Statistical Physics of Ecological Networks: from patterns to principles
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Talk that I gave in Leeds at the school of Mathematics on 26/11/2014. It is an overview of my recent on research on mutualistic ecological networks by using tools and approaches from statistical physics.
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Statistical Physics of Ecological Networks: from patterns to principles
- 1. From Patterns to Principles Statistical Physics of Ecological Networks @SamirSuweis
- 2. Outline •Statistical Physics & Ecology ! •Architecture of the species interaction network ! •Stability of ecological networks ! •From patterns to principles: explaining nested architectures in mutualistic ecological community
- 3. Stochastic approaches for microbial mobility. Interacting particle models Neutral Theory & Ecological Patterns Population dynamics & metapopulation models Ecological Networks & Optimization Stability in Ecological Communities Statistical Inference (Biodiversity) Everything :-) Visiting Ph.D. Master student Ph.D. Post-Doc Post- Doc Master student Ph.D. Not mixing our expertise, but summing them up
- 4. Emergent Pattern in Ecology: RSA 0 2 4 6 8 10 15 10 5 0 Number of species Coral Reefs Abundance classes [log scale] Tropical Forests 40 Number of species 0 30 20 10 0 1 2 3 4 5 6 7 Abundance classes [log scale]
- 5. Complex Patterns from Simple Rules All species are equivalent Single trophic level Basic (random) ecological processes Birth & death Master Equation dPn(t) dt = bn1Pn1(t) + dn+1Pn+1(t) (bn + dn)Pn(t) Parameters: bn/dn and m = b0 Functional form of bn • Density dependent effects Coral Reefs 0 2 4 6 8 10 15 10 5 0 Number of species Abundance classes [log scale] Tropical Forests 40 Number of species 0 30 20 10 0 1 2 3 4 5 6 7 Abundance classes [log scale] Volkov et al., Nature 2007
- 6. Darwin’s entangled bank […] and to reflect that these elaborately constructed forms, so different from each other in so complex a manner, have been all produced by laws acting around us. (Darwin, Origin of Species)
- 7. Our approach: “Make everything as simple as possible, A. Einstein but not simpler.” “You don’t really understand something unless you can explain it to your grandmother.”
- 8. The architecture of mutualistic species interactions network From patterns to principles
- 9. Ecological Networks 10/14/2014 Web of Life: ecological networks database Networks All Data All Species 0 10000 Interactions 0 10000 Reset Results Download(89) Help
- 10. Find the pattern :-)
- 11. A closer look to the nested structure Plant Pollinator web in Chile Arroyo, et al. Random same S,C Random same S,C Avian fruit web in Puerto Rico Carlo, et al. 1 5 10 15 20 1 10 20 32 1 5 10 15 20 25 1 10 20 30 36 NODF=0.424 NODF=0.192 1 5 10 15 20 25 1 10 20 30 36 NODF=0.072 1 10 20 32 1 5 10 15 20 NODF=0.133 Bascompte et al., PNAS 2003
- 12. Quantitative measures of nestedness :-( The number of common partners the i-th and the j-th plant have Overlap NODF measure Almeida et al., Oikos 2008
- 13. Network data vs Randomization 1 Null model 1: we keep fixed S and C, and place at random the edges # Species [S] Nestedness [NODF] Data 20 40 60 80 100 120 140 160 180 200 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 Random
- 14. Network data vs Randomization 2 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.7 0.6 0.5 0.4 0.3 0.2 0.1 NODF DATA NODF Null Model Null model 2: we keep fix p(k) while randomizing the edges
- 15. Why this recurrent topological structure? Does the nested structure give more stability to these ecological communities?
- 17. Are nested networks more stable?
