Metapopulation-based approaches to spatial food webs

François Massol

Patch dynamics models in ecology Two general classes Metapopulation

Mainland/island

Patch dynamics models in ecology Two general classes Metapopulation

dp  cp 1  p   ep dt

Mainland/island

dp  c 1  p   ep dt

Patch dynamics models in ecology • To answer different questions: – species occupancy on a given landscape – species diversity (TIB) – conditions for deterministic species coexistence (CC trade-off) – effects of fragmentation and/or habitat destruction on species occupancy –…

The food web challenge

The food web challenge

The food web challenge Order of colonization events Chain extinctions

†

†

Two models

Gravel et al. 2011: The model Structuring assumptions: 1.

2.

a species cannot colonize unless one prey species is already present a species that loses its last prey species gets extinct

Gravel et al. 2011: The model Xi Yi i

random variable for the occurrence of species i (= 0 or 1)

pi  E  X i 

indicator for the occurrence of at least one prey of species i

qi  E Yi | X i  0

rate at which species i loses its last prey species

dpi  cqi 1  pi    e  dt

i

 pi

Gravel et al. 2011: The model our model

dpi  cqi 1  pi    e  dt

i

 pi

MacArthur & Wilson’s

dp  c 1  p   ep dt

Gravel et al. 2011: Analysis Structuring assumptions: 1.

2.

a species cannot colonize unless one prey species is already present a species that loses its last prey species gets extinct

Approximation for analysis: 1. 2. 3.

consumers are structured by their diet breadth (g) preys of the same predator occur independently prey presence is independent of predator presence

Gravel et al. 2011: Analysis species i

qi

pi

i

Gravel et al. 2011: Analysis species i

qi

pi

i

before approximations

pi 

cqi /  e 

i

1  cqi /  e 

  qi  1  E  1  X j  | X i  0  jGi 

i

 i



  E   e   jG  i

Gi

 j X j  1  X k  | X i  1 kGi  k j

set of prey species for species i

Gravel et al. 2011: Analysis species i

qi

pi

i

after approximations

pi 

qi  1  e

Gi log1 p• 

cqi /  e 

i

1  cqi /  e 

P

i

 i



 • p•   Gi  1  p•  x•

Gi

P

 Gi  e P 

log 1 p• 

P

average of x among regional species # of prey species for species i

Gravel et al. 2011: Analysis diet breadth g

qg

pg

g

after approximations g log 1 pg    P  c / e  1  e    pg  g log 1 pg    g log 1 pg    P P 1   c / e  1  e 1  ge    

x•

P

average of x among regional species

Gravel et al. 2011: Analysis p 1.0

pB 0.8

p•   pg

0.6

g  1.5

 g2  0.05

0.4

p1

PB / P  0.5

0.2

0

5

10

15

20

c/e

Gravel et al. 2011: Analysis p 0.6

pB

0.5 0.4

g  1.5

0.3

 g2  0.05 p• PB p/g P  0.5

0.2 0.1

p1 0.0

0.5

1.0

1.5

2.0

c/e

Gravel et al. 2011: Empirical support? • • • • • • •

dataset: Havens (1992) 50 Adirondack lakes 210 species (13-75) 107 primary producers 103 consumers 2020 links (17-577) low connectance (0.09)

Gravel et al. 2011: Empirical support Estimation of c/e for each lake by maximum likelihood

Model

log likelihood

Classic TIB (Intercept)

- 2428.2

Trophic – TIB (Analytical)

- 2416.8

Trophic – TIB (Simulations)

- 2392.4

Gravel et al. 2011: Empirical support Estimation of c/e for each lake by maximum likelihood

Model

log likelihood

Classic TIB (Intercept)

- 2428.2

no trophic structure

Trophic – TIB (Analytical)

- 2416.8

with diet breadth

Trophic – TIB (Simulations)

- 2392.4

complete structure

Gravel et al. 2011: Empirical support • • • • •

second dataset: Piechnik et al. (2008) 6 islands (Florida keys) sampled  before  total  defaunation  in  the  60’s 250 species (arthropods only, 15-38 per island) no primary producer, but 120 taxa (herbivores & detritivores) are not constrained • 130 consumers • 13068 feeding links (32-331 per island) • high connectance (0.21)

Gravel et al. 2011: Empirical support Second data set (Piechnik et al. 2008) Model

log likelihood

Classic TIB (Intercept)

- 259.3

no trophic structure

Trophic – TIB (Analytical)

- 259.9

with diet breadth

Trophic – TIB (Simulations)

- 260.0

complete structure

poorer fit (high connectance, partial food web data)

Gravel et al. 2011: Conclusions & Perspectives Conclusions: – richer/more precise predictions than TIB with no additional parameter – captures phenomena occurring in low connectance webs – integrates interactions in dispersal-based model

Perspectives: – application to other biological networks in space – refining approximations – testing against other models (e.g. group-dependent rates)

Calcagno et al. 2011 •each trophic level only occupies a fraction 0) •a predator population cannot occupy a patch without preys

Calcagno et al. 2011 •each trophic level only occupies a fraction 0) •a predator population cannot occupy a patch without preys

Calcagno et al. 2011 •each trophic level only occupies a fraction 0) •a predator population cannot occupy a patch without preys

Calcagno et al. 2011 •each trophic level only occupies a fraction 0) •a predator population cannot occupy a patch without preys

Calcagno et al. 2011 •each trophic level only occupies a fraction 0) •a predator population cannot occupy a patch without preys

Calcagno et al. 2011 •each trophic level only occupies a fraction 0) •a predator population cannot occupy a patch without preys

Calcagno et al. 2011: Questions • Is spatial structure a strong constraint on food chain length? – if yes, in which conditions?

• How does the 'patchy prey' hypothesis interact with other processes? – bottom-up control of extinction rates – top-down controls – foraging

Calcagno et al. 2011: model colonization - includes TD and foraging effects





dpi  ci qi pi 1  ei 1 qi 1    eii* pi  ci 1 qi 1 pi dt

intrinsic extinction

- naturally includes BU regulation effects - also includes TD effects

perturbation

Calcagno et al. 2011: some results Maximum food chain length

Relative specific extinction rate, e/c 1-2

2-3

3-4 4-5 …

Relative perturbation rate, µ/c

Calcagno et al. 2011: some results

Metapopulation-based approaches for networks • Calcagno, V., Massol, F., Mouquet, N., Jarne, P., David, P., 2011. Constraints on food chain length arising from regional metacommunity dynamics. Proc R Soc Lond B Biol Sci 278, 3042-3049. • Gravel, D., Massol, F., Canard, E., Mouillot, D., Mouquet, N., 2011. Trophic theory of island biogeography. Ecol Lett 14, 1010-1016. • Pillai, P., Loreau, M., Gonzalez, A., 2009. A patch-dynamic framework for food web metacommunities. Theor Ecol, 1-15. • Pillai, P., Gonzalez, A., Loreau, M., 2011. Metacommunity theory explains the emergence of food web complexity. Proc Natl Acad Sci U S A 108, 19293-19298.

Thank you! Datasets: J. Dunne, D. Piechnik, M. Zalewski Comments & discussions C. Albert, D. Alonso, J. Chase, J. E. Cohen, C. de Mazancourt, S. M. Gray, R. D. Holt, O. Kaltz, M. Leibold, M. Loreau