emergence of complex structure through co-evolution

Collaborators:Paul Anderson, Kim Christensen, Simone A di Collobiano, Matt Hall, Domininc Jones, Simon Laird, Daniel Lawson, Paolo Sibani

Emergence of complex structure through co-evolution: The Tangled Nature model of evolutionary ecology

Henrik Jeldtoft Jensen Institute for Mathematical Sciences & Department of Mathematics

What are we after ?

>> Connecting micro to macro

Micro level: stochastic individual based dynamics always running at clock speed one

Macro level: intermittent dynamics networks, structure

Collective adaptation

Henrik Jeldtoft Jensen Imperial College London

List of content:

• Motivation:

• The Tangled Nature Model

• Co-evolution and the evolved networks

Phenomenology: ❆ intermittency

❆ emergent networks ❆ connectance

❆ evolving correlations Explanation Simple mean field Fokker-Planck equation

Motivation:☻How far can a minimal model go?

☻Input: mutation prone reproduction at level of interacting individuals.

☻Output: Species formation, macro dynamics, decreasing extinction rate + SAD, SAR, etc.

☻Check: Trend in broad range of data.


Phenomenology

Interaction and co-evolution

The Tangled Nature model• Individuals reproducing in type space

• Your success depends on who you are amongst


Type - S

n(S)= Number of individuals

Type - S

n(S)= Number of individuals

Definition Individuals

, where

and

L= 3 Dynamics – a time step

Annihilation Choose indiv. at random, remove with probability


Reproduction:

► Choose indiv. at random ► Determine

occupancy at the location


The coupling matrix

Either consider to be uncorrelated

or to vary smoothly through type space.

J(S, S′)

J(S, S′)


from reproduction probability

1


Asexual reproduction:

by two copies

with probability

Replace


Mutations

Mutations occur with probability

, i.e.


Time dependence

Total population N(t)

Diversity

Time in Generations

90

100

0

5104×

5104×

700

800

(Average behaviour)

Origin of drift? Effect of mutation Let

convex

Dynamics: The functional form of reproduction probability

pkill

poff

pkill

poff

4

1 2 3 4 5 6 7 8 9 100.8

1

1.2

1.4

1.6

1.8

2

2.2

2.4x 10

!9

Time(105 gens)

Mutu

al In

form

ation (

avera

ged o

ver

250 r

ealis

ations):

Random

netw

ork

s

FIG. 3:

100

101

102

103

104

950

1000

1050

1100

1150

Avera

ge p

opula

tion

Time (100 generations)

100

101

102

103

104

1350

1400

1450

Avera

ge p

opula

tion

Time (100 generations)

FIG. 4:


Dominic Jones


Intermittency:

# of transitions in window Matt Hall

1 generation =

Non Correlated


Intermittency at systems level:Correlated

Simon Laird


Complex dynamics:Intermittent, non-stationary

Jumping through collective adaptation space

Transitions

Record dynamics



Record dynamics: the record

stochastic signal

Paolo Sibani and Peter Littlewood:

exponentially distributed


Record dynamics:

► Poisson process in logarithmic time

► Mean and variance

►Rate of records constant as function of ln(t)

► Rate decreases

exponentially distributed

Record dynamics:Ratio

remains non-zeroCumulative Distribution (tk − tk−1)/tk−1

Paul Anderson


Consequences of record dynamics. Statistics of transition times independent of

underlying “noise mechanism”.

Evolution: same intermittent dynamics in micro- as in

macro-evolution.

Decreasing extinction rate.


Other systems• Ants:

exit times

• Earthquakes:

After shocks - Omori law (?)

• Magnetic relaxation:

temperature independent creep rate

• Spin glass:

exponential tails


Dynamics - correlations:


Dynamics - correlations:


The evolution of the correlations

I =∑

J1,J2

P (J1, J2) log[P (J1, J2)

P (J1)P (J2)]

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!"$

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"%&#

!

!%"#

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!%'#

!%(

!%(#

!%)*+!"

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,-./+01/2/345-62789:5:4;+<2=63.45-62

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%

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'

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(&#

#)*!"

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+,-.*/0.1.234,5167

89.230.*:;4;3<*=1>52-34,51

Mutual information of core Mutual information of all

Dominic JonesDominic Jones

Networks emerge


Simon Laird

Time evolution of Distribution of active coupling strengths

Correlated


The evolved degree distribution Correlated

Simon Laird

Exponential becomes 1/k in limit of vanishing mutation rate

Copyright ©2002 by the National Academy of Sciences

Dunne, Jennifer A. et al. (2002) Proc. Natl. Acad. Sci. USA 99, 12917-12922


The evolved connectance Correlated

Simon Laird

#edges

D(D − 1)

Connectance 4.3. Network properties CHAPTER 4.

