Upload
kathryn-gradin
View
213
Download
0
Embed Size (px)
Citation preview
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
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
Question
When is critical slowing down likely to be a useful leading indicator of a critical transition in ecological models?
Outline
• Smithsonian Tropical Research Institute• Background: critical slowing down• Competition model• Predator-prey model• Grasslands model• Future work
Outline
• Smithsonian Tropical Research Institute• Background: critical slowing down• Competition model• Predator-prey model• Grasslands model• Future work
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?
Smithsonian Tropical Research Institute
Panama
Smithsonian Tropical Research Institute
50 ha plot
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)
Smithsonian Tropical Research Institute
sciencedaily.com
Photo: Marcos Guerra, STRI
Photo: Leonor Alvarez
Center for Tropical Forest Science
Forest resilience
Staver et al. 2011 Science
Chisholm, Condit, et al. in prep
Outline
• Smithsonian Tropical Research Institute• Background: critical slowing down• Competition model• Predator-prey model• Grasslands model• Future work
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
Critical transitions
May 1977 Nature
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
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.
Critical slowing down
van Nes & Scheffer 2007 Am. Nat.
Critical slowing down
van Nes & Scheffer 2007 Am. Nat.
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?
Outline
• Smithsonian Tropical Research Institute• Background: critical slowing down• Competition model• Predator-prey model• Grasslands model• Future work
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.
Competition modelCase 1: Interspecific competition greater than intraspecific competition
Stable
Stable
Unstable
Unstable
Chisholm & Filotas 2009 J. Theor. Biol.
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?
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.
Competition model
Chisholm & Filotas 2009 J. Theor. Biol.
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.
Competition modelCase 2: Interspecific competition less than intraspecific competition
Stable
Stable
Unstable
Stable
Chisholm & Filotas 2009 J. Theor. Biol.
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.
Outline
• Smithsonian Tropical Research Institute• Background: critical slowing down• Competition model• Predator-prey model• Grasslands model• Future work
Predator-prey model
Rosenzweig 1971 Science
V = prey abundanceP = predator abundance
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
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
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.
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.
Predator-prey model
Scheffer 1998 The Ecology of Shallow Lakes
Hopf bifurcation occurs when K = Kcrit :
Critical slowing down begins when K = Kr :
Predator-prey model
Predator-prey model
Chisholm & Filotas 2009 J. Theor. Biol.
Predator-prey model
Chisholm & Filotas 2009 J. Theor. Biol.
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.
Predator-prey model
Chisholm & Filotas 2009 J. Theor. Biol.
Multi-species models
van Nes & Scheffer 2007 Am. Nat.
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.
Outline
• Smithsonian Tropical Research Institute• Background: critical slowing down• Competition model• Predator-prey model• Grasslands model• Future work
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.
Western Basalt Plains Grasslands
Western Basalt Plains Grasslands
Western Basalt Plains Grasslands
Williams et al. 2005 J. Ecol.; Williams et al. 2006 Ecology
Grasslands invasion model
Nativegrassbiomass
Nutrient input rate
Agricultural fertiliser run-off
Sugar addition
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
Grasslands invasion model
Relatively safe, but higher control
costs.
Riskier, but lower control costs.
Nutrient input
B2
B1
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