The Tangled Nature Model of Evolutionary Ecology: (Is the approach of Statistical Mechanics relevant to the New Ecology Systems Perspective project.

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Allan Owens
5 years ago
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Institute for Mathematical ciences and Dept of Mathematics United Kingdom

1 The Tangled Nature Model of Evolutionary Ecology: Is the approach of tatistical Mechanics relevant to the New Ecology ystems Perspective project. Institute for Mathematical ciences and Dept of Mathematics United Kingdom Collaborators: Daniel Lawson, imon Laird, Paul Anderson, Kim Christensen, Matt Hall, imone A di Collobiano,

2 Evolutionary ecology: Interacting organisms + Evolution Evolving bio-net Each type will see an ever changing environment Focus on system level properties stability mode of evolution nature of the adaptation ecological characteristics: AD, AR, Connectance,...

3 Interaction and co-evolution The Tangled Nature model Individuals reproducing in type space Different types influence the livelihood of each other

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

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

6 The coupling matrix J, Either consider J, to be uncorrelated or to vary smoothly through type space and sparse or dense Results unoccupied site. Links are deactivated positive interaction negative interaction

7 from reproduction probability 1

8 Asexual reproduction: Replace by two copies with probability

9 Mutations Mutations occur with probability, i.e. ee also work on similar models by Rikvold et al.

10 REULT

11 egregation in genotype space Initiation Only one genotype Jn term = 0 Total population Diversity Nt adjusts Time steps

12 Macro dynamics: Non correlated Graph courtesy to Matt Hall Type label 1 generation = # generations

13 Time dependence Average behaviour Total population Nt Diversity Time in Generations

14 Intermittency: # of transitions in window Matt Hall 1 generation =

15 Complex dynamics: Exergy?? Intermittent, non-stationary Jumping through collective adaptation space: quake driven Transitions ft Logt

16 Macro dynamics - the transitions Non correlated Graph courtesy to Matt Hall

17 tability of the q-e: Consider simple adiabatic approximation. tability of genotype assuming: Consider tationary solution Fluctuation Fulfil i.e. stability

18 Transitions between q-e caused by co-evolutionary collective fluctuations

19 Macro dynamics: Correlated imon Laird Fig. 1 An occupation plot of a single run for a system with R = 10,000. For each timeslice a point appears where a phenotype is in existence but as the full space is in 16 dimensions a projection onto a single trait is used.

20 Time evolution of Distribution of active coupling strengths Non correlated Low connectivity High connectivity From Anderson & Jensen J Theor Biol. 232, Normalised density of individuals with strength J Interaction strength, J Fig. 3. Interaction distributions. Top: Distribution of interaction strengths between individuals for y ¼ 0:005: Bottom: y ¼ 0:25: Inset: Entire distribution. olid lines, random; crosses, simulation at t ¼ 500; dotted lines, simulation at t ¼ 500; 000: All plots are normalized so that their area is one. For high y; a significant increase in positive interactions is seen. For low y; a change is seen but for trivial reasons.

21 Time evolution of Distribution of active coupling strengths Correlated High connectivity From Laird & Jensen, Ecol Compl. 3,

22 Time evolution of pecies abundance distribution Non Correlated Proportion of sites with! individuals From Anderson & Jensen J Theor Biol. 232, Number of individuals per site,! 0 Fig. 5. pecies abundance distributions. pecies abundance distributions for the simulations only. Dashed line, t ¼ 500; dashed dotted line, t ¼ 5000; solid line, t ¼ 500; 000: Low y on the left, high y on the right. The ecologically realistic log-normal form is only seen for high y: Low connectivity High connectivity

23 Time evolution of Degree distribution Non Correlated Number of sites with x interactions Low connectivity High connectivity Number of interactions, x Fig. 2. Degree histograms. Top: Degree histogram for y ¼ 0:005: Bottom: y ¼ 0:25: olid lines, random; dotted lines, simulation. From the left, the pairs of curves are for t ¼ 500; 5000 and 500,000. At later times, the number of active links increases for both the simulation and random data. From Anderson & Jensen J Theor Biol. 232,

24 The evolved degree distribution Correlated D=29 D=26 D=19 Binomial, D= Pk e Degree, k Figure 1: Degree distributions for the Tangled Nature model simulations. hown are ensemble averaged data taken from all networks with diversity, D = {19, 26, 29} over 50 simulation runs of 10 6 generations each. The exponential forms are highlighted by comparison with a binomial distribution of D = 29 and equivalent connectance, C to the simulation data of the same diversity. Exponential becomes 1/k in limit of vanishing mutation rate From Laird & Jensen, Ecol. Model. In Press ee also Laird & Jensen, EPL, 76,

25 Diversity and interaction Weight function,,, 0 t N t n J t N k t H µ α α = { a scale parameter [0,1] and with, 0 ε ε ε α α α α + = E E H H t N t n E Density dependent Density independent Fujiyama lanscape 0

26 Diversity and interaction From Lawson, Jensen & Kaneko, J Theo. Biol. 243,

27 Origin of threshold in k: A balance between inter-species and intra-species Interaction. and [0,1] where,,, 0 + = E t N t n t E H H α α ε,,, t N t n J t N k t H µ α α = J E k H H N N N n N E Jn N k H D N E D then and 2 1 Assume 2 : Weight function for H 1: Weight function for ε µ ε µ ε > > = + = = = = Mean field sketch

28 The evolved connectance Correlated C = # Edges # Possible Edges Mean connectance Diversity Figure 4: Plot of ensemble-averaged mean connectances, < C > against species diversity. Error bars represent the standard error. The lower dotted line marks the null system connectance, C J = 0.05, which the evolved systems clearly surpass. The overlaid functional form is that given by Eq.8 using the correct background connectance, C J = 0.05 and with a value of, s = 5.5 for the selection parameter. From Laird & Jensen, Ecol Compl. 3,

29 pecies area relation: pecies Diversity Real Data Real data Regression Line z=0.19 imulated data Dispersion by random walk a Area ha for data, arbitrary for simulation x plant data from Hertfordshire, see ML Rosenzweig, pecies Diversity in pace and Time, Cambridge University Press, 1995 From Lawson & Jensen, J Theo. Biol. 241,

30 The individual in ever evolving surroundings: Collective system level adaptation towards mutualistic biased webs of interactions Macro-Evolution through intermittent transitions

31 Download papers from: Collaborators imon Laird Daniel Lawson Paul Anderson Kim Christensen Matt Hall imone A di Collobiano

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