1. The Emergence of Community Structure in Metacommunities
Per Arne Rikvold (Physics, Florida State U.) Élise Filotas and Lael Parrott
(Geography, U. of Montreal),
Martin Grant (Physics, McGill U.),
Supported by Natural Sciences and
Engineering Research Council of Canada, le Fond Québécois de la Recherche sur la Nature et les Technologies, Réseau
Québécois de Calcul de Haute Performance,
and the U.S. National Science Foundation

2. Ecol/Evol and Nonequilibrium Statmech
Ecology and evolution present us with many systems that consist of large numbers of “particles” (individuals, species, …)
These “particles” interact by nonlinear and often unknown rules, or “interactions”
The systems are far from equilibrium.
Ideal playground for statistical mechanics!

3. Individual-based Coevolution Model P.A.R. and R.K.P. Zia, Phys. Rev. E 68, 031913 (2003).
Binary, haploid genome of length L gives
2L different potential genotypes …101
Considering this genome as coarse-grained, we consider each different bit string a “species.”
Asexual reproduction in
discrete, nonoverlapping generations.
Simplified version of “tangled-nature” model introduced by Hall, Christensen, et al., Phys. Rev. E 66, (2002);
J. Theor. Biol. 216, 73 (2002).

4. Selection via population dynamics
Probability that an individual of genotype I has F
offspring in generation t before dying is PI({nJ(t)}).
Probability of dying without offspring is (1-PI).
N0: Carrying capacity limits total population Ntot(t).
MIJ : Effect of species J on birth probability of I.
MIJ and MJI both positive: symbiosis or mutualism.
MIJ and MJI both negative: competition.
MIJ and MJI opposite sign: predator/prey relationship.
Here: MIJ quenched, random e [-1,+1], except MII = 0.
Abundant energy; “space” limitation N0.

5. -I Abundant resources and/or few predators Scarce resources and/or
many predators
-I

6. New species introduced through Mutations
Each individual offspring undergoes mutation to a different species with probability m per individual.
Diffusion between corners of a hypercube
L=3

7. Non-spatial simulations

8. Main quantities measured
Total population, Ntot(t)
Diversity: Total number of species.
Shannon diversity, D(t), gives the number of heavily populated species. Obtained as D(t) = exp[S(t)]
where
S(t) = - SI [nI(t)/Ntot(t)] ln [nI(t)/Ntot(t)]
is the information-theoretical entropy (Shannon-Wiener index).

9. Intermittent dynamics
Shannon
Diversity,
D(t)
Ntot(t),
normalized
nI > 1000
nI e [101,1000]
nI e [11,100]
nI e [2,10]
nI = 1
Quasi-steady states (QSS) punctuated by active periods. Self-similarity.

10. Typical community structure
Small, mutualistic, fully connected communities

11. Spatially Extended Model

12. Work with Élise Filotas, Lael Parrott, and Martin Grant
Place model on 2D 64£64 square lattice.
Include “fitness-dependent” dispersion: Individuals with PI < pd
move randomly to a neighboring site.
Metacommunity and neighborhood

13. Similarity between communities
Generalized Jaccard index ( |AÆB|/|AÇB| ) as
where
is the sum of the relative abundances over those species i 2 A that are shared with B.
Local similarity index: IA = h IAB iB neighbor of A
Global similarity index: I = h IA iMetacommunity A B

14. Similarity vs dispersal rate
pd= pd= pd=0.8

15. Cluster identification at pd=0.22
Locally similar clusters in a sea of dissimilar local
communities. Reminiscent of Potts model.

16. Low pd: Many dissimilar, species-poor communities
Average global () diversity
Low pd: Many dissimilar,
species-poor communities
yield high global diversity.
Average local () diversity pd
High pd: Relatively diverse,
but similar communities
yield lower global diversity.
Uniform metacommunity. pd

17. pd = 0.22 pd = 0.28 Diversity Shannon diversity
Neighborhood similarity
pd = 0.28
Diversity
Shannon diversity
Neighborhood similarity

18. Community structure
Pred/Prey
Competitors
Mutualists
pd = 0
pd = 0. 8
Original species pool
At low pd, the communities are heavily biased toward mutualism.
At high pd, immigrants and mutants make interaction distribution
more similar to the total gene-pool, but still biased toward
mutualism: mutualistic core

19. Species Abundance Distributions
pd = 0.8
pd = 0.8
pd = 0
At pd = 0, local communities are small and highly
mutualistic. Other species are unsuccessful mutants
of the main species.
At pd = 0.8, local communities are larger and
have a much broader distribution of rare
immigrant species with various kinds of interactions.

20. Heterogeneous landscape
Carrying capacity
Vertical landscape coordinate
200
2000
3800

21. Community structure pd = 0 pd = 1 N0 = 200 N0 = 3800
Original species pool

22. Species Abundance Distributions
pd = 0
pd = 1
Filled: N0 = 200
Open: N0 = 3800
At low dispersal rates, a typical community is formed of a core of
2 (N0 = 200) to 4 (N0 = 3800) highly abundant species.
At high dispersal rates the SAD consists of a continuous range
between the most common and the rarer species.

23. Conclusions
Nonspatial model yields intermittent dynamics and highly connected, mutualistic communities.
Spatially extended model has phase transition with respect to dispersion threshold.
Low dispersion: Isolated local communities.
High dispersion: Uniform metacommunity.
Low carrying capacity and dispersion threshold promote mutualism.

24. Some Publications
• P.A.R. and R.K.P. Zia, Phys. Rev. E 68, (2003)
• R.K.P. Zia and P.A.R, J. Phys. A 37, 5135 (2004)
• V. Sevim and P.A.R., J. Phys. A, 38, 9475 (2005)
• P.A.R., J. Math. Biol. 55, (2007)
P.A.R. and V. Sevim, Phys. Rev. E 75, (2007)
E. Filotas, M. Grant, L. Parrott, and P.A.R,
J. Theor. Biol. 266, (2010)
Ecol. Modell. 221, (2010)