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Critical slowing down as an indicator of transitions in two-species models Ryan Chisholm Smithsonian Tropical Research Institute Workshop on Critical Transitions.

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Presentation on theme: "Critical slowing down as an indicator of transitions in two-species models Ryan Chisholm Smithsonian Tropical Research Institute Workshop on Critical Transitions."— Presentation transcript:

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 Photo: Christian Ziegler Green iguana (Iguana iguana) Keel-billed Toucan (Ramphastos sulfuratus) Pentagonia macrophylla 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 Jaguar (Panthera onca)

10 Smithsonian Tropical Research Institute sciencedaily.com Photo: Marcos Guerra, STRI 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 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 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 N i = abundance of species i K i = carrying capacity of species i r i = intrinsic rate of increase of species i α ij = 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 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 N i = abundance of species i K i = abundance of species i r i = intrinsic rate of increase of species i α ij = 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 Competition model Chisholm & Filotas 2009 J. Theor. Biol.

28 Competition model N i = abundance of species i K i = abundance of species i r i = intrinsic rate of increase of species i α ij = 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 Competition model Case 2: Interspecific competition less than intraspecific competition 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 Rosenzweig 1971 Science V = prey abundance P = predator abundance

33 Predator-prey model 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 h ( V ) = per-capita rate at which predators kill prey h(V) > 0; h ’ (V) > 0; h ’’ (V) < 0; h(0) = 0 Rosenzweig 1971 Science, Chisholm & Filotas 2009 J. Theor. Biol. f(V)f(V) h(V)h(V) V

34 Predator-prey model Rosenzweig 1971 Science, Chisholm & Filotas 2009 J. Theor. Biol. Equilibria: Unstable Stable for K ≤ J Exists for K ≥ J Stable for J ≤ K ≤ K crit 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 h ( V ) = per-capita rate at which predators kill prey h(V) > 0; h ’ (V) > 0; h ’’ (V) < 0; h(0) = 0

35 Predator-prey model Predator isocline Prey isoclines 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 h ( V ) = per-capita rate at which predators kill prey h(V) > 0; h ’ (V) > 0; h ’’ (V) < 0; h(0) = 0 Rosenzweig 1971 Science, Chisholm & Filotas 2009 J. Theor. Biol.

36 Predator-prey model Unstable equilibrium Stable 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 h ( V ) = per-capita rate at which predators kill prey h(V) > 0; h ’ (V) > 0; h ’’ (V) < 0; h(0) = 0 Rosenzweig 1971 Science, Chisholm & Filotas 2009 J. Theor. Biol.

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

38 Hopf bifurcation occurs when K = K crit : Critical slowing down begins when K = K r : Predator-prey model

39 Chisholm & Filotas 2009 J. Theor. Biol.

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

41 Predator-prey model K r and K crit 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 Chisholm & Filotas 2009 J. Theor. Biol.

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

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

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

47 Western Basalt Plains Grasslands

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49 Williams et al J. Ecol.; Williams et al Ecology

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

51 Grasslands invasion model A = plant-available N pool B i = biomass of species i ω i = N-use efficiency of species i ν i = N-use efficiency of species i μ i = N-use efficiency of species i α ij = 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 model Relatively safe, but higher control costs. Riskier, but lower control costs. Nutrient input B2B2 B1B1

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

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