Emergence of complex structure through co-evolution:

Transcription

1 Emergence of complex structure through co-evolution: The Tangled Nature model of evolutionary ecology Institute for Mathematical Sciences & Department of Mathematics Collaborators: Paul Anderson, Kim Christensen, Simone A di Collobiano, Matt Hall, Domininc Jones, Simon Laird, Daniel Lawson, Paolo Sibani

2 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

3 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

4 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.

5 Phenomenology

6 Interaction and co-evolution The Tangled Nature model Individuals reproducing in type space Your success depends on who you are amongst n(s)= Number of individuals n(s)= Number of individuals Type - S Type - S

7 Definition Individuals and, where Dynamics a time step L= 3 Annihilation Choose indiv. at random, remove with probability

8 Reproduction: Choose indiv. at random Determine occupancy at the location

9 The coupling matrix J(S, S ) Either consider J(S, S ) to be uncorrelated or to vary smoothly through type space.

10 from reproduction probability 1

11 Asexual reproduction: Replace by two copies with probability

12 Mutations Mutations occur with probability, i.e.

13 Time dependence (Average behaviour) Total population N(t) Diversity Time in Generations

14 Origin of drift? Effect of mutation Let convex

15 Dynamics: The functional form of reproduction probability pof f Average population pkill Dominic Jones pkill 2 10 Time (100 generations) Average population pof f Time (100 generations) FIG. 4:

16 Intermittency: Non Correlated # of transitions in window Matt Hall 1 generation =

17 Intermittency at systems level: Correlated Simon Laird

18 Complex dynamics: Intermittent, non-stationary Jumping through collective adaptation space Transitions

19 Record dynamics

20 Record dynamics: stochastic signal the record Paolo Sibani and Peter Littlewood: exponentially distributed

21 Record dynamics: exponentially distributed Poisson process in logarithmic time Mean and variance Rate of records constant as function of ln(t) Rate decreases

22 Record dynamics: Ratio (t k t k 1 )/t k 1 remains non-zero Cumulative Distribution Paul Anderson

23 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.

24 Other systems Ants: exit times Earthquakes: After shocks - Omori law (?) Magnetic relaxation: temperature independent creep rate Spin glass: exponential tails

25 Dynamics - correlations:

26 Dynamics - correlations: The evolution of the correlations I = J 1,J 2 P (J 1, J 2 ) log[ P (J 1, J 2 ) P (J 1 )P (J 2 ) ] # )*!"!$ Dominic Jones!%) *+!"!$ Dominic Jones!%(# *:;4;3<*=1>52-34,51 (&# ( '&# ' %&# 9:5:4;+<2= !%(!%'#!%'!%!#!%!!%"#! "%&# %!" #!" $ +,-.*/ ,5167 "%&!" #!" $,-./+01/2/ Mutual information of core Mutual information of all

27 Networks emerge Time evolution of Distribution of active coupling strengths Correlated Simon Laird

28 The evolved degree distribution Correlated Simon Laird Exponential becomes 1/k in limit of vanishing mutation rate

29 Copyright 2002 by the National Academy of Sciences Dunne, Jennifer A. et al. (2002) Proc. Natl. Acad. Sci. USA 99,

30 The evolved connectance Correlated #edges D(D 1) Simon Laird

31 Connectance Montoya JM, Sole RV Topological properties of food webs: from real data to community assembly models, OIKOS 102, (2003) Williams RJ, Berlow EL, Dunne JA, Barabasi AL, Martinez ND Two degrees of separation in complex food webs, PNAS 99, (2002) Simon Laird

32 Coarse Grained Description Node and Edge Model

33 Analysis approach: Tangled Nature IBM Node model Simple Mean Field analysis Fokker-Planck equation

34 Focus only on whether a type is occupied or not Dynamical Rules Removal: Duplication: with probability 1/D place edge between parent and child copy existing edge with probability introduce new edge with probability P p P e P n

35 Self-consistent Mean Field Degree Dynamics Resulting evolution equation for degree distribution n k (t + 1) = n k (t) n k(t) D + k n k+1 n k D Removed node Adjacent node looses an edge + [P e k + P n (D 1 k )] n k 1 n k D n k 1 + P p D + (1 P p) n k D Daughter node Adjacent gains an edge

36 Mean field Degree Dynamics Stationary solution n(k) = n(0) exp[ k/k 0 ] with k 0 as P n 0 Qualitative agreement with simulations of Node Model (and TaNa)

37 Effect of adaptation on connectance Underlying 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

38 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

39 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 Fraction Fluctuations in E E = E(D, C) + sσ(d, C) = E m C + s[e m C(1 C)] 1 2 Max,i.e., Em=D(D-1)

40 Effect of selection on connectance The resulting estimate for the connectance, E/Em, of the adapted sub-net C Adap = C + s = C + s [ C(1 C) E m [ 2C(1 C) D(D 1) ] 1 2 ] 1 2. Qualitative agreement with simulations of Tangled Nature model

41 The evolved connectance Correlated #edges D(D 1) Simon Laird

42 Fokker-Planck equation

43 Analytical result n k (t + 1) = n k (t) + Γ R (D, k, t) + Γ Du (D 1, k, t). Γ R (D, k, t) = Γ d R(k) + Γ N R (k + 1) Γ N R (k). Γ Du (D 1, k, t) = Γ P Du(k 1) Γ P Du(k) + Γ C Du(k) +Γ N Du(k 1) Γ N Du(k).

44 Fokker-Planck eq. iterated Linear-Log Log-Log 1 1 n(k) n(k) k 1 10 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-log plot to the left. The approach towards a 1/k dependence in the limit of P e 1 can be seen in the log-log plot to the right. The two straight lines have slope -1. The parameters are D = 20, P e = 0.01, 0.1, 0.3, 0.5, 0.7, 0.9, 0.95 and P p = P n was chosen to be P n = P p (1 P e )/(1 P p ) ) HJ Jensen, PRS A 2008

45 Limiting behaviour In limit mutations 0 Implies P e = 1 and P n 0 Fokker-Planck eq. reduces to n k (t + 1) = n k (t) + n k ( + D 2 k 1 k 1 =1 q=1 1 D 1 1 D ) + 2P p D 1 (n k 1 n k ) qn k1 [ 1 D P Ed(k 1, k + 1, q) 1 D 1 P Ed(k 1, k, q) 1 D P Ed(k 1, k, q) + 1 D 1 P Ed(k 1, k 1, q)]

46 Limiting behaviour Include only leading terms from k 1 = 1 and q = 1 n k (t + 1) = n k + n k D(D 1) + 2P p D 1 [n k 1 n k ] + n 1 M [ 1 D {(k + 1)n k+1 kn k } 1 D 1 {kn k (k 1)n k 1 }]. Stationary solution n k 1/k

47 Summary

48 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

49 Thank you for the chance to speak Papers from Collaborators: Paul Anderson Kim Christensen Simone A di Collobiano Matt Hall Dominic Jones Simon Laird Daniel Lawson Paol Sibani