- 18. Many ways to quantify stability (13 definitions !) + no analytical results Persistence dPi dt = ↵Pi IPiP2 i + XNa j=1 ijAjPi h1 ij + P k,hji0 Ak dAi dt = ↵Ai IAiA2i + XNp j=1 jiAiPj h1 ji + P k,hji0 Pk . Model Individual survival Fig. 2. Numerical analysis of species persistence as a function of model parameterization. This figure shows the simulated dynamics of species abundance and the fraction of surviving species (positive abundance at the end of the simulation) using the mutualistic model of (6). Simulations are performed by using an empirical network located in Hickling, Norfolk, UK (table S1), a randomized version of this network using the probabilistic model Persistence Bastolla et al., Nature 2009 Rohr et al., Science 2014 of (32), and the network without mutualism (only competition). Each row corresponds to a different set of growth rate values. It is always possible to choose the intrinsic growth rates so that all species are persistent in each of the three scenarios, and at the same time, the community persistence defined as the fraction of surviving species is lower in the alternative scenarios. Persistence 0 10 20 1 0.5 0 r2 = 0.60 r2 = 0.35 Partners Strong mutualism 0 0.2 0.4 0.6 1 0.5 0 r2 = 0.87 r2 = 0.77 Connectance 1 0.5 0 r2 = 0.77 102 104 Network magnitude a b c SCIENCE sciencemag.org 25 JULY 2014 • VOL 345 ISSUE 6195 1253497-3 James et al., Nature 2012
- 19. Eigenvalues of Random Matrix dx Random = x dt !ij ⇠ N(0, ) c = 1 pSC 20 10 0 -10 -20 -20 -10 0 10 20 0.6 0.8 1.0 1.2 1.4 1.0 0.8 0.6 0.4 0.2 0 Stability [Resilience] Max[σ SC Re()] P(stability) Reh Im h R. May Random Structure Real Imaginary A B −7 −6 −5 −4 −3 −2 −1 0 1 2 3 4 5 0.5 1.0 −0.5 0.5 1.0 −0.5 Mutualistic Nested Structure LETTER doi:10.1038/nature10832 Stability criteria for complex ecosystems Stefano Allesina1,2 Si Tang1
- 20. Nestedness reduces system resilience!
- 21. Beyond Resilience: Localization Localized Not Localized 0 50 100 150 200 0.8 0.6 0.4 0.2 0.0 Component @SpeciesD »vi rIPR = * PS i=1 ## v1(i)|4 PS i=1 |vran 1 (i)|4 + !x(t) = XS ↵=1 ⇠⇠⇠ · u↵ u↵ · v↵ e↵tv↵
- 22. ASSIGN INTERACTION max 0 0.5 ! !1 STRENGTHS !ij = aij 0 k i Null Model ) aran
- 23. Ecological networks are localized! A B ! ! ! ! !! ! !!! ! # #### ! # !! ! ! ! ! ! ! ## ! !! ! ! !! ! ! ! ! ## ## ! # # ! ! ! ! ! ! !!!! !! ! !! !!! ! ! ! # # # # ## # ## # # # # #### # # # # ######## # # # ## ### # # 0 200 400 600 800 50 20 10 5 2 1 Size !S rIPR left # ! # # # # # # !! ! ! ! ! # ! ! ! # ## # # ! !! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! # ! ! ! ! ! ! ! ! !! !! ! ! !! ! !! !! ! ! ! ! # # # # # # # # # ## # # # # # # # # # ## # # # # # # # # ## # # # # # # 0 200 400 600 800 15.0 10.0 7.0 5.0 3.0 2.0 1.5 1.0 Size !S rIPR right Localization attenuates perturbations A1 = |⇠0|( X j v1,j |)2 Max|{v1}A1 = 1/pS Min|{v1}A1 = i,j⇤ [] / Trade Off with May! Suweis et al., 2014
- 24. Localization occurs on the hubs lmax=-0.0813779 lH=0.145052 »v1 »u1 »wH k kmax 0 20 40 60 1.2 1.0 0.8 0.6 0.4 0.2 0.0 Species Suweis et al., 2014
- 25. Ï Localization Á Ê Ù Ê ‡ Ú Ù NODF Á Á Á Á Á Á Á Ù Ù Ú Ú Ê Ê ‡ Ù Á Á Ù Ú Ú Ù Á Á Ú Ú Ú Ú Ê Ú Ê Ê ‡ Ï Ï Ï Ï Ï Ï Ï Ï Ï Ú Ï Ú Ú Ú Ú Ú Ê Ú Ú Ê ‡ ‡ Ù Ù Á Ú Ú Ú Ú Ú Ê Ú Ï Ê Ê Ï Ê ‡ Ê Ê ‡ Ê Ê Ê Ê Ê Ê Ê Ê Ê ‡ ‡ Ê Ê Ê Ú Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ú Ú Ú Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï ÏÏ Ï Ï Ï Ï Ï Ï Ï Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú ÚÚ Ú Ú Ú Ú Ú Ú ÚÚ Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ù Ù Ù ÙÙ Ù Ù Ù Ù Ù Ù ÙÙ Ù Ù Ù Ù Ù ÙÙ Ù Ù Ù Ù Ù Ù Ù Ù Ù Ù Ù Ù Ù Ù Ù Ù Ù Ù ÙÙ Ù Ù Ù Ù Ù Ù Ù Ù Ù Á Á Á Á Á Á Á Á Á Á Á Á Á Á Á Á Á Á Á Á Á Á Á Á Á Á Á Á Á Á Á Á Á Á Á Á Á Á Á Á 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 5 4 3 2 1 0 Nestedness 6= Localization Suweis et al., 2014
- 26. Back to nested patterns… Simple mechanism driving mutualistic community to nested network architectures? Adaptive/foraging strategy?
