1. Critical slowing down as an indicator of transitions in two-species models
Ryan Chisholm
Smithsonian Tropical Research Institute
Workshop on Critical Transitions in Complex Systems
21 March 2012 Imperial College London

2. Acknowledgements
Elise Filotas, Centre for Forest Research at the University of Quebec in Montreal
Simon Levin, Princeton University, Department of Ecology and Evolutionary Biology
Helene Muller-Landau, Smithsonian Tropical Research Institute
Santa Fe Institute, Complex Systems Summer School 2007: NSF Grant No

3. Question
When is critical slowing down likely to be a useful leading indicator of a critical transition in ecological models?

4. Outline Smithsonian Tropical Research Institute
Background: critical slowing down
Competition model
Predator-prey model
Grasslands model
Future work

5. Outline Smithsonian Tropical Research Institute
Background: critical slowing down
Competition model
Predator-prey model
Grasslands model
Future work

6. Smithsonian Tropical Research Institute
“…dedicated to understanding biological diversity”
What determines patterns of diversity?
What factors regulate ecosystem function?
How will tropical forests respond to climate change and other anthropogenic disturbances?

7. Smithsonian Tropical Research Institute
Panama

8. Smithsonian Tropical Research Institute
50 ha plot

9. Smithsonian Tropical Research Institute
1500 ha
2551 mm yr-1 rainfall
381 bird species
102 mammal species (nearly half are bats)
~100 species of amphibians and reptiles
1316 plant species
Green iguana
(Iguana iguana)
Keel-billed Toucan (Ramphastos sulfuratus)
Pentagonia macrophylla
Jaguar (Panthera onca)
Photo: Christian Ziegler

10. Smithsonian Tropical Research Institute
Photo: Marcos Guerra, STRI
sciencedaily.com
Photo: Leonor Alvarez

11. Center for Tropical Forest Science

12. Forest resilience
Staver et al Science

13. Chisholm, Condit, et al. in prep

14. Outline Smithsonian Tropical Research Institute
Background: critical slowing down
Competition model
Predator-prey model
Grasslands model
Future work

15. Transitions in complex systems
Eutrophication of shallow lakes
Sahara desertification
Climate change
Shifts in public opinion
Forest-savannah transitions
Scheffer et al Nature, Scheffer 2009 Critical Transitions in Nature and Society

16. Critical transitions
May 1977 Nature

17. Detecting impending transitions
Decreasing return rate
Rising variance
Rising autocorrelation
=> All arise from critical slowing down
Carpenter & Brock 2006 Ecol. Lett., van Nes & Scheffer 2007 Am. Nat.,
Scheffer et al Nature

18. van Nes & Scheffer 2007 Am. Nat.
Critical slowing down
Recovery rate: return rate after disturbance to the equilibrium
Critical slowing down: dominant eigenvalue tends to zero; recovery rate decreases as transition approaches
van Nes & Scheffer 2007 Am. Nat.

19. van Nes & Scheffer 2007 Am. Nat.
Critical slowing down
van Nes & Scheffer 2007 Am. Nat.

20. van Nes & Scheffer 2007 Am. Nat.
Critical slowing down
van Nes & Scheffer 2007 Am. Nat.

21. Question
When is critical slowing down likely to be a useful leading indicator of a critical transition in ecological models?  What is the length/duration of the warning period?

22. Outline Smithsonian Tropical Research Institute
Background: critical slowing down
Competition model
Predator-prey model
Grasslands model
Future work

23. Competition model Ni = abundance of species i
Ki = carrying capacity of species i
ri = intrinsic rate of increase of species i
αij = competitive impact of species j on species i
Equilibria:
* Two saddle-node bifurcations
Lotka 1925, 1956 Elements of Physical Biology; Chisholm & Filotas 2009 J. Theor. Biol.

24. Chisholm & Filotas 2009 J. Theor. Biol.
Competition model
Case 1: Interspecific competition greater than intraspecific competition
Stable
Unstable
Chisholm & Filotas 2009 J. Theor. Biol.

25. Question
When is critical slowing down likely to be a useful leading indicator of a critical transition in ecological models?  What is the length/duration of the warning period?

