Emergent phenomena in large interacting communities

Transcription

1 Emergent phenomena in large interacting communities Giulio Biroli Institute for Theoretical Physics, CEA Saclay, France Statistical Physics Lab- ENS Paris Joint works with G. Bunin, C. Cammarota, V. Ros, F. Roy

2 Ecosystems Communities formed by individuals belonging to different species. Interactions between individuals intra and inter species. Competition for resources- - Cooperation. Abundances of species vary dynamically due to the births and deaths.

3 Traditional ecosystem

4 Modern ecosystems

5 Modern ecosystems Abundance proniles Time dependence Human microbiome project hundred- thousand species in a given ecosystem

6 Questions & Motivations From a few species to many STATISTICAL PHYSICS Emergent properties of complex ecosystems How many equilibria for the same ecosystem? In some ecosystems many, Bashan et al. Nature 016 Response to perturbations? Memory, hysteresis, Dethlefsen, Relman PNAS 011 Equilibria or chaotic dynamics? Chaos in plankton ecosystem, Beninca et al Nature 008 What are the factors determining diversity (number of surviving species)?

7 Lotka- Volterra equations for ecosystems dn i = N 4r i i (K i N i ) N i 0 abundance of species i S is the number of species j,(j6=i) 3 ij N 5 p j + N i i (t)+ i i =1,...,S Well- mixed population: no- space dependence Dynamics due to intra- and inter- species interactions Properties of the community reached dynamically

8 Lotka- Volterra equations for ecosystems dn i = N i 4r i (K i N i ) N i abundance of species i S is the number of species j,(j6=i) 3 ij N 5 p j + N i i (t)+ i i =1,...,S dn i = r i N i (K i N i ) A species alone self- regulates to the abundance K i

9 Lotka- Volterra equations for ecosystems dn i = N i 4r i (K i N i ) N i abundance of species i S is the number of species j,(j6=i) 3 ij N 5 p j + N i i (t)+ i i =1,...,S dn i = N i 4r i (K i N i ) j,(j6=i) 3 ij N 5 j interaction between species

10 Lotka- Volterra equations for ecosystems dn i = N i 4r i (K i N i ) N i abundance of species i S is the number of species j,(j6=i) 3 ij N 5 p j + N i i (t)+ i i =1,...,S dn i = N i 4r i (K i N i ) j,(j6=i) 3 ij N 5 p j + N i i (t) Demographic Noise to model Nluctuations in births and deaths h i (t) j (t 0 )i =! (t t 0 )

11 Lotka- Volterra equations for ecosystems dn i = N i 4r i (K i N i ) N i abundance of species i S is the number of species dn i = N i 4r i (K i N i ) 3 ij N 5 p j + N i i (t)+ i j,(j6=i) j,(j6=i) i =1,...,S 3 ij N 5 p j + N i i (t)+ i Immigration rate

12 Lotka- Volterra equations for ecosystems dn i = N i 4r i (K i N i ) N i abundance of species i S is the number of species j,(j6=i) 3 ij N 5 p j + N i i (t)+ i i =1,...,S Large number of species (S~ is large)

13 Lotka- Volterra equations for ecosystems dn i = N i 4r i (K i N i ) N i abundance of species i S is the number of species j,(j6=i) 3 ij N 5 p j + N i i (t)+ i i =1,...,S Main assumption: complex - > random (May in ecology & Wigner in physics) (Determining interactions network: a key inference problem) h ij i = µ S h iji c = S ij Gaussian RVs i.i.d. h ij ji i c = h iji c =1 symmetric 1 apple apple 1 Small noise and small immigration rate (Ki=1) Representative model, see Barbier et al. PNAS to appear

14 Ecosystems Phase Transitions =1 multiple equilibria Complex phase < 1 Chaos Similar for symmetric and non symmetric interactions (here =0) Related works and phase diagrams Sompolinsky, Crisanti, Sommers 88 ; Diederich, Opper 89; Fisher, Mehta 14; Kessler, Shnerb 15; Bunin 16 EACT SOLUTION G. B., G. Bunin and C. Cammarota ariv: and works in progress (F. Roy, V. Ros)

15 Unique Equilibrium Phase

16 Unbounded Growth Phase

17 Complex Phase =1 symmetric interactions: multiple equilibria < 1 non- symmetric interactions: chaos

18 Transition to Chaos < 1

19 dn i Symmetric interactions 3 = N i 4r i (K i N i ) ij N j 5 + p N i i (t)+ i j,(j6=i) =1 E = i dn i = N Ni E({N i })+ p N i i (t) V i (N i )+ 1 i6=j ij N i N j + i (! i) log N i Stochastic dynamics of a disordered system (~spin- glass) Langevin equation small or zero noise Low temperature physics T =!

20 The phase diagram & the energy landscape Spin- glass (replica method)

21 The Two Phases One equilibrium Multiple Equilibria (Critical Spin- Glass Phase) One single stable equilibrium Inverse eigenvalue distribution of the j Eigenvalue distribution of the Hessian All equilibria marginally stable

22 Critical Multiple Equilibria Phase Marginal stability Nixes dynamically the diversity S = 100 S = 00 S = c marginal stability May s bound Analogous phenomenon in jamming of hard spheres: isostaticity of packings Extreme susceptibility to perturbations (memory only in the one equilibrium phase) Large Nluctuations- correlations 4(t, t 0 )= 1 S [h N i (t) N i (t 0 ) N j (t) N j (t 0 )i ij h N i (t) N i (t 0 )ih N j (t) N j (t 0 )i] 4 (t-t') t - t' (t,t) c

23 Dynamics and Transition to Chaos < 1 Chaos All equilibria are unstable (Kac- Rice Method) Chaotic dynamics (dynamical mean- Nield theory) Ongoing: charaterize chaotic dynamics, properties of the transition to chaos Related works: Sompolinsky, Crisanti, Sommers 88 ; Kessler, Shnerb 15

24 Emergent phenomena in interacting communities Different phases of ecosystems from the exact solution of the Lotka- Volterra model of ecosystems An entire region with multiple equilibria poised at the edge of stability: - marginal phase, extreme susceptibility to perturbations, large correlations,... - diversity is dynamically Nixed by the requirement of being marginal stable: May s bound is saturated Chaotic phase where all equilibria are unstable Generality beyond the particular model we studied: emergent properties as for phases of matter. Many perspectives: Chaotic dynamics, slow dynamics, avalanches, other functional responses, retardation effects, space dependence,...