- 27. My foraging strategy :-) Same idea!
- 28. Theoretical Framework • Abundances = {x1,x2,...,xS} ! ! ! • σΩ , σΓ so that x* is stable • Community population dynamics
- 29. Implementation of the Optimization Principle Start with xi ~N(1,0.1) and random M (α, S, C fixed) T T+1 i j l k j l swap bWil Foraging Strategy i Mil M ) M0 if x0,⇤ i x⇤i x⇤ = M−1 · ↵
- 30. Let’s play!
- 31. Why does it works ?? 1) Relation between optimization of single species and community abundance 2) Relation between species abundance and nestedness
- 32. Cooperation in mutualistic community! 11.8 11.6 11.4 11.2 T=n T=n+1 swap : : Population 11 10.8 10.6 10.4 10.2 9.8 STEPS [T] |γij|=0 0 200 400 600 800 100012001400160018002000 10 T T+1 0 200 400 600 800 100012001400160018002000 11.8 11.6 11.4 11.2 11 10.8 10.6 10.4 10.2 10 9.8 STEPS [T] Population Averaged over 100 realizations Mil 22.5 22 21.5 21 20.5 20 19.5 0 200 400 600 800 100012001400160018002000 STEPS [T] Totoal Population mean 1 realiz i j l k j l swap bWil x⇤ = M−1 · ↵ n+1 i j i l j l |γij|=0.0017 |γij|=0 i j |γij|=0.0017 i l 0.803522 1.08178 1.05803 1.05014 0.977939 1.01422 0.958128 1.13397 1.04078 1.0356 0.9664 1.02013 1.00682 0.67361 1.10131 1.07571 1.10289 0.959658 0.996913 0.918892 1.15298 1.03813 1.0223 1.01314 0.958794 1.00217 x* = x* = x⇤ + !x⇤ = (M + !M)−1 · ↵
- 33. Overlap and community abundance are correlated! x⇤ = M−1 · ↵ M = M0 + V = I + ⌦ O O I+ ⌦ + O T O xtot = K + Co ) o / C1xtot + constant 0.2 0.3 0.4 0.5 0.6 0.7 0.8 66 62 58 54 50 Nestedness [NODF] C Abundance [x]
- 34. Stability and Localization in Optimal Mutualistic Networks c 0.05 0.04 0.03 0.02 0.01 0 −0.05 −0.04 −0.03 −0.02 −0.01 0 Max[Re(λ)] rarest species [x] b R2=0.999 4321 0 5 10 15 20 25 5 4 3 2 1 0 number of connections [k] species abundance ‹x› si=|Σjγij| a ‹x› pdf Max[Re(λ)] 0 1 2 5 0 - 0.8 - 0.7 - 0.6 - 0.5 - 0.4 25 20 15 10 5 right left
- 35. Conclusions ! Emergent ecological patterns may be described using simple models: learning processes from patterns ! Trade-off between resilience/ecological complexity and localiziation: measuring stability from different perspectives ! Emergent nested species interaction network: explaining patterns using simple principles
- 36. Thanks for your attention! Questions? Neutral Theory: PNAS 2011, JTB 2012 Optimization: Nature 2013 Stability: Oikos 2014 Localization: soon in Arxiv @SamirSuweis impactstory.org/ SamirSuweis