Figure 4.11: Field data showing connectance versus diversity for a variety of eco-logical systems. Data is taken from articles by Montoya et al [18] and Williams etal [19] mostly representing the same ecosystems. The connectances of the originalarticles have been doubled to make their measures consistent with the connectancedefinition in this thesis (see text).

4.3.2 May-wigner criterion for ecosystem stability

It was a commonly held view that the stability of an ecological network was en-

hanced by increased complexity. There are many notions of complexity [73], but

here we primarily mean the connectance. The perception was that the larger num-

ber of interaction paths between species acts to dampen any natural fluctuations or

environmentally-sourced perturbations. This seemed reasonable and is intuitive if we

think of the concurrent effects of the feedback loops as being averaged out. This

intuition can be misleading though. If we consider, as an approximation, that the

interaction effects occur as random normally distributed fluctuations each identically

distributed and independent then the dampening viewpoint is inappropriate. The

volatility of these fluctuations is measured by the standard deviation of the normal

distribution. As the summed effects of multiple sources of the fluctuations leads to

a variance which is a sum of the individual variances we have an overall standard

87

REFERENCES REFERENCES

[15] Raup DM Biological extinction in earth history, Science, 231, 1528-1533 (1986)

[16] Sole RV, Manrubia SC, Benton M, Bak P Self-similarity of extinction statisticsin the fossil record, Nature, 388, 764-767 (1997)

[17] Drossel B Biological evolution and statistical physics, Adv. Phys. 50, 2, 209-295, (2001)

[18] Montoya JM, Sole RV Topological properties of food webs: from real data tocommunity assembly models, OIKOS 102, 614-622 (2003)

[19] Williams RJ, Berlow EL, Dunne JA, Barabasi AL, Martinez ND Two degreesof separation in complex food webs, PNAS 99, 12913-12916 (2002)

[20] May RM Stability and complexity in model ecosystems: monographs in popu-lation biology, Princeton UP, (1974)

[21] Pimm SL Lawton JH, Cohen JE Food web patterns and their consequences,Nature 350, 669-674, (1991)

[22] Warren PH Making connections in food webs, Trends Ecol. Evol. 4 136-140(1994)

[23] McCann KS The diversity stability debate, Nature 405, 228-233, (2000)

[24] Dunne J, Williams R J, Martinez N D Food web structure and network theory:The role of connectance and size, PNAS, 99, 12917-12922 (2002)

[25] Kaufmann SA, The origins of order: Self-organization and selection in evolu-tion, Oxford UP, (1993)

[26] Derrida B Random energy model: An exactly solvable model of disordered sys-tems, Phys. Rev. B 24, 5, 2613-2626, (1981)

[27] Kaufmann SA, Johnsen S Coevolution to the edge of chaos: Coupled fitnesslandscapes, poised states and coevolutionary avalanches, J. Theor. Biol. 149,467-505, (1991)

[28] Havens K Scale and structure in natural food webs, Science 257, 5073, 1107-1109, (1992)

[29] Eigen M Self-organization of matter and the evolution of biological macro-molecules, Naturwissenschaften 58, 465-523, (1971)

[30] Eigen M Schuster P The hypercycle: A principle of natural self-organization,Berlin: Springer, (1979)

142

Simon Laird

Coarse Grained Description

Node and Edge Model

Analysis approach:

Fokker-Planckequation

Simple Mean Field analysis

Node model

Tangled Nature IBM


Dynamical Rules

Removal: with probability

Duplication:

place edge between parent and child

copy existing edge with probability

introduce new edge with probability

Focus only on whether a type is occupied or not

1/D

Pp

Pe

Pn


Self-consistent Mean Field Degree Dynamics Resulting evolution equation for degree distribution

nk(t + 1) = nk(t) − nk(t)D

+ 〈k〉nk+1 − nk

D

+ [Pe〈k〉 + Pn(D − 1 − 〈k〉)]nk−1 − nk

D

+ Ppnk−1

D+ (1 − Pp)

nk

D

Removed node

Adjacent node looses an edge

Adjacent gainsan edge

Daughter node


Mean field Degree Dynamics

Stationary solution

with

n(k) = n(0) exp[−k/k0]

Qualitative agreement with simulations of Node Model (and TaNa)

k0 →∞ as Pn → 0


Effect of adaptation on connectanceUnderlying type space is a binomial net - place a sub-net of size D

Some regions of this space will, due to fluctuations, locally have an above average conenctance. It is beneficial for the evolved configurations to enter into these regions

With increasing size, D, of the adapted sub-net; it becomes increasingly difficult to confine the sub-net to within the above average regions

Increasing D

Effect of selection on connectance Consider a binomial net of size D and connectance C (= edge probability).

Assume that adapted sub-net is located in a region of the master-network in which the total number of edges E is larger than the global average.