26. Chisholm & Filotas 2009 J. Theor. Biol.
Competition model
Ni = abundance of species i
Ki = abundance of species i
ri = intrinsic rate of increase of species i
αij = competitive impact of species j on species i
Recovery rate: When species 1 dominates, recovery rate begins to decline at:
Chisholm & Filotas 2009 J. Theor. Biol.

27. Chisholm & Filotas 2009 J. Theor. Biol.
Competition model
Chisholm & Filotas 2009 J. Theor. Biol.

28. Chisholm & Filotas 2009 J. Theor. Biol.
Competition model
Ni = abundance of species i
Ki = abundance of species i
ri = intrinsic rate of increase of species i
αij = competitive impact of species j on species i
Recovery rate begins to decline at: More warning of transition if the dynamics of the rare species are slow relative to those of the dominant species
Chisholm & Filotas 2009 J. Theor. Biol.

29. Chisholm & Filotas 2009 J. Theor. Biol.
Competition model
Case 2: Interspecific competition less than intraspecific competition
Stable
Unstable
Chisholm & Filotas 2009 J. Theor. Biol.

30. Chisholm & Filotas 2009 J. Theor. Biol.
Competition model
Case 2: Interspecific competition less than intraspecific competition
More warning of transition if the dynamics of the rare species are slow relative to those of the dominant species
Chisholm & Filotas 2009 J. Theor. Biol.

31. Outline Smithsonian Tropical Research Institute
Background: critical slowing down
Competition model
Predator-prey model
Grasslands model
Future work

32. Predator-prey model V = prey abundance P = predator abundance
Rosenzweig 1971 Science

33. Rosenzweig 1971 Science, Chisholm & Filotas 2009 J. Theor. Biol.
Predator-prey model
h(V)
V = prey abundance
P = predator abundance
r = intrinsic rate of increase of prey
k = predation rate
J = equilibrium prey population size
A = predator-prey conversion efficiency
K = carrying capacity of prey
f(V) = effects of intra-specific competition among prey
f(V) > 0; f ’(V) < 0; f(K) = 0; df/dK > 0
h(V) = per-capita rate at which predators kill prey
h(V) > 0; h’(V) > 0; h’’(V) < 0; h(0) = 0
f(V) V
* Transcritical bifurcation and Hopf bifurcation
Rosenzweig 1971 Science, Chisholm & Filotas 2009 J. Theor. Biol.

34. Rosenzweig 1971 Science, Chisholm & Filotas 2009 J. Theor. Biol.
Predator-prey model
Equilibria:
Unstable
Stable for K ≤ J
V = prey abundance
P = predator abundance
r = intrinsic rate of increase of prey
k = predation rate
J = equilibrium prey population size
A = predator-prey conversion efficiency
K = carrying capacity of prey
f(V) = effects of intra-specific competition among prey
f(V) > 0; f ’(V) < 0; f(K) = 0; df/dK > 0
h(V) = per-capita rate at which predators kill prey
h(V) > 0; h’(V) > 0; h’’(V) < 0; h(0) = 0
Exists for K ≥ J
Stable for J ≤ K ≤ Kcrit
* Transcritical bifurcation and Hopf bifurcation
Rosenzweig 1971 Science, Chisholm & Filotas 2009 J. Theor. Biol.

35. Rosenzweig 1971 Science, Chisholm & Filotas 2009 J. Theor. Biol.
Predator-prey model
Predator isocline
V = prey abundance
P = predator abundance
r = intrinsic rate of increase of prey
k = predation rate
J = equilibrium prey population size
A = predator-prey conversion efficiency
f(V) = effects of intra-specific competition among prey
f(V) > 0; f ’(V) < 0; f(K) = 0; df/dK > 0
h(V) = per-capita rate at which predators kill prey
h(V) > 0; h’(V) > 0; h’’(V) < 0; h(0) = 0
Prey isoclines
Rosenzweig 1971 Science, Chisholm & Filotas 2009 J. Theor. Biol.

36. Rosenzweig 1971 Science, Chisholm & Filotas 2009 J. Theor. Biol.
Predator-prey model
Unstable equilibrium
V = prey abundance
P = predator abundance
r = intrinsic rate of increase of prey
k = predation rate
J = equilibrium prey population size
A = predator-prey conversion efficiency
f(V) = effects of intra-specific competition among prey
f(V) > 0; f ’(V) < 0; f(K) = 0; df/dK > 0
h(V) = per-capita rate at which predators kill prey
h(V) > 0; h’(V) > 0; h’’(V) < 0; h(0) = 0
Stable equilibrium
Rosenzweig 1971 Science, Chisholm & Filotas 2009 J. Theor. Biol.

