Critical slowing down as an indicator of transitions in two-species models

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 2012Imperial 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. 0200500

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 (Ramphastossulfuratus) Pentagoniamacrophylla Jaguar (Pantheraonca) 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 Staveret al. 2011 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 Schefferet al. 2009 Nature, Scheffer 2009 Critical Transitions in Nature and Society Eutrophication of shallow lakes Sahara desertification Climate change Shifts in public opinion Forest-savannah transitions

16. Critical transitions May 1977 Nature

17. Detecting impending transitions Carpenter & Brock 2006 Ecol. Lett., van Nes & Scheffer 2007 Am. Nat., Schefferet al. 2009 Nature Decreasing return rate Rising variance Rising autocorrelation => All arise from critical slowing down

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

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

20. 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: Lotka 1925, 1956 Elements of Physical Biology; Chisholm & Filotas 2009 J. Theor. Biol.

24. Competition model Case 1: Interspecific competition greater than intraspecific competition Stable Stable Unstable 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. 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 Chisholm & Filotas 2009 J. Theor. Biol. Recovery rate: When species 1 dominates, recovery rate begins to decline at:

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

28. 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 Chisholm & Filotas 2009 J. Theor. Biol. 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

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

30. 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. 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 Rosenzweig 1971 Science,Chisholm & Filotas 2009 J. Theor. Biol.

34. 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 Rosenzweig 1971 Science,Chisholm & Filotas 2009 J. Theor. Biol.

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

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

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

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

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

44. Multi-species models Chisholm & Filotas 2009 J. Theor. Biol. 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

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. 2008 PNAS

47. Western Basalt Plains Grasslands

48. Western Basalt Plains Grasslands

49. Western Basalt Plains Grasslands Williams et al. 2005 J. Ecol.; Williams et al. 2006 Ecology

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