critical slowing down as an indicator of transitions in two-species models
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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 in Complex Systems 21 March 2012 Imperial College London. Acknowledgements. - PowerPoint PPT PresentationTRANSCRIPT
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Critical slowing down as an indicator of transitions in two-species models
Ryan ChisholmSmithsonian Tropical Research
Institute
Workshop on Critical Transitions in Complex Systems
21 March 2012Imperial College London
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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
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Question
When is critical slowing down likely to be a useful leading indicator of a critical transition in ecological models?
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Outline
• Smithsonian Tropical Research Institute• Background: critical slowing down• Competition model• Predator-prey model• Grasslands model• Future work
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Outline
• Smithsonian Tropical Research Institute• Background: critical slowing down• Competition model• Predator-prey model• Grasslands model• Future work
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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?
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Smithsonian Tropical Research Institute
Panama
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Smithsonian Tropical Research Institute
50 ha plot
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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)
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Smithsonian Tropical Research Institute
sciencedaily.com
Photo: Marcos Guerra, STRI
Photo: Leonor Alvarez
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Center for Tropical Forest Science
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Forest resilience
Staver et al. 2011 Science
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Chisholm, Condit, et al. in prep
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Outline
• Smithsonian Tropical Research Institute• Background: critical slowing down• Competition model• Predator-prey model• Grasslands model• Future work
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Transitions in complex systems
• Eutrophication of shallow lakes• Sahara desertification• Climate change• Shifts in public opinion• Forest-savannah transitions
Scheffer et al. 2009 Nature, Scheffer 2009 Critical Transitions in Nature and Society
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Critical transitions
May 1977 Nature
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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. 2009 Nature
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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.
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Critical slowing down
van Nes & Scheffer 2007 Am. Nat.
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Critical slowing down
van Nes & Scheffer 2007 Am. Nat.
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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?
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Outline
• Smithsonian Tropical Research Institute• Background: critical slowing down• Competition model• Predator-prey model• Grasslands model• Future work
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Competition modelNi = abundance of species iKi = carrying capacity of species iri = intrinsic rate of increase of species iαij = competitive impact of species j on species i
Equilibria:
Lotka 1925, 1956 Elements of Physical Biology; Chisholm & Filotas 2009 J. Theor. Biol.
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Competition modelCase 1: Interspecific competition greater than intraspecific competition
Stable
Stable
Unstable
Unstable
Chisholm & Filotas 2009 J. Theor. Biol.
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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?
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Competition modelNi = abundance of species iKi = abundance of species iri = intrinsic rate of increase of species iαij = 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.
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Competition model
Chisholm & Filotas 2009 J. Theor. Biol.
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Competition model
Ni = abundance of species iKi = abundance of species iri = intrinsic rate of increase of species iαij = 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.
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Competition modelCase 2: Interspecific competition less than intraspecific competition
Stable
Stable
Unstable
Stable
Chisholm & Filotas 2009 J. Theor. Biol.
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Competition modelCase 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.
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Outline
• Smithsonian Tropical Research Institute• Background: critical slowing down• Competition model• Predator-prey model• Grasslands model• Future work
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Predator-prey model
Rosenzweig 1971 Science
V = prey abundanceP = predator abundance
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Predator-prey model
V = prey abundanceP = predator abundancer = intrinsic rate of increase of preyk = predation rateJ = equilibrium prey population sizeA = predator-prey conversion efficiencyK = carrying capacity of preyf(V) = effects of intra-specific competition among preyf(V) > 0; f ’(V) < 0; f(K) = 0; df/dK > 0h(V) = per-capita rate at which predators kill preyh(V) > 0; h’(V) > 0; h’’(V) < 0; h(0) = 0
Rosenzweig 1971 Science, Chisholm & Filotas 2009 J. Theor. Biol.
f(V)
h(V)
V
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Predator-prey model
Rosenzweig 1971 Science, Chisholm & Filotas 2009 J. Theor. Biol.
Equilibria:
Unstable
Stable for K ≤ J
Exists for K ≥ JStable for J ≤ K ≤ Kcrit
V = prey abundanceP = predator abundancer = intrinsic rate of increase of preyk = predation rateJ = equilibrium prey population sizeA = predator-prey conversion efficiencyK = carrying capacity of preyf(V) = effects of intra-specific competition among preyf(V) > 0; f ’(V) < 0; f(K) = 0; df/dK > 0h(V) = per-capita rate at which predators kill preyh(V) > 0; h’(V) > 0; h’’(V) < 0; h(0) = 0
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Predator-prey modelPredator isocline
Prey isoclines
V = prey abundanceP = predator abundancer = intrinsic rate of increase of preyk = predation rateJ = equilibrium prey population sizeA = predator-prey conversion efficiencyf(V) = effects of intra-specific competition among preyf(V) > 0; f ’(V) < 0; f(K) = 0; df/dK > 0h(V) = per-capita rate at which predators kill preyh(V) > 0; h’(V) > 0; h’’(V) < 0; h(0) = 0
Rosenzweig 1971 Science, Chisholm & Filotas 2009 J. Theor. Biol.
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Predator-prey modelUnstable equilibrium
Stable equilibrium
V = prey abundanceP = predator abundancer = intrinsic rate of increase of preyk = predation rateJ = equilibrium prey population sizeA = predator-prey conversion efficiencyf(V) = effects of intra-specific competition among preyf(V) > 0; f ’(V) < 0; f(K) = 0; df/dK > 0h(V) = per-capita rate at which predators kill preyh(V) > 0; h’(V) > 0; h’’(V) < 0; h(0) = 0
Rosenzweig 1971 Science, Chisholm & Filotas 2009 J. Theor. Biol.
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Predator-prey model
Scheffer 1998 The Ecology of Shallow Lakes
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Hopf bifurcation occurs when K = Kcrit :
Critical slowing down begins when K = Kr :
Predator-prey model
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Predator-prey model
Chisholm & Filotas 2009 J. Theor. Biol.
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Predator-prey model
Chisholm & Filotas 2009 J. Theor. Biol.
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Predator-prey modelKr 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
Þ Prey controlled by predators rather than intrinsic density dependenceÞ Increases tendency for oscillationsÞ Larger K makes oscillations larger and hence rates of return slower
Chisholm & Filotas 2009 J. Theor. Biol.
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Predator-prey model
Chisholm & Filotas 2009 J. Theor. Biol.
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Multi-species models
van Nes & Scheffer 2007 Am. Nat.
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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.
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Outline
• Smithsonian Tropical Research Institute• Background: critical slowing down• Competition model• Predator-prey model• Grasslands model• Future work
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Practical utility of critical slowing down
Biggs et al. 2008 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.
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Western Basalt Plains Grasslands
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Western Basalt Plains Grasslands
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Western Basalt Plains Grasslands
Williams et al. 2005 J. Ecol.; Williams et al. 2006 Ecology
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Grasslands invasion model
Nativegrassbiomass
Nutrient input rate
Agricultural fertiliser run-off
Sugar addition
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Grasslands invasion model
A = plant-available N poolBi = biomass of species iωi = N-use efficiency of species iνi = N-use efficiency of species iμi = N-use efficiency of species iαij = light competition coefficientsI = abiotic N-input fluxK = soil leaching rate of plant-available Nδ = 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. 2008 PNAS
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Grasslands invasion model
Relatively safe, but higher control
costs.
Riskier, but lower control costs.
Nutrient input
B2
B1
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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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