1. Accounting for food web
information in island
biogeography
Dominique Gravel, François Massol,
Elsa Canard, David Mouillot, Nicolas Mouquet

2. Introduction Outline
1. Introduction
2. The model
3. Analysis
4. Fit to existing data
5. Conclusions & perspectives
Journée Bioinformatique et Biodiversité 2011 – Jun 29th

3. Introduction The question of diversity
http://mrbarlow.wordpress.com/
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4. Introduction The question of diversity
Dispersal
Interactions
Diversity
Environment
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5. Introduction Island biogeography
Island Mainland
MacArthur & Wilson 1967
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6. Introduction Island biogeography
c
e
†
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7. Introduction Island biogeography
dp c
= c (1 − p ) − ep
dt
e
†
Journée Bioinformatique et Biodiversité 2011 – Jun 29th

8. Introduction Island biogeography
dp
= c (1 − p ) − ep
dt
c/e
p =
*
1+ c / e
islands closer to the mainland are larger islands are less prone to
easier to colonize species extinctions
Journée Bioinformatique et Biodiversité 2011 – Jun 29th

9. Introduction Island biogeography
Journée Bioinformatique et Biodiversité 2011 – Jun 29th

10. Introduction The food web challenge
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11. Introduction The food web challenge
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12. Introduction The food web challenge
Order of colonization events
Chain extinctions
† †
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13. The model
Structuring assumptions:
1. a species cannot colonize unless one prey species is already
present
Model
2. a species that loses its last prey species gets extinct
Journée Bioinformatique et Biodiversité 2011 – Jun 29th

14. The model
Xi random variable for the occurrence of species i (= 0 or 1)
pi = E [ X i ]
Model
Yi indicator for the occurrence of at least one prey of species i
qi = E [Yi | X i = 0]
εi rate at which species i loses its last prey species
dpi
= cqi (1 − pi ) − ( e + εi ) pi
dt
Journée Bioinformatique et Biodiversité 2011 – Jun 29th

15. The model
our model
dpi
Model
= cqi (1 − pi ) − ( e + εi ) pi
dt
MacArthur & Wilson’s
dp
= c (1 − p ) − ep
dt
Journée Bioinformatique et Biodiversité 2011 – Jun 29th

16. Analysis
Structuring assumptions:
1. a species cannot colonize unless one prey species is already
present
2. a species that loses its last prey species gets extinct
Analysis
Approximation for analysis:
1. consumers are structured by their diet breadth (g)
2. preys of the same predator occur independently
3. prey presence is independent of predator presence
Journée Bioinformatique et Biodiversité 2011 – Jun 29th

17. Analysis
species i
qi pi εi
Analysis
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18. Analysis
species i
qi pi εi
before approximations
cqi / ( e + εi )
Analysis
pi =
1 + cqi / ( e + εi )
⎡ ⎤ ⎡ ⎤
qi = 1 − E ⎢ ∏ (1 − X j ) | X i = 0 ⎥ εi = E ⎢ ∑ ( e + ε j ) X j ∏ (1 − X k ) | X i = 1⎥
⎢ j∈G ⎥
⎣ j∈Gi ⎦ ⎢
⎣
i k∈Gi
k≠ j ⎥
⎦
Gi set of prey species for species i
Journée Bioinformatique et Biodiversité 2011 – Jun 29th

19. Analysis
species i
qi pi εi
after approximations
cqi / ( e + εi )
Analysis
pi =
1 + cqi / ( e + εi )
Gi log (1− p• ) ⎛ ε• p• ⎞ Gi log (1− p• )
qi ≈ 1 − e P εi ≈ ⎜ Gi
⎜ 1 − p•
⎟e
⎟
P
⎝ P ⎠
x• P
average of x among regional species
Gi # of prey species for species i
Journée Bioinformatique et Biodiversité 2011 – Jun 29th

20. Analysis
diet breadth g
qg pg εg
after approximations
Analysis
⎛1 − e g log(1− pg ) P ⎞
(c / e) ⎜ ⎟
pg ≈ ⎝ ⎠
⎛1 − e g log(1− pg ) P ⎞ ⎛ 1 + ge g log(1− pg ) P ⎞
1 + (c / e) ⎜ ⎟⎜ ⎟
⎝ ⎠⎝ ⎠
x• P
average of x among regional species
Journée Bioinformatique et Biodiversité 2011 – Jun 29th

21. p
Analysis
1.0
pB
0.8
0.6 p• ± σ pg g = 1.5
Analysis
0.4
σ g = 0.05
2
p1 PB / P = 0.5
0.2
0 5 10 15 20
c /e
Journée Bioinformatique et Biodiversité 2011 – Jun 29th

22. p
Analysis
0.6
pB
0.5
0.4
g = 1.5
Analysis
0.3
σ g = 0.05
2
0.2 σ
p• ±PB p/g P = 0.5
0.1
p1
0.0 0.5 1.0 1.5 2.0
c /e
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23. Empirical support?
• dataset: Havens (1992)
• 50 Adirondack lakes
• 210 species (13-75)
• 107 primary producers
• 103 consumers
• 2020 links (17-577)
Data fitting
• low connectance (0.09)
Journée Bioinformatique et Biodiversité 2011 – Jun 29th

24. Empirical support
Estimation of c/e for
each lake by maximum
likelihood
Model log likelihood
Classic TIB (Intercept) - 2428.2
Data fitting
Trophic – TIB (Analytical) - 2416.8
Trophic – TIB (Simulations) - 2392.4
Journée Bioinformatique et Biodiversité 2011 – Jun 29th

25. Empirical support
Estimation of c/e for
each lake by maximum
likelihood
Model log likelihood
Classic TIB (Intercept) - 2428.2 no trophic structure
Data fitting
Trophic – TIB (Analytical) - 2416.8 with diet breadth
Trophic – TIB (Simulations) - 2392.4 complete structure
Journée Bioinformatique et Biodiversité 2011 – Jun 29th

26. 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
Data fitting
• 130 consumers
• 13068 feeding links (32-331 per island)
• high connectance (0.21)
Journée Bioinformatique et Biodiversité 2011 – Jun 29th

27. 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
Data fitting
poorer fit
(high connectance, partial food web data)
Journée Bioinformatique et Biodiversité 2011 – Jun 29th

28. 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)
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29. Complexity-diversity?
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30. Thank you!
Dataset: J. Dunne
Comments on paper
C. Albert, D. Alonso, J. Chase, J. E. Cohen, S. M. Gray, R. D. Holt,
O. Kaltz, M. Loreau
The End Journée Bioinformatique et Biodiversité 2011 – Jun 29th