Stability, dispersal and ecological networks. François Massol

Transcription

1 Stability, dispersal and ecological networks François Massol June 1 st 2015

2 General theme Evolutionary ecology of fluxes o Evolution & ecology of dispersal o Spatial structure, networks of populations o Food webs & other interaction networks

3 General theme Evolutionary ecology of fluxes o Evolution & ecology of dispersal o Spatial structure, networks of populations o Food webs & other interaction networks o Ultimately: (spatial) network of (interaction) networks

4 Spatially structured networks

5 Spatially structured networks Space

6 Spatially structured networks Space

7 Spatially structured networks

8 Today s outline CONSEQUENCES OF DISPERSAL IN ECOSYSTEMS 1. Ecosystem stability 2. Patch dynamics on networks

9 ECOSYSTEM STABILITY

10 Ecosystem stability The question: Does species diversity and the variability of species interactions stabilize ecosystems? time

11 Diversity stabilizes ecosystems Dominant view until the 70 s a larger number of paths through each species is necessary to reduce the effects of overpopulation of one species MacArthur 1955

12 or does it?

13 Mathematical stability

14 Formalization Assume a feasible equilibrium X * of dx dt GX ( ) where X =abundance vector for all the S species G(X) = dynamics of the system (competition, predation, mutualism )

15 Linearization Assume a feasible equilibrium X * Linearize the dynamics around the equilibrium d * ( X X ) dt * * G( X ).( X X ) Jacobian matrix J Stable equilibrium (locally) All eigenvalues of J have negative real parts

16 The Jacobian matrix of a random ecosystem Assume that the system is random and properly scaled, i.e. the Jacobian looks like J m 2 B( c) Ν 0, m 2 B( c) Ν 0, m where B (c) = Bernoulli distribution N (0,σ²) = Gaussian distribution

17 Empirical spectral distribution 10 5 Eigenvalues of J in the complex plane = empirical spectral distribution (ESD) σ c( S 1) m

18 Empirical spectral distribution 10 Eigenvalues of J in the complex plane 5 = empirical spectral distribution (ESD) For large S, the system is stable if and only if 1 0 c S m 5 May σ c( S 1) m

19 Sequels to May s paper Three classic lines of investigation after May s paradox : 1. Rephrasing the stability criterion 2. Jointly studying feasibility & stability 3. Extending May s approach to more detailed cases

20 A recent example Following line (3) : dissected May s arguments by interaction type o predation (-/+) o mutualism (+/+) o competition (-/-) Allesina & Tang 2012

21 A recent example Main result from Allesina & Tang Empirical spectral distribution (ESD) changes by interaction type Allesina & Tang 2012

22 Our own sequel Dominique Gravel Mathew Leibold S species S species n patches dispersal among patches

23 Principle of the analysis 10 What we need radius centre

24 Principle of the analysis 10 What we need Random part of the Jacobian = radius centre = Deterministic part of the Jacobian

25 Principle of the analysis Support of the ESD of X = A + B (size = n) with A random, mean = 0, sd = σ B deterministic, ESD = µ B = z s that verify B/ n z u ( du) 2 1 Tao et al. 2010

26 Spatial structure in the Jacobian ( m d) I A 1 d ( n1) I ( 1) d n I d ( n1) I ( m d) I Ak d ( n1) I d ( n1) I ( 1) d n I ( m d) I An Among patches Within patches

27 c S 1 m Deterministic part ( md) I d ( n1) I ( 1) d n I d ( n1) I ( md) I d ( n1) I d ( n1) I ( 1) d n I ( md) I Among patches Within patches

28 c S 1 m Deterministic part Eigenvalues of the deterministic part of the Jacobian change from (-m,-m,,-m) to S times (-m,-m,,-m,-m-dn/(n-1),,-m-dn/(n-1)) S times (n-1)s times The deterministic effect of d is to push a fraction of the ESD to the left of the complex plane

29 cs 1 m Random part A A k A n Among patches Within patches

30 cs 1 m Random part Connectance goes from c to c/n System size goes from S to ns Variance?

31 Computing the variance: large d Depends on the correlation ρ among A s V V A / ne n n / 1 n 1 e c S 1 / n m e

32 Computing the variance: small d No change in variance, but change in ESD centre When d is small, a different approximation: c S 1 m d approximately valid whatever the value of ρ

33 What it looks like 10 5 d = 0 n = 20 S = 100 m = 2 ρ = σ c( S 1) m

34 What it looks like 10 5 d = 1 n = 20 S = 100 m = 2 ρ = σ c( S 1) m d

35 What it looks like 10 σ c( S 1)( n 1) / n m nd / ( n 1) 5 d = 8 n = 20 S = 100 m = 2 ρ = m σ c( S 1) / ne

36 Extensions Works with non-complete (but regular) spatial graphs Works with species-specific dispersal rates Simulations (with feasibility constraints) show the same results One thing you can t study from J alone is the feedback between d and the homogeneity of A

37 Feedback between d and A

38 Feedback between d and A Predicted decline due to heterogeneous patches High dispersal = synchronizing dynamics = more homogeneous A High dispersal => larger biomass => stronger feedback