37. Scheffer 1998 The Ecology of Shallow Lakes
Predator-prey model
Scheffer 1998 The Ecology of Shallow Lakes

38. Predator-prey model Hopf bifurcation occurs when K = Kcrit :
Critical slowing down begins when K = Kr :

39. Chisholm & Filotas 2009 J. Theor. Biol.
Predator-prey model
Chisholm & Filotas 2009 J. Theor. Biol.

40. Chisholm & Filotas 2009 J. Theor. Biol.
Predator-prey model
Chisholm & Filotas 2009 J. Theor. Biol.

41. Chisholm & Filotas 2009 J. Theor. Biol.
Predator-prey model
Kr and Kcrit converge as:
More warning of transition when:
Predator-prey conversion efficiency (A) is high
Predation rate (k) is high
Prey growth rate (r) is low
Prey controlled by predators rather than intrinsic density dependence
Increases tendency for oscillations
Larger K makes oscillations larger and hence rates of return slower
Note that Lotka-Volterra model has no intrinsic density dependence on the prey and therefore exhibits completely undamped oscillations.
Chisholm & Filotas 2009 J. Theor. Biol.

42. Chisholm & Filotas 2009 J. Theor. Biol.
Predator-prey model
Chisholm & Filotas 2009 J. Theor. Biol.

43. van Nes & Scheffer 2007 Am. Nat.
Multi-species models
van Nes & Scheffer 2007 Am. Nat.

44. Chisholm & Filotas 2009 J. Theor. Biol.
Multi-species models
Expect that multi-species models will exhibit longer warning periods of transitions induced by changes in resource abundance when:
Dynamics of rare species are slow relative to those of the dominant species
Prey species are controlled by predation rather than intrinsic density dependence
Chisholm & Filotas 2009 J. Theor. Biol.

45. Outline Smithsonian Tropical Research Institute
Background: critical slowing down
Competition model
Predator-prey model
Grasslands model
Future work

46. Practical utility of critical slowing down
Chisholm & Filotas 2009 J. Theor. Biol.
“…even if an increase in variance or AR1 is detected, it provides no indication of how close to a regime shift the ecosystem is…”
Biggs et al PNAS

47. Western Basalt Plains Grasslands
What we really want to know is how close we are to the transition
If we increase K and find that the return rate decreases, we know that we’ve passed the peak and that we’re approaching the transition. But how close are we? No idea.
Presumably management goal is to increase K as much as possible without risking a transition.
Could argue that we want a shorter warning period, so that we can get closer to the transition without actually crossing it. But then risk of accidentally crossing transition.

48. Western Basalt Plains Grasslands

49. Western Basalt Plains Grasslands
Williams et al J. Ecol.; Williams et al Ecology

50. Grasslands invasion model
Agricultural fertiliser run-off
Native
grass
biomass
Sugar addition
Nutrient input rate

51. Grasslands invasion model
A = plant-available N pool
Bi = biomass of species i
ωi = N-use efficiency of species i
νi = N-use efficiency of species i
μi = N-use efficiency of species i
αij = light competition coefficients
I = abiotic N-input flux
K = soil leaching rate of plant-available N
δ = proportion of N in litterfall lost from the system
Parameterized so that species 2 (invader) has a higher uptake rate and higher turnover rate.
Chisholm & Levin in prep.; Menge et al PNAS

52. Grasslands invasion modelB2
Relatively safe, but higher control costs. B1
Nutrient input
What we really want to know is how close we are to the transition
If we increase K and find that the return rate decreases, we know that we’ve passed the peak and that we’re approaching the transition. But how close are we? No idea.
Presumably management goal is to increase K as much as possible without risking a transition.
Could argue that we want a shorter warning period, so that we can get closer to the transition without actually crossing it. But then risk of accidentally crossing transition.
Riskier, but lower control costs.

53. Conclusions & Future work
Critical slowing down provides an earlier indicator of transitions in two-species models where:
Dynamics of rare species are slow relative to those of the dominant species
Prey species are controlled by predation rather than intrinsic density dependence
But utility of early/late indicators depends on socio-economic considerations