1. Characterizing Phase Transitions in a Model of Neutral Evolutionary Dynamics
Adam David Scott
Department of Physics & Astronomy
University of Missouri at St. Louis
APS March Meeting – Baltimore, MD
18 March 2013

2. Acknowledgements Dr. Sonya Bahar Dawn King Dr. Nevena Marić
McDonnell Foundation Complex Systems Grant
UMSL Department of Physics & Astronomy

3. Obtaining critical exponents for each
Model
Phase transitions
Evidence for two separate behaviors
Obtaining critical exponents for each

4. Model ONLY phenotypic evolution Asexual
No genetics & no physical space
2D Continuous with absorbing boundaries
Asexual
Branching & coalescing random walks
Reaction-diffusion process
Mutability, μ
Maximum possible offspring mutation from parent
Control parameter
Clustering
Dawn’s: Random death percentage for each organism
Control parameter is random death percentage
THIS: Random death percentage of population
Control parameter is mutability

5. Model: Clustering Phenetic species Cluster seeds
Speciation by phenotypes
Cluster seeds
Reference
Nearest neighbor
Second nearest neighbor
Closed sets of cluster seeds

6. Generations
Mutability
Scott et al. (submitted)

7. Phase Transitions: Order Parameters
<Population>
<Number of Clusters>
0.34
0.38
0.34
0.38
Scott et al. (submitted)

8. Evidence: Times to extinction

9. Evidence: Nearest Neighbor Index
Clark & Evans
Average nearest neighbor distance / random expectation
0.34
0.38
Scott et al. (submitted)

10. Phase Transitions Temporal 2nd order non-equilibrium
Absorbing state: extinction
Directed percolation
Spatial
2nd order equilibrium
Isotropic percolation
Pick a time, change mu, does either state trap the system? Yes = non-equilibrium, No = equilibrium

11. Directed Percolation: Critical Exponents
Density scaling, ρ~ (𝑝− 𝑝 𝑐 ) −𝛽
Density decay rate, ρ~ 𝑡 −α
Correlation length, ξ ~ (𝑝− 𝑝 𝑐 ) −ν
Decay rate, ξ ~ 𝑡 1 𝑧
Correlation time, ξ ~ (𝑝− 𝑝 𝑐 ) −ν
Need 2 for classification
𝛼= 𝛽 ν 𝑧= ν ν
(Henkel, Hinrichsen, Lübeck)

12. 0.340
0.335
0.330
0.325
0.310
0.315
0.320 μ
0.310
0.315
0.320
0.325
0.330 α
3.489
3.326
2.589
1.773
0.462

13. Directed Percolation Density Decay Rate
This model
𝛼~0.462±0.033
Compared with others
𝛼= 𝛽 ν ~ (Hinrichsen)
2.6%
𝛼= 𝛽 ν ~ (Lauritsen)
2.4%

14. Isotropic Percolation: Critical Exponents
Static measures
Fisher exponent 𝑛 𝑠 ~ 𝑠 −𝜏
Distribution of cluster mass, ns, for size, s
Cluster fractal dimension N ~ 𝐿 𝑑 𝑓
Cluster mass (N)
Number of organisms in cluster
Cluster diameters (L)
Longest of the shortest paths
Still running!
(Lesne)

15. Absorbing State Extinction
Conclusions
Absorbing State Extinction
Surviving
Surviving
Two 2nd order phase transitions
Non-equilibrium (directed percolation)
Equilibrium (isotropic percolation)
Possibility for this behavior in real biology?
First experimental directed percolation observed in 2007 for liquid crystals (Takeuchi et al.)
Aggregated
Spanning
𝜇~0.335
𝜇~0.38

16. Absorbing State Extinction
Thank you!
Absorbing State Extinction
Surviving
Surviving
Aggregated
Spanning
𝜇~0.335
𝜇~0.38

17. References
Henkel, Hinrichsen, Lübeck, Non-Equilibrium Phase Transitions: Vol 1: Absorbing Phase Transitions, 2009
Hinrichsen, “Nonequilibrium Critical Phenomena and Phase Transitions into Absorbing States”, arXiv, 2000
Lauritsen, Sneppen, Marošová, Jensen, ”Directed percolation with an absorbing boundary”, 1997.
Lesne, Renormalization Methods: Critical Phenomena, Chaos, Fractal Structures, 1998
Scott, King, Marić, Bahar, “Clustering and Phase Transitions on a Neutral Landscape”, submitted
Takeuchi, Kuroda, Chaté, Sano, “Directed percolation criticality in turbulent liquid crystals”, 2007, 2009