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