

Ê

‡

‡‡

‡

‡‡‡

‡

‡ ‡

‡

‡

‡

‡

‡

‡ ‡‡

‡‡

‡‡

‡

‡

‡

‡

‡

‡

‡‡‡

‡‡

‡

‡

‡

‡

‡

‡ ‡‡‡

‡‡

‡

‡

‡‡

‡

‡ ‡

‡‡

‡

‡

‡

‡

‡

Ï

Ï

Ï

Ï Ï

Ï

Ï

Ï Ï

Ï

ÏÏ

Ï

Ï

Ï Ï

ÏÏ

Ï

Ï Ï

Ï

Ï ÏÏ

Ï

Ï

Ï

Ï

Ï

Ï

ÏÏ ÏÏ

ÏÏ

Ï

Ï

ÏÏ

Ï

ÏÏ

ÏÏ

Ï

Ï

Ï

Ï

Ï

ÚÚÚ

ÚÚ ÚÚÚÚ Ú

Ú

Ú ÚÚÚÚ Ú

Ú ÚÚÚÚÚ Ú

ÚÚ

ÚÚ

Ú

ÚÚÚ

ÚÚ ÚÚ

ÚÚÚ Ú Ú

ÚÚÚ

ÚÚÚ

ÚÚ

Ú

Ú

Ú

Ú ÚÚÚ

Ú

Ú

Ú

ÙÙ

ÙÙÙ

ÙÙÙÙ Ù

Ù

Ù ÙÙÙ

Ù Ù

Ù ÙÙÙÙ

Ù

Ù ÙÙ

ÙÙÙ Ù

ÙÙ Ù Ù

Ù

Ù

Ù

Ù

Ù

Ù ÙÙÙ Ù ÙÙ

Ù

ÙÙ

Ù

ÙÙ

Ù ÙÙ

ÙÙ

Ù

Ù

Á

Á Á

Á

Á ÁÁ

Á

Á Á

Á

Á

Á

Á

ÁÁ Á Á

ÁÁ

Á

Á

Á

Á

Á

Á ÁÁÁ

Á

Á

Á

Á

Á

Á ÁÁ

Á

Á

Á

Á

Á

Á

Á

Á

Á Á Á

Á

Á

Á

Á

Á

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.70

1

2

3

4

5

NODF

Localization

Nestedness Localization6=

Suweis et al., 2014

Back to nested patterns…

Simple mechanism driving mutualistic community to nested network architectures?

Adaptive/foraging strategy?

My foraging strategy :-)

Same idea!

Theoretical Framework

• Abundances = {x1,x2,...,xS}

!

!

!

• σΩ , σΓ so that x* is stable • Community population dynamics

Implementation of the Optimization Principle

T T+1i j

l

k j

l

swap

bWil

Start with xi ~N(1,0.1) and random M (α, S, C fixed)

Foraging Strategy

i

Mil

M ) M

0

if x0,⇤i > x

⇤i

x

⇤ = M�1 · ↵

Let’s play!

Why does it works ??

1) Relation between optimization of single species and community abundance

2) Relation between species abundance and nestedness

Cooperation in mutualistic community!

0 200 400 600 800 1000120014001600180020009.8

10

10.2

10.4

10.6

10.8

11

11.2

11.4

11.6

11.8

STEPS [T]

Popu

latio

n

0 200 400 600 800 1000120014001600180020009.8

10

10.2

10.4

10.6

10.8

11

11.2

11.4

11.6

11.8

STEPS [T]

Popu

latio

n

Averaged over 100 realizations

0 200 400 600 800 10001200140016001800200019.5

20

20.5

21

21.5

22

22.5

STEPS [T]To

toal

Pop

ulat

ion mean

1 realiz

T T+1i j

l

k j

l

swap

bWil

x

⇤ = M�1 · ↵

0 n n+1

0.951.001.051.101.15

Step [T]

Popu

lation

[xi] i

li

j

i

j

l

|γij|=0.0017

|γij|=0

T=n

i

li

j|γij|=0.0017

|γij|=0

T=n+1swap

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* =x

⇤ + �x⇤ = (M + �M)�1 · ↵

Mil

Overlap and community abundance are correlated!

x

⇤ = M�1 · ↵

M = M0 + V =

I+ ⌦ OO I+ ⌦

�+

O ��T O

�

x

tot = K + Co ) o / C�1x

tot + constant

0.2 0.3 0.4 0.5 0.6 0.7 0.850

54

58

62

66

Nestedness [NODF]Ab

unda

nce

[x]

C

c

−0.05 −0.04 −0.03 −0.02 −0.01 0

0

0.01

0.02

0.03

0.04

0.05

Max[Re(λ)]ra

rest

spec

ies [x

]

b

R2=0.999

0 5 10 15 20 250

1

2

3

4

5

number of connections [k]

spec

ies a

bund

ance

‹x›

si=|∑jγij|

a

‹x›

pdf

Max[Re(λ)]

0 1 2

5

0

4321

-0.8 -0.7 -0.6 - 0.5 - 0.4

5

10

15

20

25

Stability and Localization in Optimal Mutualistic Networks

rightleft

Conclusions!

Emergent ecological patterns may be described using simple models: learning processes from patterns !

Trade-off between resilience/ecological complexity and localiziation: measuring stability from different perspectives !

Emergent nested species interaction network: explaining patterns using simple principles

Thanks for your attention!

Questions?

Neutral Theory: PNAS 2011, JTB 2012 Optimization: Nature 2013 Stability: Oikos 2014 Localization: soon in Arxiv

@SamirSuweis

impactstory.org/SamirSuweis