Estimate this increase as E = 〈E(D,C)〉 + sσ(D,C)

= EmC + s[EmC(1 − C)]12

Fluctuations in E

Max,i.e., Em=D(D-1)

Fraction

Effect of selection on connectance The resulting estimate for the connectance, E/Em, of the adapted sub-net

CAdap = C + s

[C(1− C)

Em

] 12

= C + s

[2C(1− C)D(D − 1)

] 12

.

Qualitative agreement with simulations of Tangled Nature model


The evolved connectance Correlated

Simon Laird

#edges

D(D − 1)

Fokker-Planck equation

Analytical resultnk(t + 1) = nk(t) + ΓR(D, k, t) + ΓDu(D − 1, k, t).

ΓR(D, k, t) = ΓdR(k) + ΓN

R (k + 1)− ΓNR (k).

ΓDu(D − 1, k, t) = ΓPDu(k − 1)− ΓP

Du(k) + ΓCDu(k)

+ΓNDu(k − 1)− ΓN

Du(k).


Fokker-Planck eq. iterated

Again we assume PnEd(k1, k2, q) = PEd(D−2−k1, k2, q). In general it is not simple to findanalytic solutions to this somewhat involved set of equations. The result of iterating theFokker-Planck Eq. (9) using these estimates is shown in Fig. 2 for diversity D = 20, whichmakes the numerical iteration manageable. We notice good qualitative agreement withthe behaviour of simulation results presented Fig. 1. For a broad range of values of theparameters the degree distributions exhibit much of the same approximate exponentialform as produced by the individual based Tangled Nature model described in the Sec. 2.

10 20k

0.01

1

n(k)

1 10k

0.01

1

n(k)

Figure 2: The degree distribution obtained by iteration of the Fokker-Planck equation(9). The exponential form is visible for a broad range of parameter values in the linear-logplot to the left. The approach towards a 1/k dependence in the limit of Pe → 1 can beseen in the log-log plot to the right. The two straight lines have slope -1. The parametersare D = 20, Pe = 0.01, 0.1, 0.3, 0.5, 0.7, 0.9, 0.95 and Pp = 0.01. Pn was chosen to bePn = Pp(1 − Pe)/(1 − Pp) )

Let us finally mention that direct simulations [11] of the simple model introduced insection 3 show that in the limit Pe → 1, Pn → 0 and Pp # 1 the degree distribution nk

behaves like 1/k. The Fokker-Planck equation Eq. (9) confirms this result. In the limitPe = 1 and Pn → 0 (i.e. perfect replication) the Fokker-Planck equation reduces to

nk(t + 1) = nk(t) + nk(1

D − 1−

1

D) +

2Pp

D − 1(nk−1 − nk)

+D−2∑

k1=1

k1∑

q=1

qnk1[1

DPEd(k1, k + 1, q) −

1

D − 1PEd(k1, k, q)

−1

DPEd(k1, k, q) +

1

D − 1PEd(k1, k − 1, q)]. (23)

Including only the leading terms from k1 = 1 and q = 1 one obtains

nk(t + 1) = nk +nk

D(D − 1)+

2Pp

D − 1[nk−1 − nk]

+n1

M[1

D{(k + 1)nk+1 − knk}−

1

D − 1{knk − (k − 1)nk−1}]. (24)

In the limit D $ 1 and Pp # 1 this equation has the stationary solution nk ∝ 1/k.

8

HJ Jensen, PRS A 2008

Linear-Log Log-Log

Limiting behaviour In limit mutations 0

Implies

Fokker-Planck eq. reduces to

Pe = 1 and Pn → 0

nk(t + 1) = nk(t) + nk(1

D − 1− 1

D) +

2Pp

D − 1(nk−1 − nk)

+D−2∑

k1=1

k1∑

q=1

qnk1[1D

PEd(k1, k + 1, q)− 1D − 1

PEd(k1, k, q)

− 1D

PEd(k1, k, q) +1

D − 1PEd(k1, k − 1, q)]


Limiting behaviour Include only leading terms from k1 = 1 and q = 1

nk(t + 1) = nk +nk

D(D − 1)+

2Pp

D − 1[nk−1 − nk]

+n1

M[1D{(k + 1)nk+1 − knk}

− 1D − 1

{knk − (k − 1)nk−1}].

Stationary solution nk ∝ 1/k

Summary

Summary and conclusion From minimal micro-dynamics

Collective evolution and adaptation

Intermittent dynamics >> record dynamics.

Nature of the evolved networks - compares well

Dynamics (degree of overlap with parent) determines degree distribution

Adaptation/selection influences connectance


Papers from www.ma.ic.ac.uk/~hjjens

Collaborators:

Paul Anderson

Kim Christensen

Simone A di Collobiano

Matt Hall

Dominic Jones

Simon Laird

Daniel Lawson

Paol Sibani


Thank you for the chance to speak

http://www.mai.ic.ac.uk/%7C

http://www.mai.ic.ac.uk/%7C

emergence of complex structure through co-evolution

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