39 Take-home messages 1. Stabilization requires heterogeneity of feedbacks among patches and dispersal 2. Dispersal can homogenize feedbacks 3. Optimal stability is achieved at intermediate dispersal rates

40 Perspectives dispersal when not diffusive o density-dependent dispersal Dispersal Density of conspecifics, preys, predators

41 Perspectives dispersal when not diffusive o density-dependent dispersal putting together dispersal at different scales (non trans-specific definition of patches)

42 Perspectives dispersal when not diffusive o density-dependent dispersal putting together dispersal at different scales (non trans-specific definition of patches) explicit link between feasibility and stability random network of patches (not regular)

43 PATCH DYNAMICS ON NETWORKS

44 Patch dynamics on networks The question: Does metapopulation network structure affect species occupancy?

45 The initial problem: a model for seed exchange networks CESAB NetSeed Doyle McKey Francisco Laso

46 A model for seed exchange networks Simplest model to capture specificities of seed exchange networks Dynamic processes: o extinction of variety in farmer s fields, o diffusion of variety through exchange between farmers, o background diffusion (getting the variety from NGOs, markets, nature, etc.) On a directed network

47 Parallel with metapopulations Seed exchange = colonisation Extinction of variety = extinction of population Background diffusion = external source of propagules

48 Parallel with epidemics Seed exchange = infection by contact Extinction of variety = patient recovery Background diffusion = self-infection

49 Modeling approach Three processes: extinction (e), background diffusion (d), diffusion through exchange (c) t t + extinction t + 1 empty 1 empty ψ empty e 1 ψ full 1 e full 1 full 1d 1 c # occupiedneighbours

50 An approximation for occupancy Method: the N-intertwined model What does this mean? = take expectation after extinction & take expectation after colonization (i.e. not computing full proba of having n occupied nodes at time t+1 as a function of having k infected nodes at time t)

51 Deriving the approximation First, consider a complete graph of size N (i.e. all nodes are linked) t p 1 (1 t e) p t 1 1 e p 1 1 d 1 c N(1 ep ) t probability that an occupied patch stays occupied probability that a patch ends up empty after extinction episode (already empty or goes extinct) probability that the empty patch is colonized, either through c or d

52 to A network = a matrix out of N1 N2 N3 N4 N N N N Network representation Adjacency matrix

53 From complete graphs to a more general model N in a complete graph # of neighbours average degree in general? But the average degree experienced by a particle diffusing on a network the expectation of the degree among nodes!

54 The average degree

55 The average degree

56 The average degree

57 The average degree

58 The average degree average degree proba in network 4/7 1/7 1/7 1/7 13/7 1.86

59 The average degree average degree proba in network 4/7 1/7 1/7 1/7 13/ proba among edges 4/13 2/13 3/13 4/13 33/

60 The average degree This demonstration = what happens from one node to the next (i.e. the relevant quantity is the expected squared degree) Taking paths of infinite length within the network, what matters is the dominant eigenvalue / spectral radius of the adjacency matrix

61 Deriving the approximation Now consider that N can be replaced by the spectral radius ρ p t1 (1 e) pt 1 1 e pt 1 1 d 1 c N(1 ep ) t p t1 (1 e) pt 1 1 e pt 1 1 d 1 c (1 ep ) t In principle, occupancy could be deduced from the knowledge of c, d, e and ρ

62 What does this approximation say? c = 0.1 c = 0.2 c = 0.3 q = (1-e)p e/(1-e)

63 What does this approximation say? c = 0.1 c = 0.2 c = 0.3 q = (1-e)p e/(1-e)

64 What does this approximation say? q = (1-e)p Inflexion point at e 1 e c = 0.1 c = 0.2 c = 0.3 = two regimes (driven by c vs. driven by d) log 1 c e/(1-e)

65 Does the approximation work? Maxime Dubart s MSc 1 internship Topologies: regular, E-R, exponential, scale-free Set d and c at given values Vary reciprocity at a given ρ Patterns of q as a function of e/(1-e)

66 General fit r = 0.5 c = 0.1 d = ρ = 16.25

67 General fit r = 0.5 c = 0.1 d = ρ = The approximation overestimates q However, comparing topologies at fixed ρ yields a consistent ranking ( at fixed average degree)

68 Effect of reciprocity r = 0.5 c = 0.1 d = ρ = r = 1 c = 0.1 d = ρ = 16.25

69 Effect of reciprocity r = 0.5 c = 0.1 d = ρ = r = 1 c = 0.1 d = ρ = When controlling for ρ, independent of reciprocity

70 Take-home messages 1. Two phases for occupancy: c and d-driven 2. Networks with same ρ but different reciprocities yield similar occupancies 3. Networks with same ρ but different average degrees yield similar occupancies 4. At given ρ, consistent ranking of topologies (Erdős-Rényi > Exponential > Scale-free)

71 Immediate perspective Evolution of colonization capacity under the competition-colonization trade-off Erdős-Rényi Scale-free

72 Co-authors: Thank you! M. Dubart, D. Gravel, F. Laso, M. Leibold, D. McKey Working groups on networks NIMBioS Food Web MIRES & DyBRES CESAB NetSeed