Effects of correlated interactions in a biological coevolution model with individualbased dynamics

1
Effects of correlated interactions in a
biological coevolution model with
individual-based dynamics

• Volkan Sevim and Per Arne Rikvold
• School of Computational Science,
• Center for Materials Research and Technology, and
  
• Department of Physics,
• Florida State University

2
Ecological/Evolutionary Dynamics

• Reproduction/Death
• Predation
• Mutation
• Succession
• Speciation
• Extinction

Character of fluctuations? Universal features?

• MC simulation of a model ecosystem
• Reproduction
• Interactions (e.g. predator/prey)
• Mutation/Speciation
• Death/Extinction

3
ModelA model ecosystem
Just a dummy index
Organism
A bitstring of length L8 bits
Genome
2L256 possible species
4
Model A Monte Carlo Step
Begin simulation with a small population of a
particular species.
1 - Reproduce asexually with a finite probability.
Ntot(t) Total population at time t N0 Verhulst
Factor (Carrying capacity) WI Fitness of
species I
2 - Mutate with a finite probability. (Generate
new species) m10-3 per genome per generation.
5
Model Interactions Fitness
MIJ negative Species I is harmed by species J
Zero diagonals No intraspecific interactions
MJI positive Species J benefits from species I
2L x 2L Interaction Matrix, M MIJ Effect of pop.
density of species J on species I.
6
Model Construction of Interaction Matrix

• Random (Uncorrelated) Matrix
• Assign a random number to each matrix element
  from a Gaussian distribution with a st. dev. s0
  and mean 0.

Not very realistic because, interactions of a
mutant with other species can be very different
than those of its ancestor.
How to make mutants resemble to their ancestors?
Matrix Elements
7
Model Construction of Interaction Matrix

• Correlated matrix (Correlated fitness landscape)
• Start with a random (uncorrelated) matrix
• Average each matrix element over its neighbors in
  genome space, up to nth nearest neighbor. (Not
  neighbors in the matrix!!)

Think of ij as a concatenated bitstring. Look for
its neighbors at a Hamming distance n or less.
(Bitstrings with n or less bits different than
ij.)

• For example for n1, (avg. over nearest
  neighbors)

(Because the nearest neighbors of 1 are 0, 3, 5,
7, and nearest neighbors of 0 are 1, 2, 4, 8)
8
Model Construction of Interaction Matrix

• Correlated matrix (Correlated fitness landscape)
• Start with a random (uncorrelated) matrix
• Average each matrix element over its neighbors in
  genome space, up to nth nearest neighbor. (Not
  neighbors in the matrix!!)

Dividing the sum with vZn instead of Zn, keeps
the standard deviation of matrix elements
constant. M0, M Random and correlated
matrices. n Averaging radius. Include elements
up to nth nearest neighbor. Zn Total number of
elements in the sum. H(ijkl) Hamming distance
between the concatenated bitstrings ij and kl.
9
Model Correlations between matrix elements after
averaging
s0
10
Model Correlations between matrix elements after
averaging
Correlation Functions (L8 bits)
Correlation functions deviate from the theory for
large n for finite samples
Theoretical and numerical results agree for small
n
r
11
Model Complications in the averaging process
Not negligible for large Zn !
Zn for L8
12
Results Active and Quiet Periods
Diversity Index(Roughly, the number of highly
populated species)
Punctuated equilibria
Bursts of mass extinctions or rapid increase in
diversity
Diversity Index
where, S is the information-theoretical entropy
A quasi-steady state (QSS)
where, rI is the population density of species I.
Time (generations)
Sampled every 16 steps
13
Results Entropy Changes
Quiet periods mutations and reproduction
Active periods Big avalanches
Threshold for QSS.
14
Correlated Model- Some features Lifetime
Distributions
Lifetime Time interval between the creation of a
species (by mutation) and its extinction.
Lifetime Distributions (L8) 3.3x107 MC steps
each, avg. over 16 runs
Correlations lead to more pronounced 1/t2 behavior
15
Results - Correlated Model Quasi-Steady State
Duration Distributions
QSS The interval in which the change in the
entropy of the system is below a certain threshold
Quasi-Steady State Duration Distributions
(L8) 3.3x107 MC steps each, avg. over 16 runs
Correlations lead to 1/t behavior
16
Results - Correlated Model Power Spectrum
Densities
Power Spectrum Densities for the Diversity Index
(L8) 3.3x107 MC steps each, avg. over 16 runs
17
Summary

• The averaging method employed to construct a
  correlated fitness landscape creates nontrivial
  complications for correlation ranges that are a
  significant fraction of the genome length.
• Long-range correlations have noticeable effects
  on the long term behavior of the system. However,
  even the effects of unrealistically strong
  correlations appear to be quite minor, and
  qualitative features of the system remain
  unchanged. 1/t2 and 1/f behaviors in lifetime
  distributions, and PSD of diversity index,
  respectively, seem quite robust.

18

• Some recent articles on dynamics of ecological
  systems and food webs
• P.A. Rikvold and R.K.P. Zia, Punctuated
  Equilibria and 1/f noise in a biological
  coevolution model with individual based dynamics,
  Phys. Rev. E 68, 031913 (2003).
• B. Drossel, and A. J. McKane, Modeling Food Webs,
  arXiv.nlin.AO/0202034.
• M. Hall, K. Christensen, S. A. di Collobiano, and
  H. J. Jensen, Time-dependent extinction rate and
  species abundance in a tangled-nature model of
  biological evolution, Phys. Rev. E 66, 011904
  (2002).

19
Model Correlations between matrix elements after
averaging
Correlation Function (L8 bits, n7)
Deviation in the correlation function translates
into deviation in the matrix element distribution
20
Results - Uncorrelated Model Lifetime
DistributionsPower Law Behavior
Lifetime Distributions for Uncorrelated Model
Narrow distributions lead to more pronounced 1/t2
behavior
How different matrix element distributions affect
the fluctuations in the system?
21
Results - Uncorrelated Model Distributions for
the Duration of Quasi-Steady States Power Law
Behavior
Quasi-Steady State Distributions for Uncorrelated
Model
Tested range of standard deviations do not make a
difference
How different matrix element distributions affect
the fluctuations in the system?